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SPECTRUM AND PROPERTIES OF MESOSCOPIC SURFACECOUPLED PHONONS IN RECTANGULAR WIRES By STEVEN EUGENE PATAMIA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2001 Copyright 2001 by Steven E. Patamia ACKNOWLEDGMENTS I would like to thank my advisor, Prof. Pradeep Kumar, for indulging my interest in pursuing this vestigial problem. I want to acknowledge publically that I understand the risk to both our careers that accompanied the continued pursuit of a fundamental and yet elusive problem which, despite its significance, has been declared without solution. I want to thank Dr. Albert Migliori and the Los Alamos National Laboratories for supporting the completion of this dissertation and for confirming in a material way that there were physicists and scientific enterprises for whom the results truly mattered. I want to thank my children, Mandy and Sarah, for their tolerance of my determination to pursue a PhD late in life. On the chance that she will orn, i read this page, I want to express appre ciation to Michelle for reasons only she will know. TABLE OF CONTENTS ACKNOWLEDGMENTS .......... ABSTRACT . . . . . . . CHAPTERS 1 INTRODUCTION . . . . . . 1.1 Nature of the Problem ......................... 1.2 History ................... .............. 1.3 Acoustic Phonons in Quantum Wires ................. 1.3.1 Confinem ent . . . . . . . 1.3.2 Mesoscopic ElectronPhonon Interactions ........... 1.3.3 Confined Acoustic Phonons and the Deformation Potential 2 ASSUMPTIONS AND CONVENTIONS .. ......... Physical Model and Coordinate System . ..... Symbolic Consistencies and Adopted Tensor Notation . Linear Elasticity Theory . .............. Free Boundary Conditions . ............. Nonseparability of Boundary Solutions . ...... . . 19 . . 21 . . 22 . . 27 . . 30 3 THEORETICAL ASPECTS OF RECENT NUMERICAL METHODS Stationary Lagrangian . ......... Numerical Approximation . ...... Partitioning the Problem into Parity Groups . 4 MATHEMATICAL STRATEGY .. . ........... Notation and Function Extension Issues Defining the Basis and Superpositions . Dimensionless Representations . . Derivation of RayleighLamb Equation . How to Transform Superpositions .. . 33 . 39 . 42 . 45 . 49 . 54 . 55 . 62 5 DERIVING NORMAL MODES OF PROPAGATION ........ 5.1 General Considerations ......................... 5.2 Acoustic Poynting Vector of a Normal Mode ............. 5.3 Propagating Modes Involving Hy, H, Shear .. .......... 5.3.1 Deriving the Frequency Equations .............. 5.3.2 Interpretation . . . . . . . 5.3.3 M ode Dispersions .. .................... 5.3.4 Comments on the Process of Mapping Mode Dispersions 5.4 Disposing of the H, / 0 Possibility .. ............... 6 K = 0 MODES OF A RECTANGULAR WIRE .. ...... k = 0 Boundary Conditions . ........... Uncoupled (Separable) k = 0 Modes . .... Uncoupled Modes not the Limit of Propagating Modes . Coupled (nonseparable) k = 0 Modes . ..... 6.4.1 DerivationAll Parity Families . ..... 6.4.2 Manifestations By Parity Family . .... . . 102 . . 103 . . 107 . . 112 . . 112 . . 117 7 FRACTAL PHASE SPACE OF COUPLED MODES AT K  0 . 7.1 Motivation: Low Temperature Heat Conductance . . . 7.2 Effective Dimension & Density of Propagating Modes as k  0 . 7.3 Low Temperature Heat Conductance . . . 8 CONCLUSION .. ..... REFERENCES .. ....... BIOGRAPHICAL SKETCH .. ... 125 . 125 129 . 135 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SPECTRUM AND PROPERTIES OF MESOSCOPIC SURFACECOUPLED PHONONS IN RECTANGULAR WIRES By Steven Eugene Patamia December 2001 C'!i iii i'I: Pradeep Kumar Major Department: Physics This dissertation presents original analytical derivations of the propagating modes of coupled mesoscopic phonons in an isotropic rectangular wire with stress free surfaces. Incidental to the derivations, novel consequences of the derived cutoff modes are presented as they affect the lowenergy heat conductance of such wires, or indeed .niv property that depends upon the dimensionality of the phase space within which the modes reside. Owing to nonseparability of the freesurface boundary conditions, an analytic description of coupled mesoscopic modes has heretofore been presumed to be underivable. Results presented herein show that the mode spectrum of coupled mesoscopic phonons is both subtle and rich, but considerable success in their analytic derivation is achieved. Using numerical methods developed for resonance problems, at least one contemporary researcher has purported to exhibit the lowest dispersion branches of propagating mesoscopic phonon modes in GaAswhich is not isotropic. The accuracy of these branches has not been measured, but they bear a qualitative consistency with isotropic modes derived herein. Since before the beginning of the 20th century, analytical solutions have been known for the infinite thin plate and even the case of waveguides with circular cross sections. Solutions for these special cases take the form of transcendental relations among the wavenumber and boundary parameters, but the underlying wavefunctions are separable in the coordinates. The analytical results presented herein for the general rectangular case involve nonseparable solutions whose separable components do not individually satisfy the boundary conditions. These solutions also take the form of transcendental relations, but there are sets of transcendental relations for each family of the cases that partition the problem. Consequently, the eigenspectrum, while defined by exact forms, must be enumerated by identifying plotted intersections of the root families of these transcendental relations. The resulting spectrums are more complex and have less apparent order than the spectrum produced using either periodic boundary conditions or rigid boundary conditions for uncoupled phonons. CHAPTER 1 INTRODUCTION 1.1 Nature of the Problem Despite some progress in exhibiting the lowlying band structure of meso scopic phonons by numerical methods, the study of phenomena involving phonon interactions has been hampered by an inability to analytically derive phonon modes involving surface coupling within rectangular geometries [1]. In unbounded bulk media, longwavelength phonons are adequately represented by independent families of elastic plane waves distinguished by polarization. However, at nonrigid surfaces, elastic boundary conditions cannot in general be satisfied except by a coupled superposition of shear and longitudinal vibrations. In some geometries particularly those involving edges and cornersthis surface constraint cannot even be satisfied by separable wave functions let alone a derivable superposition of simple plane waves. Reflecting this difficulty, no general analytical solution has heretofore been published for the eigenvalues or wave functions of phonons in a mesoscopic wire of rectangular crosssection [24]. As contemplation shifts from bulk material to bounded samples with elongated geometries, the manifestation of longwavelength phonons will include torsion and flexing of the sample. Bulk phonon models based on longitudinal scalar potentials alone are intrinsically incapable of exhibiting any flexural or torsional behavior since these require both some mechanism for shear and specific kinds of distortion of a specified surface geometry. Torsional and flexural behavior can be modeled to some extent using bending moduli and minimizing the potential energy associated with the distortion of finite objects (See, e.g., [5] 4.12). Yet, correct torsional and flexural modes ought to be automatically included within a comprehensive phonon model which correctly identifies phonons as mixed longitudinal and shear vibrations which satisfy nonrigid boundary conditions within a specific geometry. In modern terms, the nature of the problem can be succinctly described in the following useful way. Within the limits of linear elasticity, phonons in bulk are distinguished by polarization direction and propagation velocity and they do not interact. At .,niv nonrigid surface, the reduction of external stress to zero is a boundary condition that can be satisfied in general only if the two species of modes become coupled at the surface. In effect, each species scatters into the other until an energy transfer balance is achieved. A detailed examination of the boundary conditions clarifies an important feature of this coupling. Namely, the nondegenerate longitudinal phonons polarized along their propagation direction (I am ignoring quasishear and quasilongitudinal situations which can occur in non isotropic materials) only couple to shear phonons which have a polarization component normal to the surface. The surface interaction that arises as the surfaces are permitted to distort acts as a perturbation that splits the natural degeneracy of the shear phonons. Shear phonons polarized parallel to the surface are not affected and reflect specularly subject to their displacements vanishing at the surface. With respect to the phonon band structure, this degeneracy breaking is manifest as level repulsions among phonon dispersion subbands. It should immediately be noted that degenerate perturbation techniques are not fruitful in deriving solutions for the coupled phonon problem. The reason for this is that such an approach depends upon an ability to find eigenfunctions of the perturbation itself. Pursuing this inevitably encounters the fundamental source of mathematical difficulty which plagues the problem generally. Namely, solutions that incorporate the full boundary conditions at .,i.] i:ent surfaces in rectangular geometries are easily shown to be nonseparable. Defining a perturbation that reflects coupling of.,l.i i:ent sides and then identifying a basis for its eigenfunctions become intractable. 1.2 History Early in the history of elasticity theory, vibrational modes which ac.':i' l,111 full relaxation of applied surface forces (formally, as the projection of the stress tensor onto a vector normal to the surface vanishes) were successfully derived for certain geometries. A solution was first obtained for circular, infinitelength bars by Pochammer [6] in 1875 and independently by C'! ..' [7] in 1886. In 1889 Lord Rayleigh [8] published the analogous solution for an infinite plate with stress free surfaces. In the foregoing cases, the "solutions" obtained took the form of socalled "frequency equations" which define a transcendental relationship among shear and longitudinal wave vector components. The roots of these equations, for each propagation wave number, constitute the eigenspectrum. Reflecting the difficulties inherent in analytically exposing the roots of such transcendental relations, only some .,i i >! totic roots of the PochammerC'!I . frequency equations accompanied them immediately, and .,i, i id ic solutions to Rayleigh's plate solution did not appear until it was revisited by Lamb [9] in 1917. As Lamb explores .,iiI. Iltic roots and the displacement functions, he makes perhaps the first observation that, at increasing propagation wave numbers, two of the fundamental modes converge to form Rayleigh Surface Waves. Over succeeding decades, frequency equations for plates and rods were extensively studied and numerically derived roots of the underlying transcendental relations are extensively characterized and well known in engineering applications [1014]. It might seem surprising that despite difficulties identified above, that the frequency equation for circular waveguides was found at all, let alone before, the published analogous solution for the infinite plate. The reason is that a cylindrical coordinate system readily accommodates the fact that surface coupling only involves shear waves whose polarizations result in displacements normal to the surface. For cylindrical environments, the shear displacement field is naturally decomposable into a radial component coupled to the longitudinal field and a component parallel to the surface which is not. By contrast, for surfaces with sharp corners, only an unobvious superposition of waves forms (not necessarily plane) will accommodate coupling at .,i.i i:ent faces. Making the geometry finite whether with flat or rounded endsfurther complicates this difficulty, as was noted even by C'i., Once the wave guide takes on a rectangular cross section, the resultant bound ary value problem develops pathologies linked to the fact that solutions are gen erally nonseparable in the coordinates. The lack of an analytical solution to the rectangular cross section problem has consequently been widely conceded in the literature (see e.g., [24]) In view of this perceived limitation, approximate solutions for various subsets of the modes for a bar have been developedTimoshenko [15] "beam theory" being an important example in the realm of engineering applica tions. In the absence of direct analytical solutions for the rectangular cross section, it remained possible to apply a variational approach to generate solutions that coin cidentally satisfy the boundary conditions. The possibility of this tactic emerges in the literature at least as early as 1966 when Medick [16] attempted to develop a 5 1dimensional wave propagation theory for rectangular bars. Medick reviewed the important result [17,18], already then a decade old, that solutions which render the Lagrangian stationary are precisely those which simultaneously satisfy the bulk wave and natural boundary conditions. Medick, however, did not apply the variational principle directly to numerical computation. The basic observation that the eigenfrequencies of a freely vibrating elastic body are stationary in the space of functions was deduced early on by Lord Rayleigh who incorporated it into his treatise The Theory of Sound [19] in 1877. This result is usually referred to as "Rayleigh's Principle" [20]. Note that using Rayleigh's Principle to support a variational solution can be distinguished from approaches discussed within that rely explicitly upon stationarity of the timeindependent Lagrangian as a manifestation of Hamilton's principle. Early direct application of a variational principle to compute the loworder modes of parallelepipeds was presented by Holland [21] in 1968. Beginning with stationarity of the timeindependent Lagrangian, Holland articulates an essen tially RayleighRitz approach wherein finite combinations of trial functions are minimized. Holland's particular interest is in piezoelectric materials, but his computations are not in any way restricted by this. Holland examines the resulting eigenfunctions in an attempt to describe the displacement pattern of the modes themselves along with the spectrum. Holland's trial functions for components of the displacement field are products of sines and cosines of the coordinates. Later, Demarest (below) believed that such choices could precipitate inaccuracy in the results of a variational calculation. In recognition of the importance of this convergence issue, Holland and EerNisse examine this factor critically in a later monograph [22]. Building on Holland's 1968 result, Demarest [23] in 1971 improved the accuracy of Holland's mode computations for a freesurface cubein part by changing the nature of the trial functions. The trial functions of Demarest's own Rayleigh Ritz expansion are products of Legendre Polynomials of the coordinates. Along with identifying an inadequacy in Holland's choice of trial function, Demarest makes the observation that while Legendre Polynomials exhibit a computationally useful orthogonality in the relevant space, in principle they could be replaced by a power series in the coordinates of which Legendre Polynomials are just a linear combination. Neither Holland's 1968 paper nor Demarest's 1971 paper derive the general rela tionship between satisfaction of the boundaryvalue condition and the efficacy of the variational principle utilized. Since they are able to demonstrate a satisfactory correspondence between actual measurement results and the numerical predictions, they appear to consider the theoretical, albeit important, question of boundary conditions to be moot. Demarest instead focuses on the fact that derivatives of his trial functions constitute a more "relatively complete" set of basis functions than Holland's. By "more relatively complete," he means that linear combinations of selections from this finite basis are able to converge to a broader range of possible functions. Holland, as Demarest points out, manipulates his trial function choices to favor convergence of their derivatives to zero at the boundary. Demarest's choice of Legendre Polynomials enriches the basis so that convergence both to zero displacements and zero derivatives of displacement are possible. While this is sufficient to improve the accuracy of numerical results as compared against measurements, it cannot be decisive. The actual boundary conditions which will be explored herein cannot be simply partitioned into cases involving zero displacements versus zero derivatives of displacements. Moreover, in most cases unobvious superpositions are needed to assemble boundary solutions which are provably nonseparable and the nature of their boundary behavior is impossible to discern in advance of actually exhibiting them. Whatever its incremental improvement, Demarest's choice of basis also thwarts the expectation that the final solution, and presumably its components, must be constituted of no more than two parts which separately satisfy a basic wave equationas is provable from Elasticity Theory. Both Holland and Demarest utilize a RayleighRitz kind of procedure which approximates the solution to an eigenvalue problem by finding stationary points among a combination of trial functions. The conventional RayleighRitz procedure is implemented using trial functions which individually satisfy applicable boundary conditions (see any text on mathematical physics, e.g., [24] 17.8). This, however, is not possible under circumstances where there are no available basic solutions for the geometry that are known to satisfy those boundary conditions in the first place. It thus was openly expressed by Visscher, Migliori and Bell [25] in 1991 to be "fortuitous" that displacement functions which could be found to minimize the timeindependent Lagrangian would automatically satisfy the bulkwave and free boundary conditions simultaneously. As I will point out more particularly within, this fortuity is less dramatically useful than its discovery infers. The reason is that it is only as helpful as the set of trial functions is versatile and is literally true only if the set of trial functions happens to include a combination that is a solution. Again, the central impediment is that basis functions which at least satisfy the boundary conditions do not even exist. Visscher et al., therefore, concentrated upon assembling a basis chosen for its versatility and the convenience with which numerical computations could incorporate its elements. In fact, the basis elements they choose, while linearly independent, are not orthogonal, and do not even satisfy a wave equation let alone boundary conditions. Remarkably, at least in terms of resonances of closed objects, the basis is nevertheless sufficient to produce useful results to a sometimes high degree of accuracy [26]. Possibly because engineering and physics communities do not consistently over lap, Visscher et al. were probably not aware that Medick had previously noticed the same fortuity a quartercentury before and had been reviewing results available a decade before that. Nevertheless, Visscher, Migliori and Bell offer an extremely elegant derivation of the principle and clarify greatly the theoretical underpinnings of the algorithm developed by Holland and Demarest. Moreover, equipped with access to some of the earliest tools to bear the designation "supercomputer," Visscher et al. tested directly the efficacy of a particularly useful and elegant choice of basis functions. Specifically, they implement Demarest's ,ii:. I. li that products of powers of the coordinates should constitute a set of basis functions that would be sufficiently "complete" to converge to accurate solutionsat least for the frequency eigenvalues in resonance problems. Like Holland and Demarest, Visscher et al. were motivated by the need to find accurate loworder eigenmodes which could be verified by comparison to actual measured resonances of samples. They showed that an algorithm employing a basis built from simple products of powers of the coordinates would accomplish this even for geometric shapes far more challenging then the parallelepipeds of Demarest and Holland. Subsequently, Migliori has perfected this technique into an accurate standard means of finding the microscopic elastic constants of materials having arbitrary il I1 or composite structure [26]. The directness of computability and broad applicability of the computational algorithm refined by Visscher, Migliori, and Bell has recently been adapted and applied to exhibit the modes of an infinite rectangular wire. Through the simple expedient of replacing one coordinate factor of the basis functions with a common periodic function of the long coordinate alone, i.e., "Pyr v + x"r ei q Nishiguchi, Ando and Wybourne [1] resolved the transverse cross section com ponents using the same variational methods as Visscher et al. while retaining translational invariance along the length of the infinite wire. Instead of obtaining a finite set of resonant eigensolutions for a resonator, he obtains a finite family of solutions for each value of q in a waveguide. they then generate the dispersion rela tions along the zaxis by plotting the resulting solutions which remain continuous in the parameter q. Since the wires are infinite length, there are no considerations of mixing from endeffects. There is also no need for the variational algorithm to resolve the modal pattern along the long dimension. As a consequence, convergence is limited only by the algorithm's ability to resolve the transverse displacements whose modest periodicity over a limited range is more readily approximated by a finite power expansion. Their specific results are computed for GaAs, but as Visscher et al. had emphasized, there is nothing in the nature of the numerical algorithm which inherently limits it to any material or i iI I1 structure. The mode dispersions developed by Nishiguchi et al. lack comparison with measurement. Directly measuring the eigenfrequencies of a resonator is straightfor ward, but directly measuring the longwavelength subband dispersions of phonons subject to a specific geometric boundary may not be. The technique of finding bulk phonon dispersions using neutron scattering cannot be easily applied to a geometrically confined lowwavelength environmentparticularly if the sample is microscopic. Also, full resonances are precise by definition and represent a severely constrained computational problem. It is not obvious whether a set of linearly independent, but not inherently orthogonal, trial functions remain similarly able to converge strongly to a solution where the basis is constrained conceptually, but not actually, along one direction. These concerns will be further addressed in the results presented. In general contemporary research, mesoscopic acoustic phonons have been investigated theoretically, and to some extent experimentally, in primarily two kinds of situations. The first is in respect of their interactions with phonons in quantum wires and the second is their role as carriers of thermal energy. The two topics have some overlap. In the next section, I will summarize the findings with respect to electronic interactions in quantum wires and in Ch. 7 I will take up the more emergent topic of kinetic heat transport by acoustic phonons in the lowtemperature highconfinement regime. 1.3 Acoustic Phonons in Quantum Wires 1.3.1 Confinement From the end of the 1980's and continuing into the present, mesoscopic physics has enjoi, d an obvious prominence driven largely by the relentless downscaling of electronic devices. Commercial applications aside, the emergence of an ability to fabricate devices within which quantum behavior becomes observable as a consequence of dimensional confinement itself stimulates persistent and widespread interests in the effects of small size and low temperatures. Certainly, tod i, there is no more ubiquitous device in the theoretical and experimental condensed matter inventory as the "quantum wire." In this context, socalled "phonon o. .1ii in, ni has also become a phenomenon of specific interest. Phonons, being a classical rather than quantum phenomenon per se, are not subject to the sharp dimensional confinements possible for quantum particles, like electrons. When characterizing phonons as "confined," authors are only distinguishing phonon modes subject to quantization by physical boundaries from bulk phonons quantized using periodic boundary conditions. There is no inferred loss of dimensionality of the phonons, though, of course, the lowest sub bands typically have imaginary transverse wave numbers and higher subbands may sometimes become energetically inaccessible. While "(~.'1 ii il at drastically small scales can lead to actual dimen sional reduction of quantum particles, its generic effect is to render the physical boundaries relevant. The signature feature of what is called i:: ....' I remains the commensurateness of the characteristic wavelength with the distance between boundaries. For phonons, however, actual small scales and low temperatures tend to magnify the departures from bulk behavior provoked by mesoscopic considera tions. Certainly, at any scale, mesoscopic phonons exist since some subset of the phonon spectrum consists of phonons at wavelengths comparable to the size of the physical environment. At small scales and low temperatures, however, mesoscopic phonons are the dominant ones and their actual frequencies, which scale inversely with overall size, become large enough to augment the probability of electronic interactions. Mesoscopic phonons are, by definition, of a wavelength large enough that surface effects and geometry control their dispersion and wave structure. Periodic boundary conditions and plane wave modeling give way to explicit boundary solutions for specific geometries. Surface coupling endemic to the nonrigid bound ary conditions determines the eigenspectrum and mixes longitudinal and shear contributions. 1.3.2 Mesoscopic ElectronPhonon Interactions Absent an analytical solution for surfacecoupled phonons, research into con fined phonon interactions progressed by using approximations or rough models which were believed to mimic the most important features of actual phonons. A brief survey of the results produced in this manner thus serves to identify conclusions and calculations which should be revisited in the light of analytical results from this investigation. By the 1990's, papers explicitly calculating the interaction of electrons in a quasionedimensional gas (i.e., in a quantum wire) with "confined phonons" were appearing regularly [27,28]. Initial investigations focused primarily on hot electron relaxation via emission of "confined" optical phonons. The phonon models utilized during this time were typically uncoupled symmetric and antisymmetric transverse resonances where the transverse components were taken either to be zero (denominated "guided" modes) or a maxima at the boundaries (denominated I il" modes). The latter condition is actually similar to stressfree conditions for elastic phonons. Results from the two flavors of boundary conditions are compared. Among the interesting conclusions is that boundary conditions resembling stress free surfaces lead to an order of magnitude faster relaxation rate than obtained by assuming zero displacement boundaries [27]. Longitudinal optical phonons in polar semiconductors will decay to acoustic phonons (see Ref. [29] 6.2) within some average LO lifetime which can actu ally augment dissipation by virtue of precluding LO phonons in the process of decay from being absorbed back by the carriers. However, at low temperatures in quantum wires (e.g., below 30 K for a 100 A wide GaAs wire) hotelectron scattering to LO phonons becomes exponentially weak and direct ... 1i c phonon emission is the only important dissipative process" [30] though for quantum wire wells, LO scattering may continue to dominate into lower temperature regimes [31]. Early in this period, theoretical interest in hot electron relaxation to acoustic phonons in wires was encouraged by experimental developments. In 1992 Seyler and Wybourne published a PRL [32] reporting on the detection of resonances with presumably acoustic modes in small (approximately 20 nm thick and between 30 and 90 nm wide) AuPd wires over a broad range of low temperatures (1 20 K). For the balance of the decade through to the present, this particular experimental observation is among the most consistently cited as justification for further theoretical exploration of the "importance of acousticphonon confinement in reduced dimensional electronic structures" [4]. The essential role of acoustic phonons in energy dissipation within quantum wires has been scrutinized in various vi, Senna and Das Sarma have investigated a "Giant ManyBody Ei! i ii.'. iii, i, of electronacoustic phonon coupling at low temperatures by renormalizing the phonons in the presence of the electron gas [33]. They find that at low temperature (viz. 1 K), below which direct plasma resonance cannot be significant, the quantummechanical (which they distinguish from ther mal) uncertainty in the phonon modes creates an interaction enhancement orders of magnitude greater than for bare phonons. Mickevicius, Mitin, and Kochelap used Monte Carlo calculations to investigate phonon radiation from a Quasi1D electron gas in rectangular GaAs quantum wires of 80x80 A in the neighborhood of 4 K [34]. They report that virtually the entire power dissipation is due to the transverse radiation of ballistic acoustic phonons. All of the above cited theoretical explorations involve GaAs and/or AlAs as the specific material of choice. While GaAs remained an almost predominant material chosen for theoretical studies during and since this time, Si has also been explored. In a comprehensive set of calculations, Sanders, Stanton, and C'!I 0 1; cal culated [35] a range of transport properties for a Si quantum wireincluding defor mation potential scattering by "quantum confined phonons." Although the authors explicitly reserve the term ...I iI c" for the lowest subband, the phonon model used corresponds exclusively to longitudinal elastic waves coupled to electrons via a deformation potential. Possibly to avoid intimating that higher subbands of these confined elastic modes should be called "optical" the authors describe them as being "excited quantumconfined phonons." In any case, the focus in this paper is on a secondquantization representation of the phonons without attempting to mimic specific surface boundary conditions. In a 1994 study, Yu, Kim, Stroscio, lafrate, and Ballato [4] (liberally cited else where herein) build upon the 1949 Ph.D. thesis work of Morse [36,3] to implement a more realistic confined acoustic phonon model that incorporates some surface coupling in a rectangular quantum wire environment. Upon reciting that "As is well known, there are no exact solutions for the complete set of phonon modes for a rectangular vii. Yu et al. adopt Morse's compromise strategy which was expected to be adequate when the cross sectional aspect ratio was greater than two. Morse had made progress solving the rectangular problem (which I investigate more successfully herein) by treating the closest parallel surfaces as an independent plate scenario (see section 4.4 herein) and finding what amounts to RayleighLamb modes relative to those surfaces. He then observed that two of the three surface normal stress components become small at the .,.i i:ent surfaces as the aspect ratio grows and tunes a free parameter in his plate solution to force one of the two diminished stresses to be zero. Morse's transcendental equation derived from considering the closer of the surfaces (equation (14) in Ref. [4] under discussion) is very similar to, though less general than, my own intermediate equation (5.27) which appears within my derivation of propagating coupled modes. His starting assumptions are far less general, but at that corresponding point of the derivations the situation is artificially similar. Restricting themselves to dilatational modes (see comments below), Yu et al. proceed to compute a normalization of Morse's separable solution and then compute electronphonon scattering from deformation potential interaction for a range of cross sections (principally 28x57 and 50x200 A) at 77 K. Their essential finding is that scattering rates are notably higher for such confined coupled acoustic modes than for bulk acoustic modes and increase dramatically as the overall scale is reduced. In fact, they compute that the scattering rate for a 28x57 A wire are an order of magnitude larger than for a 100x200 A cross section. 1.3.3 Confined Acoustic Phonons and the Deformation Potential Hereafter, my own use of the label *,ustic" will, absent qualification, encom pass all phonons that are a manifestation of elastic deformation. Consistent with much of the literature being reviewed (see, e.g., Ref [4] discussed above), this means that the subbands that arise from real boundary confinement of elastic waves will continue to be referred to as acoustic despite the fact that in bulk environments this term refers only to a dispersion branch that originates with zero frequency at zero k. My convention is appropriate to this study where only confined phonons are relevant, is consistent with texts and monographs devoted to acoustics of solids [5,10,37,38]and it appears consistent with emergent text books devoted to phonons in nanostructures [29]. Unfortunately, past publications are not ahv as inclusive, but in the worst case it will simply be important to notice whether acoustic modes under discussion have been restricted to a specified dispersion branch or explicitly stated to be bulklike. As is consistent in the historical literature of acoustics per se, I will distinguish modes whose dispersion goes to zero as k  0 by calling them "fundamental." Within the most inclusive use of the term, uiI c" phonons are coupled to carriers solely by the deformation potential which tracks the ion density fluctua tions. In each of the abovecited articles, only longitudinal (i.e., compressional) phonons (see, especially, [39]) participate in this coupling. Since bulk shear mode displacements can be generated by a vector potential, the divergence of their displacements is zero and they cannot alter the ion potential field via fluctuations in ion density. There are, nevertheless, circumstances where a deformation potential can be associated with noncompressional phonons. For materials with degeneracies in band structure that can be broken by shear distortions, it is long been known that deformation potential should be generalized to a tensor such that the potential change due to deformation incorporates shear as well as divergence effects [40,41]. Even if such special circumstances do not apply, there remains a more fundamental reason to revisit the exclusive focus on compressional modes once the phonons are subject to boundary confinement. In the continuum limit, acoustic phonons are merely elastic modes irrespective of whether the environment is confined or bulklike. This study is concerned, however, exclusively with elastic modes confined to rectangular waveguides. As will be developed in detail in subsequent chapters, elastic waveguide modes can be partitioned with respect to the parity patterns of their displacements relative to a coordinate systems aligned with the long axis and normal to the sides. For the fundamental modes (i.e., those confined branches which do go to zero as k  0) these patterns correspond to macroscopic motion of the waveguide dilatational, torsional, and flexural. Of these, the dilatational is so named because the displacement pattern of the fundamental modes resembles propagating density fluctuations on the scale of the width of the waveguide. It is no doubt for this reason that as phonon models in quantum wires take boundary confinement into account, there is an explicit assumption made that only the part of the model corresponding to dilatational modes need be considered. Torsional and flexural modes correspond to twisting and bending and appear, on first impression, to be dominated by shear and are assumed to have no significant deformation potential (as assumed in Ref. [4]). Transferring allegiance from longitudinal bulk modes to dilatational confined modes promotes a significant oversight. In bulk media, torsion and flexural dis placement patterns do not even exist, but once the phonons are confined by actual surfaces, these categories arise precisely because surface coupling mixes longitudi nal and shear contributions into the displacement pattern. Ignoring the torsion and flexural modesperhaps for lack of an analytical model for the surface coupling comes at the cost of ignoring definite sources of density variation. A.r.vTi, tin the potential consequences of this oversight, it will later be seen that the branch shape of some fundamental torsional, along with the fundamental flexural modes, shows a high density of states at low frequencies. Anyone who has ever wrung out a washrag realizes that fundamental torsional modes involve patterns of change in local density. Bending a foam rubber object creates a clear opportunity to observe the compressions and stretches which accom p .ni: that movement. Consistent with these reallife macroscopic observables, the derivations presented in subsequent chapters reveal that the longitudinal potential makes a necessary contribution to virtually every coupled mode. While it may turn out that some subset of modes within a subband are dominated either by shear or longitudinal patterns, there is no a priori justification for assuming that 18 any category of coupled modes can be safely ignored even if only the deformation potential from density fluctuation remains relevant. Even Demarest took care to point out that the dilatational, torsional, and flexural naming attached to parity patterns should not be taken too seriously as each contained elements of shear and dilatation [23]. CHAPTER 2 ASSUMPTIONS AND CONVENTIONS 2.1 Physical Model and Coordinate System I assume that the waveguide is composed of an isotropic material formed into an infinite bar whose rectangular cross section is invariant along its length. The stiffness tensor for an isotropic material has only two independent elastic constants and these, together with the material density, determine the bulk shear and longitudinal velocities, denoted c, and cQ respectively [5]. To keep the results completely general, I will reduce all derived observables to a dimensionless form rescaled relative to the shear velocity, the smaller of the halfwidths of the cross section, h, and the aspect ratio a between the widths. Derived results will thus be in terms of a single material characteristicthe ratio of velocities R = ce/cand a single geometric parameterthe cross sectional aspect ratio, a. When computing definite eigenvalues to dipl!iw representative quantitative results, R will be arbitrarily set to a value of 3. Some results will turn out to be independent of R. For perspective, it is a well known result of linear elasticity theory [42] that R cannot be less than /2 for isotropic materials. Commercial aluminum, a modestly soft and nearly isotropic metal, has an R of roughly 2 whereas GaAs, a nonisotropic semiconductor, has an R averaging close to the v2 limit. All phonons are contemplated in the i pi '....Ipi regime, by which I mean that their wavelengths are not less than an order of magnitude smaller than the smallest cross sectional width. I also assume that treating the material as an elastic continuum is justified by first assuming the waveguide material has a typical inter lattice spacing much smaller than the smallest phonon wavelength considered. As a practical matter, this would still permit important results to apply to quantum wires with widths on the order of a few hundred atoms. The displacements will be assumed sufficiently small that applied elasticity theory is well within the linear regime. For phonons that are thermally excited or scattered from interactions with itinerate electrons, magnons, and similar particles, this is a reasonable physical assumption consistent with remaining within the phonons' own mesoscopic regime. Calculations and representations will be rendered within a righthanded Cartesian 2ah 2h Figure 2.1: Nomenclature of Elastic Waveguide. Frequency and wave number will be rescaled relative to h (and c8). The smaller halfwidth will be denoted hz = h. The only geometric factor in final results will be a, the crosssection aspect ratio. coordinate system. The long axis of the bar will be considered the x axis to facilitate easy comparison with publications of historical significance in which this is the more common convention. The long coordinate will be embedded along the geometrical center of the bar. This placement of the long axis symmetrically divides the bar. Accordingly, the bar will be transversely bounded by hy < y < hy and h, < z < hz. For convenience, hz will be consistently taken as the smaller half width should hy / h, and h without subscript will refer to this smaller quantity. As already indicated, a will denote the cross sectional aspect ratio so that hy = ahz. 2.2 Symbolic Consistencies and Adopted Tensor Notation In addition to foregoing nomenclature, the base symbol for all quantities related to elastic displacements will be the letter u. Vectors, as opposed to their compo nents, will be bolded. When necessary to distinguish shear versus longitudinal displacement contributions, a parenthesized superscript will be used, as in u() and u(s). The displacement field will be decomposed into longitudinal and shear parts generated by a scalar potential p and a vector potential H respectively. The Greek letter TI will be consistently chosen as a base symbol for longitudinal wave numbers and a for shear wave numbers so that these associations can be perceived at first glance. Einstein notation will be used in tensor equations. Repeated indices will, unless otherwise noted, imply a summation over {x, y, z} coordinates. Commas preceding one or more indices will abbreviate derivatives taken with respect to them. However, the problem will be investigated within a Cartesian metric and so there will be no distinction between covariant and contravariant vectors and correspondingly there will be no methodical use of raised versus lowered indices. Absent the ability to restrict summations to matching upper and lower indices, summation may be assumed for all matching pairs of indices on any one side of an equationsubject to contrary commentary. The symmetric and antisymmetric parts of tensors with respect to any subset of indices will sometimes be explicitly indicated using the standard notational devices of enclosing the list of p indices that are symmetrized in parenthesis and any that are to be antisymmetrized in square brackets. A ...(j...) [A ...,j... + A ... j... + ...] A [ij ...[ ] L [A ... A... j... + .] Hz,y Hy, = 2H[,y] = 2H[y,,] The I .I !!y antisymmetric" tensor of any appropriate rank will be indicated using an c (as opposed to E), as in [V x H] = CijkHj,k 2.3 Linear Elasticity Theory The phonons in this study are modeled as elastic vibrations. It is helpful and important to identify and outline the origin of specific elements of elasticity theory essential to the derivations developed herein. Besides clarifying notation and connecting it to standard literature, this will serve the important purpose of exposing implied and explicit assumptions which underlie my own derivations and which could limit applicability of the results. The basic results of the linear theory are straightforward, but elasticity the ory has many subtleties and complexities which will not be needed in what fol lows. For full development of the topic, monographs and texts range from com prehensive classics [43], to relatively recent standard texts which are cited by most contemporary researchers who utilize an elasticity model of longwavelength phonons [37,38]. As a recommended supplement, a succinct development of ele mentary linear elasticity theory is provided within the theoretical physics series of Landau and Lifshitz [42], but some lesser known texts and monographs are more relevant to the study of elastic waveguides [5,10]. Linear elasticity can be viewed as a generalization of Hooke's Law applied to a continuum whose elastic response need not be isotropic. Hence, the basic stressstrain relation law takes the following general form: oij = Cijk Uk (2.1) The tensor Cijk is sometimes called the I Li ;" tensorin contradistinction to the "compliance" tensor whose elements are those of the inverse of the stiffness tensor's matrix representation. It is not uncommon to refer to the stiffness tensor as the elastic tensor. The tensor Uke is a dimensionless strain defined to encapsulate deformation in a form invariant under pure rotation while omitting nonlinear terms. This is accomplished by defining it to be the symmetric part of the gradient of the displacement vector (sometimes denoted Vsu [38]). [Vu](k,) = U = U(i,j) = 2(uk,e + u,k) (2.2) The potential energy density then has the following form: V = 2Uk Cijki utk (2.3) Since uij is symmetric, directional invariance of V leads ultimately to essential symmetries of the elastic tensor as follows: Cijk = Cijk = Cjik = C .,, = Cjilk and Cijk = , (2.4) These intrinsic symmetries substantially reduce the number of unique components (81, before applying the symmetries) and give rise to an abbreviated notation, ubiquitous in materials science, for denoting the, at most 21, unique elastic con stants which remain. Abbreviated indexing reflects the pattern of symmetries by denoting pairs with single digits. The strain, stress, and elastic tensor components are then expressed in terms of the correspondingly reduced number of indices. The abbreviation scheme is simply 11 1 t 222 33 3 23, 32 4 13, 31 5 12, 21 6 This allows equation (2.1) to be expressed as the following twodimensional matrix vector product relation: 91 C11 C12 C13 C14 C15 C16 U1 02 C22 C23 C24 C25 C26 U2 03 C33 C34 C35 C36 . (2.5) 4 C44 C45 C46 U4 05 C55 C56 U5 06 C66 U6 where omitted matrix elements are symmetric reflections. Various crystal symme tries will reduce further the number of unique elements of Cp,. Although I will sometimes rely upon and refer to general properties of Cpq, the important novel results depend upon the material being isotropic. Isotropy can be represented by imposing rotational invariance on an existing cubic symmetry. The full elastic matrix for a cubic material (in abbreviated notation) is (2.6) which can be rendered rotationally invariant by requiring that C44 (Cll C12) (2.7) The result is that the three components can be parameterized using only two constants (or moduli), and these can be chosen to be equivalent to the socalled Lam6 constants which emerge naturally when the theory is derived starting from an assumption of isotropy (see, e.g., [42]). A = 12 and p = c44 with Cl = + 2p (2.8) For an isotropic material, the stressstrain relationship (2.1) can then be summa rized as cij = Xij E Urr + 2/Uij (2.9) Notice that I have not been content to allow Ur, to imply summation because one of the penalties of avoiding the use of upper indices is that, under the definition (2.2), ui is proper notation for an individual diagonal component of u. Having called attention to this problem, it can be mitigated by adopting a convention that, absent clarifying comments or explicit summation, repeated indices in a rank two tensor will imply summation (i.e., contraction) only when the usual letters set aside as coordinate variables are not used. Under this rule, the use of r as a subscript above in lieu of i, j, or k would have signaled that the ambiguity should be resolved in favor of implying summation. The freebody equation of motion within the media follows immediately from Newton's law and substituting equation (2.1) into (2.3). p ui,tt = i = aijj (2.10) If the right side of equation (2.10) is expanded using equations (2.1), (2.2), and the symmetries identified in (2.4), then the equation of motion can be expressed in terms of displacements as P ui,tt = Cijk Uk,j (2.11) which, in the case of a normal mode (i.e., ei"t time dependency), immediately yields a form of vector wave equation with possibly nonfactorable i  S/2Ui + Cijk Uk,j = 0 (2.12) To exhibit the wave picture in the case of isotropy, I can use, instead, equation (2.9) in expanding equation (2.10). This leads ultimately to a useful coordinatefree equation of motion U,tt ( ) V(V ) + V2u (2.13) This expression is n, I!i!" in the sense that an assumption that the displacement field vanishes at infinity permits, via a wellknown theorem of vector algebra, u to be separated into divergenceless and solenoidal parts which, in turn, produces separated wave equations with distinct velocities as follows: u() (+ V2U) u ) V2(V x H) (2.14) Here I have begun to utilize ip as the scalar potential generating the longitudinal displacement and H as the vector potential generating the shear displacement. In general, these potential symbols may themselves represent superpositions. The foregoing clearly identifies the two isotropic bulk velocities: c, P/ 44 (2.15) + L (2.16) P P These can then be substituted back into equation (2.9) to begin utilizing the bulk velocities as the primary characteristics of an isotropic material. a = p (c2 2C2) .v. + 2p c2 (2.17) 2.4 Free Boundary Conditions The problem investigated herein is defined by taking the stress normal to the surfaces to be zero. I begin to transition toward a dimensionless form of this situation by first dividing equation (2.17) by pc2. Recall that R has been designated to be the isotropic material property defined by ce/c,. Then, in the coordinate system of the problem, the assertion that stress normal to the ith surface at xi = hxi is zero can be expressed in a form independent of p and c, as (R2 2)6Jijrr + 2uij = 0 at x, = hx, (2.18) Since ut,i = ui, the contraction urr in equation (2.18) is just V u. Since the divergence of a curl is zero, there is no shear contribution to this term. Then, since the longitudinal displacement is the gradient of the scalar potential 0, I can replace Vu with V2( Since p is solution of a wave equation, we can finally replace V2s with k,2 where k = w/c at the eigenfrequency w. Finally, since each u = 9 longitudinal strains are succinctly expressible in terms of the scalar potential in the form u) = lPij. Separating the shear and longitudinal parts and utilizing the foregoing, the boundary conditions at either of .,Ii i.ent pairs of parallel surfaces can now be restated as a set of qualities relating shear and longitudinal contributions. To emphasize the essential structure independent of chosen surface, I will here use a surfaceoriented notational scheme as follows: Denote with s the coordinate axes (i.e., y, or z) normal to these surfaces. With s as the starting coordinate, denote the next cyclic coordinate (righthanded progression) as p and the next after that as p. In terms of this notation, the boundary conditions for any transverse surface can be written in the following form: At s = h8 for all p c [hp, h,] and all p c [hp, hp]: , k2 + 2u') ( 9) ys? () (2.19) IOsp tsp where 3 = R2 2. The shear displacement vector components in explicitly: u(s) Vx Hz,y Hx,z Hy,. terms of H components are Hy, H,,, H1,y (2.20) The shear strain tensors are then: l(Hz,yy 1(Hz,zz i(Hz,zy Hz,xy Hy,zz Hx,yz Hz,xy Hy,zz H,iyz Hy,yz + Hx,xzZ Hz,zz + H, 1y Hy,zz + Hy,xx H ,,xx) H 11v) (2.21) with the order of differentiation having been arranged to highlight patterns and symmetries. When tensors from equations (2.21) are substituted into surface manifestations of equation (2.19), the overall cyclic pattern of the subscripts is fully apparent and the boundary conditions can be resummarized completely in terms of potentials as follows: At s = h, for all p c [hp, hp] and all p c [hp, hp]: 3 kp p + 29p,ss 'P,sp P,sp 2(Hp,  +1 [(H,,pp Hp,p) ,s Hp,,ss) + (Hs,s Hp,ss) + (Hs,s Hp,p),p] Hp,p),p] (2.22) 2.5 Nonseparability of Boundary Solutions Utilizing equation (2.19), I will directly demonstrate that assuming the exis tence of separable potential functions in this scenario leads to an essential contra diction. The zboundary and yboundary conditions are explicitly at z = h for all y E [h, h]: 03 k2j + 2y ,zz u() (2.23) and at y = h for all z E [h, h]: kj2 o+ 2,yy=2us (8) (P,yz lyz) ,y (8) 'PIXY "aY (2.24) Suppose that p is separable wave function representable as the product of wave functions such as p(kxI,' (rqyy ,' (T1yz). Since longitudinal strains are symmetrical in the derivatives taken to form them from o, separability of p will perpetuate to the longitudinal strains. The x dependencies of the potentials will be assumed to be common because that is the basis choice appropriate to propagating modes. I can thus anticipate that the xdependencies of each boundary condition will cancel and will henceforth omit them from the remainder of this proof. Assume also that H generates a u(s) such that shear strains formed by com bining their derivatives can be represented as separable products such as u(=) ,,,,,(yy),," (,az). Again, since H must be a wave function satisfying the bound ary conditions, the separable products making up the strains it generates will ultimately also be wave equations for each direction. If the shear components were individually separable, the best possible case would be that the shear strains which combine derivatives of these, nevertheless produce individually separable functions. If they do not, the shear strains will immediately be nonseparable and there will be no point in going further since the case will have been made. It is sufficient, therefore, to assume this best possible case. Substituting these assumptions into equation (2.23) at +hz: k(P3A; + r2T1 ,' (rTyy) '' (TI ) = 2k, (nayy) (7, h, ) ST) (lh) ,yii (ayy)i (a, h,) k' U ( ) (9z)z) ( = h (agy) uyz,(ah,) (2.25) where the tilde's over functions conveys that they have been differentiated. It will not matter to the proof precisely what the separable product or its derivative is because the only facts essential to the result are that they are wave functions with definite wave numbers fixing their directional dependencies. For the foregoing equations to be valid, they must each be invariant in y. By inspection, this will only be true if rq = ay including the possibility that they are both zero. Given the cyclic symmetry connecting the boundary conditions at .,.i i:ent sides, substituting the same assumptions into equation (2.24) will similarly result in the symmetrical requirement that rI, = z. 32 Separability of the potentials together with their satisfaction of wave equations implies that 2 + 2 k2 2 + a + + Cs If ry = ay and %, = cr, this will require that c = c,which is impossible for real isotropic materials since it has already been pointed out that c > V2 cs. This calamity can be cured by (a) some superposition of solutions, (b) eliminating one side by extending it to infinity (the infinite plate scenario as solved by Rayleigh), or (c) special modal situations involving only shear (9 = 0). Each of these exceptions will be encountered in the sequel. CHAPTER 3 THEORETICAL ASPECTS OF RECENT NUMERICAL METHODS 3.1 Stationary Lagrangian Holland [21], Demarest [23], and Visscher et al. [25] each construct a theoretical justification for their numerical approximation methods by identifying solutions with the stationary points of a Lagrangian. More precisely, the starting point is stated to be Hamilton's principle, but the stationary points of the timeindependent Lagrangian were asserted to be equivalent. Whether or not this is a valid substitu tion, however, depends on whether the function space within which the stationary functions are found is restricted to functions with a separable harmonic time dependence. (Demarest makes some attempt at varying a more general class of trial functions, but fails to properly factor the variation in his equation (4).) This is not an inconsequential suspension of formality. Stationarity of the Lagrangian within a function space of elastic normal modes follows immediately from stationarity of the Hamiltonian precisely because the kinetic and potential functions have the same harmonic time dependency. Integrating any such function over time produces a factor of 1/2 so that stating variations to be zero is equivalent to stating that variation of the timeindependent part is zero. 2 L 2L = 6J L(r,t)dt =6L(r) at2 2 However, if the function space is not in some way restricted to those functions whose spatial parts are at least consistent with their having harmonic time depen dencies, then it is elementary that stationarity of their spatial parts is not a priori equivalent to the satisfaction of Hamilton's principle. The desired harmonic dependency will be manifest by satisfaction of a wave equation. After reviewing, below, the actual Lagrangian variational result, the point to be emphasized will be that restricting the set of trial functions to those which satisfy wave equations is indeed the most important prerequisite to validity of the result. The Lagrangian variational analysis within each of the aforementioned numer ical approximation attempts is well represented by that contained in the paper by Visscher et al. The reasoning is elegant and clear, and so it is helpful to reproduce it while making relevant observations. The derivation by Visscher et al. takes place within a closed domain of arbitrary shape. It therefore is, in essence, a resonance problem. The same reasoning can be extended to propagating modes by considering such a mode virtually bounded within a rectangular waveguide by surfaces perpendicular to the sides and sepa rated by a phase difference of 27. If any surface integration is performed, reversal of the outward normals of these virtual surfaces would cancel their contributions to the overall surface integral insuring the same result as for a physically bounded region. Visscher et al. preincorporate harmonic time dependence by writing the kinetic energy density as P LUiUi (3.1) They represent the potential energy density as CijkeUijUk,e (3.2) which can be shown to be equivalent to equation (2.3) by expanding the defini tions of the strain tensors and exploiting symmetries of the elastic tensor. The Lagrangian, then, is just the integral over the domain of the difference of these densities. L ~= J p Up2 Ui Cijki U, k,) dV (3.3) I will assume, as do the authors, that the reader is able to complete the exercise by doing the algebra and applying the divergence theorem to find the conditions for the first variation of this Lagrangian to be zero. I will point out that their particular choice of how to represent the potential energy density appears not to have been accidental since it guides the algebra and makes the process rather more transparent. The result, as is expected from any Lagrangian minimization, is the equation of motion. However, the particular form of the surface term which must be zero is important and so I will write the full statement of the result to include it. 6L = 0 [= (p2 i Cijke Uk,j)6ui dV + (cijW Uk,e)6ui rj dSj 0 (3.4) where n is the outward normal surface vector and nj dSj is the magnitude of the outward area vector. The parenthesized parts of the integrands are the elastic wave equation and a representation of stress. These parenthesized terms should be separately zero when the first variation of the Lagrangian is zero. The same tensor algebra used to show that equation (2.3) is equivalent to Visscher's choice of representation for the potential energy density can be reversed to show that the integrand of the surface term in equation (3.4) can be replaced with Cijk Uk which is just the right hand side of equation (2.1) by which stress is defined. Surface normal stress, by definition, is just oij nj and so the surface term in equation (3.4) is zero if the surface normal stress is zero. With respect to the wave equation aspect of this condition, the assumption of harmonic time dependence guaranteed this in advance as can be seen from the fact that in section 2.3 I derive the same wave equation from that assumption and the application of Newton's Law. Lagrangian formalism is simply a substitute for Newton, but leads to the same result. What is new is the appearance of the stressfree boundary condition as a prerequisite. There is nothing in section 2.3 that will directly permit me to infer that modal functions formulated to satisfy the elastic wave equation correspond to a stationary Lagrangian. What Visscher et al. conclude from this outcome is that "...the displacement functions u, which are solutions to the elastic wave equation with free boundary conditions on S, are just those points in function space at which L is stationary." I want to emphasize yet again that this result is partly selffulfilled by starting with an assumption of harmonic time dependence and that the satisfaction of the wave equation is actually an implicit starting point rather than a condition on the outcome. In view of the foregoing, I draw a slightly different conclusion than that expressed by Visscher's team nearly a decade ago. What I see in the result is the following implication: Some superposition of functions which are guaranteed to satisfy the elastic wave equation will also satisfy free boundary conditions when their combination makes the Lagrangian stationary. Having rephrased the conclusion it becomes easier to see how limitations can arise from the basis choices employ, 1 in actual numerical calcualtion. Visscher et al., like Demarest before them, choose a basis for computational convenience which does not in fact consist of functions which satisfy any wave equation. Resonances which involve a combination of components which are hyperbolic together with some having low periodicity can be accurately determined. It becomes more difficult to compute resonances for shapes, such as those having high aspect ratios, which require more components having extended periodicities. It is harder to assemble superpositions of waves from a basis devoid of wave solutions than it is from a basis made up of wave solutions. In any case the paper continues with a logical inversion. It is claimed that finding a stationary point in some robust function space is paramount to finding a function that is a good approximation to a normal mode. If, of course, the basis is not made up of wave solutions, the stationary point must often simultaneously produce a wave solution, but failure to constrain the basis means that more is expected of the process than need be. It can only be said that concern for placing extra burdens on the process is lost in the fact that resonance problems usually converge nicely and give results which appear to be accurate for situations of practical interest [26]. That convergence of some kind does occur in the case of resonance problems is well documented in the literature cited. Indeed, the process has now been in used to determine elastic constants for a decade. I have personally observed the process and discussed it with Dr. Migliori. From his unpublished remarks, however, it is clear that the numerical process will sometimes fail to converge when attempting to match the actual eigenspectrum of objects with high aspect ratios against estimates generated by the numerical process. This difficulty appears not to have been the subject of published methodical analysis, but I have personally experimented with rectilinear copper samples having roughly a 1x1x3 aspect and was unable to find elastic constants using which the numerical algorithm could produce a resonance spectrum that matched what was directly measurable. This author can therefore confirm that whatever its success, there are clear indications that the numerical algorithm premised on the Lagrangian formalism just reviewed is not reliable outside of some convenient set of situations despite remarkable accuracy when confined to that set of situations. My purpose is not to bury the numerical techniques, nor even their theoretical underpinnings. Rather, I wish to praise them since they are extremely helpful and useful despite the limitations they operate under. It is important to repeat that whatever the precise limits of the numerical algorithms turn out to be, the fact is that they are demonstrably accurate within the domain they are typically deploy, 1 in. Moreover, the somewhat circular logic of the Lagrangian formulation carries within it the inspiration for a renewed investigation of whether analytical solutions are possible at least for the rectilinear geometry. Specifically, the Lagrangian technique highlights that the most promising basis ought to be one made up of functions which satisfy a consistent wave equation. By "< i1I. i~i wave equation, I simply mean any whose wave vector magnitude, and thus frequency, matches that of the normal mode in a propagating problem. It is interesting that my observations on the limitations of Lagrangian mini mization actually vindicates Holland's initial approach in that he purposely chose products of sines and cosines as basis functions. If Holland had found a way to implement his numerical approximation by separating his basis to expand the longitudinal and shear contributions separately, his algorithm would have been an appropriate numerical analog of the analytic approach which I develop herein. 3.2 Numerical Approximation Equation (3.3) can be written as a difference of distinct functionals in the form L = T[u] V[u] (3.5) I am now using u in the sense of timeindependent displacement functions which inhabit a vector space equipped with the usual innerproduct (i.e., vector dot product). T and V are linear functionals operating on the space of u's and I invoke a theorem of Functional Analysis (see Ref. [44] 30), usually taken for granted, that permits me to assume the existence of operators such that I can rewrite (3.5) in the form L (ulTlu) (ulVu) (3.6) where the elements of a matrix representation of the operators are defined, as usual, by the results of their operation on a basis for the u's. This meticulous formality allows me to emphasize that this is a situation in which that basis is not actually known and that the situation is thus far abstract. Actual numerical computation is accomplished using a finite and definite basis. Following Visscher et al., I will expand the displacements and estimate the oper ators using basis functions of the form xpyqzr. This is just a computationally convenient realization of the more general use of products of functions of the coordinates such that each element is a separable function. These are not, in general, orthogonal, but they may be chosen to be linearly independentas is the case here where simple powers of the coordinates are used. It will follow that a RayleighRitz like minimization of their combination is reducible to a generalized eigenvalue problem. One distinct and useful feature of this particular basis will be that its elements have definite parity. Equation (3.2) which will form the basis for computing matrix elements of the potential energy operator, involves a mixing of different components of the displacement vector functions. I therefore expand the components of the displace ment vector functions in bases of their ownrather than expand the displacement vectors in a basis of vector functions. Correspondingly, there are multiple (though perhaps not distinct) basis sets {1 i)} (i = x, y, z) and these result in an expansion of the form Z X) (X) Ux Zk Ck k Ly J Ek "Ckk In terms of these finite bases, approximations to the kinetic and potential operators have the following representations: T%(%(o) = Vo p 4 ) i dV (3.7) VA(it(m) / 9 k) c aj 9 dV (sum over j & ) (3.8) These matrix element definitions are in the same form as used by Visscher. They are elements of approximate, not exact, operators. There is no assumption that basis functions individually satisfy the bulk wave equations of the elastic media, let alone the boundary conditions of the geometry. There is no reliance upon orthogonality properties of the basis functions in any geometry whatsoever. One way to express that the Lagrangian is stationary, is to assert that the deriv atives of equation (3.6) with respect to components of u are all zero. When the u's are discrete, this certainly is equivalent to the following (functional differentiation will lead to the same result): U2Tu) Vu) = 0 (3.9) This is just a statement of a generalized eigenvalue problem for which computer algorithms are widely available. The numerical algorithm generates a basis to some limit on the powers of the coordinates, calculates the matrix elements of the operators by integrating over the sample pursuant to equations (3.7) and (3.8), and then invokes standard software to calculate the eigenvalues and, if desired the eigenvectors. This eigenvalue problem form of a minimization problem is expected to display a convergence of eigenvalues toward some limit as the order of the operators increases. It is common experience that this convergence is ah,v of a decreasing nature so that the estimates are ah,v upper bounds of the true values. However, when the basis functions used in the numerical procedures are merely products of powers of the coordinates and when the physical reality being modeled does not involve the converging eigenfunctions, or their derivatives, converging uniformally to zero at the boundaries, the expectation that eigenvalues are upper bounds may not be provable. A typical proof that a RayleighRitz procedure will consistently produce upper bounds depends upon the trial functions themselves vanishing at the boundaries [45]. In addition, although the numerical basis consists of elements that are linearly independent, they are not in general orthogonal as is assumed, for example, in a more elegant approach to a similar theorem [46] proposed by Peierls. Of course, it may happen that some of the modes are well represented by functions that vanish at the boundaries. In Ch. 6 I will derive all of the k = 0 modes of a rectangular waveguide and a subset of these will be uncoupled and defined explicitly by the boundary zeros of sines and cosines. Powers of the coordinates certainly converge rapidly to isolated sine and cosine functionsa feature endemic to numerical computations of their values. If the solutions consist of superpositions of sines and cosines, however, and if the boundary values are not wellrepresented, there are no such regularities to assist the rate of convergence and no vanishing of the wave functions or their derivatives to converge toward. While it is beyond the scope of this research to pursue these concerns further, I have attempted to point out that the choice of a nonorthogonal basis coupled with the fact that the boundary conditions do not generally involve vanishing of the basis elements, creates inherent difficulties for the efficacy of prevailing numerical methods. This strengthens the motivation for developing an analytical solution. 3.3 Partitioning the Problem into Parity Groups Holland [21] carefully organizes his basis, made up of products of sines and cosines, into parity groups related to a coordinate system centered in his paral lelepiped with axes normal to the faces. Demarest [23], and Visscher et al. [25] adhere to the same system. The same parity pattern classification system p1 i, a pivotal role in my own analytical derivations which follow. The importance of this classification scheme derives from the interplay of a rectilinear geometry with the symmetry properties of the elastic tensor (see section 2.3) within the potential energy operator defined in equation (3.8). The kinetic energy operator can be diagonalized independently, but careful examination of equation (3.8) shows that the potential energy operator will block diagonalize where the blocks are defined by parity patterns of any basis whose elements have definite parity. Among the consequences of this fact is that the eigenvalue problem stated in equation (3.9) can be divided into a distinct eigenvalue problem for each block. From an analytical standpoint, however, the real value is that the symmetry classifications are an organizing principle for the entire analytical approach. Although the parity classification scheme was first identified in the context of closed geometries, the potential energy operator does not change form in the case of a rectangular waveguide and so the classification scheme remains equally valid and applicable. The only distinction is that the parity along the long direction is often arbitrary as will be so indicated. When turning a waveguide into a resonator, the resonator modes symmetric along the former long axis will simply divide into even and odd groups. For the sake of comprehensiveness, I therefore exhibit the full parity pattern using the nomenclature of Holland. Let E and O denote even and odd parity for functions relative to Cartesian coordinates aligned normal to the surfaces and along the central axis of the wire. An unspecified, but consistent, parityas may be associated with the long axis of a waveguideis denoted with a P. Parity complementation of an otherwise unspecified P (as, for example, resulting from differentiation) is denoted by P. Directional dependencies are implied by juxtaposition in a productin the order x, y, z. Symmetry patterns of displacement functions are then limited to only the families shown in Table 3.1shown as column vectors of the x, y, z components. The last two of these (flexural) are a degenerate pair in the case of square rectangular cross sections, but as pointed out by Nishiguchi [1], they are distinct for the general rectangular case. Each of these parity patterns can be generated by a single product representing the parity pattern of the scalar potential whose gradient produces the longitudinal part of the displacement functions. The shear part must have a matching parity pattern and so the pattern of the scalar potential fixes that of the mode generally. In the case of Dilatation, for example, observe that the gradient of a product Table 3.1: Mode Parity Patterns Families in Rectangular Geometry Defined by Displacement EE OE EO Dilatation group: D ^i .T, I Torsion group: { T, t IS1, I Flexion of zx plane: { Flexion of yx plane: ( with parity pattern PEE automatically produces vectors with the parity pattern of the Dilatation group. Similarly, the gradient of POO will generate the Torsion groupand so on. = E = O s F3, S3, F2, P=E P=0 CHAPTER 4 MATHEMATICAL STRATEGY 4.1 Notation and Function Extension Issues In section 3.3 I reviewed how, for rectangular geometries, the potential energy operator block diagonalized to partition the problem into independent solution families based on parity patterns of the basis. For this to be the case, it is only necessary that the basis functions individually display a definite parity in their separable parts. From this, any superposition of basis functions from the same family will also exhibit the same definite parity pattern even though the superposition becomes nonseparable. My approach involves solving the boundary value problem in a way that takes advantage of the fact that solutions are partitioned by parity family. However, this does not imply that it is necessary to do a distinct derivation for each such family. Rather, as much as possible, each derivation will encompass all parity families in such a way that one algebraic result can be converted into a realization for each distinct parity family by a straightforward substitution. It thus is a distinct result that the solutions for each family are shown to be manifestations of a single theoretical result. In order to be able to transcend distinctions between parity families for each derivation, it will be necessary to utilize a specialized notation. The notation will not violate other conventions of mathematical notation. It will sharpen rather than obscure important relationships. It will considerably shorten the expression of individual relationships, dramatically shorten derivations, and preclude redundant derivations. Results expressed in this notation will unify the manifestations among parity families. In its simplest characterization, the basis functions defined below will all be products of functions that are coincident with sines and cosines inside of the waveguide. It should be immediately pointed out that extending these functions to infinity is a tempting, but inadvisable option. It is true that vanishing of the density beyond the boundary would serve to keep the physical description realistic even if the displacement per se were to be so extended. However, there are distracting adverse mathematical consequences of indulging such an extension that should be avoided. One of these is that the Helmholtz Theorem, which is crucially relied upon to separate the displacement field into parts generated by a scalar potential and a vector potential, becomes problematic when the fields do not vanish. Either they should vanish totally at the boundary, or if extended, some kind of convergence factor should be inserted. This, however, leads to other complications. Ultimately, all such complications will be neatly avoided by a judicious choice of transform, but it remains the case that I will need the freedom to assume that the displacement fields have a behavior beyond the surfaces which will not contravene assumptions of the Helmholtz Theorem and at the same time I wish to avoid having to specify what that behavior is specifically. Henceforth, the functions multiplied together to form displacement basis func tions will themselves be defined as cosine or sine functions only between the surfaces. With respect to the stipulated coordinate system, these need to have definite parity, and so they will never have constant offsets to their phase at least in the transverse directions. It will prove an asset to intuition if I choose to simply call such functions E and 0 in respect of their being either even or odd. Scosqq q < h E, = (4.1) undefined ql > h Ssin q q undefined q > h where h is the boundary limit and q is y, z, or some coordinate value (such as h). To minimize notation, subscripts and superscripts can be used to indicate wave number and directional dependencies. Distinguishing x, y and z dependencies will sometimes be inferred by position if there is no risk of ambiguity. For example, E(qx)E(,y)E(lz) = EEY EE = ExEyEz = EEE. To implement generality in the derivations, I will need to denote functions of definite, but unspecified, parity by using a function variable. In general, the letter P, with appropriate subscripting to distinguish variables, will be used for this purpose. For example, PyPz could take on the specific function values of 00, EE, OE, or EO. Often, the mathematical structure of relationships will depend upon the rela tive parity of juxtaposed functions. I will accommodate this by denoting parity complementation of a function variable by placing a bar over it. For example, PP could be EO or OE. Differentiation of E or 0 with respect to coordinate will be the dominant operation. Because of the sign change introduced by differentiation of a E within sample boundaries, it is not .i. li correct to assume, for Pi as an abbreviation of P(qTixi), that Pi,i = riPi. To nevertheless permit differentiations to be unam biguously specified at the highest level of abstraction, parity complementation resulting from differentiation per se will be denoted by placing a tilde over a function variable. Specifically, E= 0 while 0= E, and it is thus alv,i correct to write Pi,i = Ti Pi. Upon eventual substitution of E or O for Pi, the appropriate sign changes can be made. However, the symmetries of the problem ultimately result in the cancelation of sign distinctions and there are also cases of successive differentiation which invoke the parityinvariant rule P= P. The derivations will eventually reveal that the sign distinctions inchoate in notations like P are eliminated in the final results which can invariably be stated strictly in terms of parity variables and their simple compliments. Solutions specific to parity families will be realized simply by choosing E vs O assignments for at most two function variables (viz. Py and Pz in the case of a waveguide with P, being an additional variable only if the waveguide is capped to become a resonator). In order for this to be resolved in the derivations, the following additional notational device will be needed: x\ x, P,=E (4.3) yP, Pi Some simple examples that illustrate how this can be applied are: 'PY P4 P { (4.4) 1 t1 Therefore, differentiation of any P or P can be expressed without recourse to the tilde notation by using Pi,i = ri Pi Pi,i /i Pi (4.5) l1 1 Pi Pi 4.2 Defining the Basis and Superpositions My key physical strategy is to assume that some superposition of tractable fundamental basis functions will assemble a tractable nonseparable function that manages to satisfy the boundary conditions. While this is clearly an obvious approach to the predicament, I have been unable to find examples in the literature that reveal any attempt to actually apply it to the analytical solution of this problem. Though the assumption that some kind of superposition is needed can be inferred from various discussions, a failure to even realize the possibility is sometimes clearly evident (see, e.g., Ref. [3] IV). I can only speculate that the unavailability of basis functions which themselves satisfy the boundary conditions has been viewed as such a serious departure from typicality that it has been more tempting to conclude unsolvability than to pursue superposition in spite of it. A more courageous view has been exhibited by researchers looking for theoret ical underpinnings of numerical approximation attempts. I therefore give credit for reinvigorating an analytic pursuit of superpositions to those who developed numerical approximation methods. These have been reviewed in the prior chapter in part because they create a framework in which superpositions and notions of how their elements should be structured take concrete shape. In fairness, it should thus be r.ii. I. l that the results achieved herein are the result of adapting the progress made in numerical methods to a revisit of the presumably intractable analytic problem. My own approach to formulating the basis functions is straightforward. I accept that the longitudinal and shear contributions should be expanded in separate bases. I depart from recent numerical approaches in that I require all basis functions to at least satisfy the bulk wave equation with respect to longitudinal or shear expansions. In fairness, it should be pointed out that Holland [21], by using products of sine and cosine functions as trial functions, approached the problem similarly, but he did not segregate the trial functions into longitudinal and shear contributions. Neither did his successors, Demarest and the Visscher team, each of whom abandoned any efforts to constrain trial functions to those which satisfied a wave equation. Numerical approaches to date have sought an overall convergence of displacement arrangements guided by a variational principle and so the trend has been to impose increasing arbitrariness on the structure of the trial functions. Analytically, this is fruitlessor worse. The reason is that the trial bases used by Holland and his successors is not in an analytically "(~.i Il !. 1. representation. The three key issues surrounding the construction of superpositions are (1) which fundamental basis functions to use, (2) how to represent the superpositions of these, and (3) how to transform the equations written in terms of the superpositions so that they can be solved as a finite set or solved by some recursive process operating on an infinite set. I have already indicated that I will expand both longitudinal and shear contributions in basis functions which individually represent wave solutions. What remains is to specify more concretely both the representa tion of these basis elements and the representation of their superpositions. In a subsequent section of this chapter, I will take up the transformation issuealso proposing a simple approach which has not appeared heretofore in the literature. In derivations which follow, fundamental basis elements will .1. 'i take the form of a separable product in the form PPyPz where the P's (see preceding section on notation) stand for particular even or odd functions which, within the boundaries of the sample, are coincident with cosine and sine functions, respec tively. The eight parity patterns corresponding to the eight mode families defined in table 3.1 correspond to the eight possible values of PPyP,. Note, however, that as long as each component of a superposition has a common parity pattern, any superposition of them will exhibit the same parity pattern even though it is itself not a separable function. Also, since I am concerned with propagating modes characterized by translational invariance, P, will naturally cancel out among the relations. (For a rectangular resonator, mode families could be generated by superimposing propagating modes of varying xdependencies.) Since the mode family is defined by a parity pattern of the overall displacement, the parity pattern of the shear components is constrained by the necessity that the parity pattern of V x H be identical to the parity pattern of V(p. It is easily checked that for X P P,P, this constraint will be satisfied so long as the components of H are made up of fundamental elements which have parity patterns in terms of these as follows: H, ~ PPy P, H, ~ PPy P H, ~ P PyP, Fundamental basis elements (either longitudinal or shear) are linearly inde pendent, in the intervals defined by the medium, for the obvious reason that sines and cosines with distinct wave numbers along any coordinate form a linearly independent set. Any linear combination of the basis elements must satisfy the bulk wave equation and this will be achieved so long as the elements individually satisfy "a" wave equation for the same magnitude wave numberwhich then factors. This is accomplished by requiring the wave numbers of terms in each basis element to satisfy the bulk dispersion relations: k2 k2 + 2 + I = for components of o (4.6) c2 ^2 k2+ a + a = for components of each Hi (4.7) Cs This constraint reduces the degrees of freedom by one. I contemplate a mode with a fixed k, and choose one transverse direction as representing a degree of freedom while constraining the second according to equations (4.6, 4.7). These functions can be treated as orthogonal with respect to each coordinate, but this will only be exploited obliquely in the sense that this property is deeply buried in the nature of the transform that will be applied to their superpositions. A superposition can be modeled as discrete or continuous. On physical grounds, however, the discrete distribution is the correct one in this case. The physical reality being modeled consists of elastic waves in bounded media. Although the boundary conditions will be viewed through the lens of mathematical abstraction, the reality is that elastic waves refract at the boundaries and any superposition models the summation process that superimposes all of their reflections. Since there are a finite set of surfaces, each refraction is a discrete event. The superpo sition is thus a discrete summation. Accordingly, the scalar potential will have the form o = P1(k x) diPy(li y)P,(rl z) (4.8) where the superscript denotes a "longitudinal conjugation" defined by Tr = /2/c,k2 positive root (4.9) For the shear superpositions, the expansions take the form H, = P,(k x) aP,(a y)P (j z) J Hy P1(kx)ZbPy(,ajy)P, (fz) H, P((kx)ZlcPy(ay)P, (, z) (4.10) where the + superscript denotes a "shear conjugation" defined by j+ = w/c2k22 positive root (4.11) Note that, in the foregoing, both li's and ai's can range over real and imaginary values within the same superposition. There is a further, important, observation. The foregoing sums contain weighted terms that are solely products of sines and cosinesat least within their boundary domains. Since each such term has definite parity with respect to the sign of the wave number in its arguments, the effect of the sign of the wave number is ahl,v factorable, in the sense of 1 Pi (k x) = P}(k xi) Pi A series of terms which differ only in the combination of the signs of the wave numbers in the arguments will ah i< factor into a single term with positive wave number arguments multiplying a sum and difference of coefficients. The combination of coefficients can ahbi be absorbed into a single coefficient. The result is that in each of the foregoing sums, I ahbii choose a representation in which only positive (albeit real or imaginary) wave numbers are summed over, but for which some of the coefficients may be negative. 4.3 Dimensionless Representations In section 2.1 the groundwork was laid for reporting the results of derivations in a dimensionless form. The specific scale factors and notation for this are as follows: First, recall that of the rectangular halfwidths, the h, will be arbitrarily notated as the smallest one whenever hy / hz. The appearance of an unsubscripted h will refer to h,. The dimensionless aspect ratio a will be hy/h,. Of the two isotropic velocities, c, will used to rescale results and the ratio ce/c8 is, as noted, denoted R. Frequency will be rescaled by the rule Q 2 where 0 = (4.12) Lo, h This is a common rescaling in the historical literature of the problem and, in addition, it is common to report f in units of 7/2. Consistent with this scheme, wave numbers will be put into compatible dimen sionless units by multiplying them by h. There is no absolute rule on the choice of symbols for longitudinal versus shear wave numbers in dimensionless units, but the general attempt will be to show dimensionless forms by converting Latin letters from lower to upper case and choosing distinct Greek symbols to convert existing Greek symbols to dimensionless form. For example, K = kh, a = a h and 3 = h are common choices. The bulk wave dispersion relations in dimensionless terms would then take forms as follows: a2 + a2 + K2 2 a+ +K2 R2 /3;+/3+^2^(4.13) In some derivations, a free wavenumber variable, A, will be used. In its dimen sionless form, it will be denoted A. The ease with which dimensionless forms of results can be written from inspec tion of dimensional ones will become apparent as examples appear. 4.4 Derivation of RayleighLamb Equation In the Introduction, it was pointed out that in 1889 Lord Rayleigh was able to derive a transcendental "frequency equation" which defines the propagating modes of an isotropic infinite plate. This equation is now universally referred to as the "RayleighLamb Equation" and the dispersion patterns it generates are often called "Lamb \\ i. Derivations, and introductions designed to promote reader completed derivations, appear often in the literature, but most of these expositions are considerably more cumbersome than the derivation about to be demonstrated (see Rayleigh [8], Miklowitz [10], and Ch. 10 of Auld [38]). Nevertheless, the form of this equation is of fundamental importance in the results to follow and it will be helpful to demonstrate how it can be rederived succinctly. Besides producing a needed result, this exercise will provide a clarifying example of the devised specialized notation at the same time it introduces the basic pattern for novel derivations which follow. The physical scenario consists of an isotropic elastic material sandwiched between infinite planes at z = h. The surfaces are stress free. A planewave system propagates along the x direction. At any x position, there are no variations along the y directions. Referring to boundary conditions (2.22), s  z, p  x, p  y: At z = hz for all y and all x: k3 k p + 2 ,z2 PwZ ,OX 2(Hy,  i [(H/,y  2 [(HYzXX Hx,y ),z Hx,zz) + (HZ,z  Hyz,) + (H,, (4.14) However, since Oy  0, the foregoing will simplify dramatically. In addition, the simplest vector potential that will generate a shear wave with no y displacement, is just: and so I can set H= H, 0 H H 0 0. The boundary conditions above now collapse to 3 k2 p + 2o, 2Hyxz (Hy,xxzz Hyzz) (4.15) Assuming that I will not need superpositions, individual basis elements that pro duce potentials able to satisfy the wave equation are (according to my already reasoned basis characterization above) simply p = DP,(kx) Py(ry) Pz,(l*z) H, = AP,(kx) Py(ay) Pz,(*z) With D and A as unknown constants. But if Oy  0, it must be that r = a = 0, and Py(O) E(O) = 1 and so the potentials can be further simplified to D P,(kx) Pz(r*z) A P,(k x) Pz(a+z) (4.16) Hy,y),z]  Hx,x),,] Substituting equations (4.16) into (4.15) and performing the differentiations at z = h produces the following simultaneous equations: D (p/3k + 2(T*)2) P1(k x) P,(T* h) 1 Pz (T* h) 1 2Ak J+ YPz (,h) 1 t p A (k2 (+)2) P1(kx)P,(~+ h) (4.17) 2 At this point it is trivial to divide one equation into the other to eliminate the unknown coefficients as well as all of the xdependencies. The result is (pk/ + 2(q*)2) P (T* h) 4 ka+ P (a+h) kl* P,(* h) (k2 (7+)2) P,(7+ h) (4.18) It is gratifying to notice that the two conflicting sign contingencies for P, simply each resolve to an assured minus signand the Pz sign contingencies neutralize each other. This is a pattern that will repeat itself in the more involved derivations. The result is already in the form of a "frequency equation," but besides some rearrangement, there is a final analytical step to be performed. Since the wave systems are made pII ii. "by virtue of rl = a 0, the conjugations are: (T*)2 L2/c2 k2 L2/c2 k2 (4.19) Recall that 3 = R2 2 with R = ce/cs. Applying these to the parenthetical on the left of equation (4.18), it can be restated more usefully. (k + 2(*) (2 2) + 2  (jOk + 2(q/*)2) (R2 2) R' +2(R' k2) R2 C 2a(, R2(, (k2 (+)2) Dk 2k  SC 2I2 \ s The details of such steps are indulged here because they are prototypical of rearrange ments that recur in subsequent derivations, but which will not be hereafter pre sented in detail. With this particular reexpression substituted, and following some rearrangement of terms, equation (4.18) can be written in the form P (* h) P (a h) 4k2 a+ (4.20) P, (* h) P (a+h) (k2 (+)2)2 Equation (4.20) is, in fact, the RayleighLamb frequency equation in a repre sentation which encapsulates both its alternative forms. The socalled symmet ric form follows from setting Pz = E = cos in which case the left hand side becomes tan(q*h)/tan(a+h). If Pz = O = sin, the left hand side becomes tan(a7+h)/tan(q*h) and the equation is said to be in its antisymmetric form. Accordingly, the RayleighLamb equation is often written in the form tan(qr*h) 4 k2 a+ I tan(7+h) (k2 (7+)2> (4.21) In this representation, the symmetric and antisymmetric forms correspond to the exponent on the right being positive or negative respectively. The names given these forms obviously match the parity of Pz in my specialized notation, but that is not why they were sonamed. If the displacement patterns corresponding to these equations are mapped, it is readily seen that the symmetric form corresponds to dilatationss" in which the plate surfaces are either extended or indented together at each x position. In the antisymmetric case, the sides of the plate are alternately extending or indentinggiving rise to a ripple effect. Indeed, the l iin., i," solutions are flexural in nature. Finally, the RayleighLamb equation would be P,() P(a) P 3) P (a) in dimensionless form (see section 4.3) 4K2f 3 (K2 a2)2 6. 5. 4.4 4. o rl a/ r1 a 1. R2 =3 Pz =E 0. I I I 1 1 1 1 1 1 1 1 1 I I II I 5i 4i 3i 2i ii 0 1 2 3 4 5 6 K [hkx] Figure 4.1: RayleighLamb Dilatational Modes. In the historical development of the solution, these are referred to as the "symmetric" modes. The roots of this equation can be mapped as dispersion curves using contour plotting. There will be a set of branches for P, = E and another for P, = O. A rearrangement of the equations is necessary to preclude fatal divergences in the numerical computation involved in the plotting. In terms of the dimensionless representation, numerical plotting is based on finding the roots of [(K2 a212 +4 K2 P ) P(a) =0 PI P, (4.22) (4.23) Figure 4.1 shows branches of the dilatational or iii,,. I i c" modes resulting from substituting P, = E = cos into dimensionless RayleighLamb equation. Figure 4.2 shows the branches of the flexural or il Ii1in,. i inc" modes resulting from substituting P = O = sin into the equation. 6.  5.  44 4. 0 r1 3. r1 *% 2. 1. R2 =3 Pz =0 0. I I I I I I I I I I I I 6i 5i 4i 3i 2i li 0 1 2 3 4 5 6 K [hkx] Figure 4.2: RayleighLamb Flexural Modes. In the historical development of the solution, these are referred to as the ii:i iiiii ii." modes. The RayleighLamb equation defines coupled modes of the infinite plate, but there are also a set of uncoupled modes that can propagate in this geometry. Recall that surface coupling is an interaction between longitudinal and that polarization of shear waves having displacement components perpendicular to the surface. I can set p = 0 in the boundary equations (4.14) and contemplate plane waves that only have displacement parallel to the z = h surfaces. This can be realized simply by setting H, = Hy= 0. The boundary conditions then reduce simply to 0 Hzy 0 = Hz,, at z = h (4.24) Shear waves with polarization resulting in displacements solely parallel to encoun tered surfaces are often called SH waves (for "shear, with displacements horizontal to the suin ) in contradistinction to SV waves (for !h. ir, with dispacements vertical to the sui .. . ) that are coupled to longitudinal waves at surfaces. 4. 3. 44 0 .4J H 2. H m 0 1 2 3 4 K [hkx] Figure 4.3: Infinite Plate SH Modes. Shear waves with displacements parallel to the surfaces, and which vanish there, form an uncoupled propagating system in an infinite plate. The lowest even and odd subbands are shown together. Now, Hz, by my chosen basis representation, must have the form Hz = B P(kx)P,(y) P (a+z) Even & Odd SH Modes Independent of R . . . . . . Thus, it is easy to see that the nontrivial solutions for this SH system follow simply by setting the zderivative of Pz(a+z) at z = h to zero. P,(ah)= 0 sin(v=22 K2)= 0 (4.25) cos(V 2 K2 I will make comparison to these SH solutions in the sequel. Subbands of this solutions set are shown in Figure 4.3. They are analagous to torsional modes of a waveguide whose dominant displacement pattern are also characterized by displacements parallel to the surfaces. 4.5 How to Transform Superpositions The rederivation of the RayleighLamb equation was an exercise in organizing the problem into an algebraic form amenable to the simple elimination of unknown constants. The three independent boundary value equations constitute at most three constraints. Absent superpositions, the scaling constants for the potentials will constitute one degree of freedom for the scalar potential and as many additional degrees of freedom as there are distinct components of the vector potential to resolve. However, at each surface only one directional coordinate will be fixed and so there is possibly one additional degree of freedom to be resolved with respect to the other. The RayleighLamb scenario reduces enough degrees of freedom to balance the constraints. Specifically, restricting the scenario to plane waves eliminates the directional degree of freedom at each surface and reduces the number of constants from a maximum of four to a manageable two. With the nontrivial constraints reduced to the same number, a solution follows. In the rectangular waveguide case, the directional degree of freedom at each sur face persists by the fact that plane waves no longer suffice. Barring some fortuity, the constraint equations will go to sixthree for each surfacewith no eliminations. A question arises: how many independent vector potential components must there be and thus how many degrees of freedom due to them? Two directional degrees of freedom plus the scalar plus at least one vector potential component makes the minimum degrees of freedom to be four. If I couple the .,.1] i:ent sides, the independent constraints may be reduced and if I include additional vector potential components I can increase the degrees of freedom. But as I have demonstrated (section 2.5) abstractly, nonseparability compels the introduction of superpositions and a new problem arises of how to resolve their expansion coefficients. The goal of devising a transformation is to deal with this latter complication. Once transforms are applied to a superposition substituted into a rectangular system, it will be seen that there are elementary RayleighLamb relationships between components of the coupled potentials. This result might be anticipated qualitatively by noticing that many dispersion subbands revealed by numerical approximations of the rectangular case bear uncanny resemblance to Rayleigh Lamb dispersions of a plate. (This was noticed somewhat by Nishiguchi [1] 3). Given that I have explicitly chosen a representation of the potentials as sums of products of sines and cosines (over only positive wave numbers), the boundary conditions are readily visualized as qualities among sums of exponentials. A Fourier transform is then naturally ,'1; 1. .1 The domain of the transformed terms is, however, finite, and so it is necessary to either periodically extend the functions beyond the boundary or to modify the definition of the functions so that they vanish or converge towards zero beyond the boundary. A simple periodic extension would have the desirable consequence of producing delta functions under a Fourier transform corresponding to each termbut only if the wave numbers were real and the coordinate was being mapped to real transform variablesor imaginary and being mapped to imaginary transform variables. However, it is to be expected that a given potential is represented by a sum containing both real and imaginary wave numbers with the result that a standard Fourier transform would diverge. The inescapability of this expectation comes from examining Rayleigh Lamb solutions for which the fundamental modes distinctly involve transverse wave numbers which are pure imaginary. In addition, as anticipated in section 4.1, a periodic extension would, at least abstractly, render the assumptions supporting application of the Helmholtz Theorem invalid. Preservation of the ability to transform into delta functions, despite divergence of the Fourier transform and other issues, can be secured by a simple expedient. Instead of the general Fourier transform, I use a IiIl!!., '1" transform which is valid only on exponentials (though without regard to whether the wave numbers are real or imaginary) and which avoids the divergence problem by virtue of the details of its defined domain. Moreover, since the domain of functions to be transformed is explicitly constrained by the physical boundaries of the sample, there is nothing illogical or restrictive in defining the applicable domain for the transform to include only exponentials defined on the coordinate intervals that measure the sample and thus there need not be any concern over periodic extension. The transform to be used then has a simple definition determined by the element mapping aeV 2a6(7 r) (4.26) hq < q < hq (q = x, y, z) 7,i TE R U 9 a E C The factor of 2 is a convenience to dispose of the factor I in the exponential representation of the sine and cosine. It is almost selfevident that the transform is 11 between the set of exponentials and set of delta functions so restricted. Since a can be zero, an additive identity exists on each side and we have an isomorphism between two groups. Because the range of r] and 7 is defined to encompass both real and pure imaginary values, the transform operates without difficulty on any combination of exponentials with real or imaginary wave numbers. It is troublesome to write down an integral form of this transform which smoothly adapts to whether the argument is real versus imaginary and which limits itself to the coordinate boundaries. Of course, the underlying mechanism is a trivial Fourier transform. Fortunately, since the domain of functions is strictly limited and the element mapping from that domain to the transform domain is clear and unambiguous, the transform can be performed without difficulty. It is inil i. I i1 therefore, that the inverse transform can be trivially written down in an integral form that is not troublesome and which applies adaptively to transforms involving imaginary delta arguments as well as real ones. One example would be 1 rX=+o \ =+ioo f (y) {2A f (A) e CAYdA + e AdA 2 VXoo Jioo Of course, this is only valid for f(A) that are produced by the :iiIlIII. ,I" trans form in the first place. The left or right hand side of any boundary value constraint will involve one or more sums in the following general form (which omits common factors of P (k x) which are also subject to differentiation): Saj f (p, pj) P, (pj qi)P2 (p q2) (4.27) 3 Here, I have generalized the various cases: ql, q2 stand for distinct coordinates y or z; P1, P2 are distinct function variables, or derivatives of them: Py or P,; p will be an r or a for representations of the scalar or component of the vector potential respectively; pt is an abbreviation for the conjugate wave number based upon the applicable velocity: p* = /z2/ck2p2 p+ = /2/c2k2p2; f (pj, pt) will be a prefactor resulting from one or more derivatives taken. These will ahlv, be a single product or sums of products of pj and/or p>. To characterize the effect of transforms on the boundary expressions, it will be sufficient to contemplate f(pj, p1) as being a single such product since a sum of such terms can be distributed to produce sums of summations. By summarizing below how the ilphi.d" transform affects boundary value terms generically defined by equation (4.27), it will be possible to immediately write down the transforms of the actual boundary conditions without elaboration. First, with y  A chosen to make the example concrete, consider the general effect of the transform on a function variable. 1 1 y{P( y)} 6(A ) + 6(A + ) (4.28) i\ 1 I can now write down the transform of equation (4.27) with respect to ql 7. Note that 7 stands for whichever transform dimension variable is matched to ql. My convention henceforth will be that y  A and z  In the rendering of transforms, ql and q2 could be either y or z, though alviv distinct in a given case. While the reader is presumed capable of writing down Fourier transforms of sines and cosines on his/her own, the usual results are somewhat simplified and adjusted in this case by virtue of the stipulation that superpositions will alv be chosen to involve only positive (albeit possibly imaginary) values of whatever wave number variable p designates. The result, therefore, is that only one of the two delta terms survive in each case and the specific form of the result depends upon the sign of the transform variable in a way that can be neatly summarized. So, for convenience I list the results in detail: ufl { Ej aj f (pj, pj) K(pji qi)Pqp (p q2)} a f (, 7t) 1 ( ) P (7t q2) for 7 > 0 (4.29) Pq1 aj f(O, Ot) 1 2 6p) (t q2) for 0 (4.30) i p 0 ajf( 7) 6 (171 ) (7q2) for/71<10 (4.31 ) i 1 Similarly, I can now write down the transform of equation (4.27) with respect to q2. There are again the same three cases depending upon the sign or 7. q27 { Ej a f (pj, pj) Pq (pj qi)lPq(p q2)} Sa f (Qt) 1 ( P ) q 1(7t ) for7>0 (4.32) {f2 a f(0t,0) 1 2 6(0op) P,(0tqi) for7 0 (4.33) I P2 IP2 Zaj f(t7 7{) } 1 } P p) (7t qi) for 7 < 0 (4.34) Si 1 2 Having been meticulous in motivating, then ju I iii: and now demonstrating the effects of this iiiiphjii' i transform. It can be drastically simplified in practice with the following observations: The prior stipulation that expansions will only be over positive, though possibly including imaginary, wave numbers has allowed the results of transforms in the context of the problem to be more easily summarized in terms of single delta functions. One additional fact can now literally trivialize the use of this transform in practice. Namely, the fact that any given derivation takes place in the context of a specific parity pattern guarantees that within any derivation the parity pattern on each side of a boundary equation will be identical. This parity agreement guarantees that when either coordinate is transformed, the P function transformed on each side will be the same. That being the case, all of the prefactors which depend on the parity of P will cancel between the sides in all cases where that P function is common to all additive terms. In general, this condition nearly alv fulfilled. Moreover, with those distinctions gone, an examination of equations (4.294.34) will reveal that replacing 7 with 7 throughout is sufficient to cover all cases. Therefore, as a practical matter, the only rule that will be needed is just Replace every Pq(pq q) to be transformed with 6(171 pq). It will not matter whether pq is a wave number or its conjugate. It will not matter whether pq is real or imaginary (it is guaranteed to be positive). The triviality of this rule is a direct consequence of having imposed a meticulous series of specific choices. It is neither fortuitous nor could it have been readily anticipated that it would reduce to this. 69 Once a boundary equation that has had a superposition substituted into it becomes transformed in this way, it becomes an equality in terms of the transform variable. Given the behavior of delta functions, each value of the transform variable will, on each side of the summation, select out either a specific term of the sum, or be identically zero. This collapses the equality of functions of sums into a constraint between components of sums from each side. Correlating transforms over y with those over z is a remaining issue, but how this must be done will be developed in the derivations which follow. CHAPTER 5 DERIVING NORMAL MODES OF PROPAGATION 5.1 General Considerations The full boundary conditions revealed generically by equations (2.22) infer that all three components of the shear vector potential are mixed together in satisfying the stressfree surface boundary constraint. If, however, I restrict attention to just those modes which propagate, it is not immediately clear whether all three vector components are needed. Some inspiration can be drawn from the RayleighLamb derivation of section 4.4. That derivation involves an infinite plate bounded in the z directions and only requires the Hy component of the vector potential. The intimation is that bounding also in the y directions might simply invoke the need for the H, component, but there is no a priori reason to expect to need an H, component as well. The foregoing motivates an attempt to find the essential relationship between H, and the other vector components which participate in satisfying the boundary conditions along the surface of a waveguide in which normal modes propagate in the x direction. Unless some constraint can be found that eliminates components or establishes some dependency among them, there will be more degrees of freedom in the propagating problem then constraints available to resolve them. I thus proceed to investigate this relationship among components in a way that is independent of the boundary conditions per se so as to confidently narrow the approaches used in solving the boundary problems. 5.2 Acoustic Poynting Vector of a Normal Mode Propagating modes carry energy. In analogy with electrodynamics, there will be a vector that indicates both the direction and magnitude of the energy flux. This vector, the acoustic Poynting vector, describes a physical reality whose invariances under manipulation of the coordinate system constrain its mathematical form. The components of H participate in the construction of a mode via antisymmetries arising from the fact that shear displacement is V x H. Moreover, satisfying the boundary conditions ostensibly involves mixing up the components of H. This i _ I I look for constraints on the relationship between components of H that might be required to maintain invariances of the Poynting vector while preserving the antisymmetries built into the shear contributions to the mode. Some familiarity with the acoustic Poynting Vector reveals it to be a complicated object when expressed in terms of strain and this provokes a curiosity over whether it may harbor such constraints. It is difficult to articulate further what is, in the end, an intuition that this might be so. The intuition will ultimately be justified by the result. A representation of the acoustic Poynting Vector itself can be readily derived. The Poynting Vector will be denoted herein as J since P's have been extensively used for another purpose. It its defined by the property that integrating it over a surface S produces the energy flux through that surface. E,t Jinids (5.1) where n is the surface normal vector. Components of the force density at a point on the surface are  ij ni (5.2) These are the internal forces which are a response to strain. Though inconsequen tial in what follows, the minus sign which appears above propagates to one in the expression for J where it is likely to seem counterintuitive. Force density times displacement is just work density and thus the measure of energy transport per unit volume. Taking the force density as constant over infinitesimal displacements, the time rate of change of Fjuj is just Fjujt. The result is that power density is simply the scalar product of force density and velocitya result familiar from elementary mechanics. Applying this to equations (5.1) and (5.2), the expression for flux and Poynting vector components it infers are Et /(Jijnj')utdS ? Ji = ij uj,t (5.3) I shall be concerned only with invariance of directionality and symmetry prop erties and, since only normal modes are relevant, I will dispense with the time derivative and ignore the minus sign. For an isotropic material, dividing equation (2.17) by material constants c, and p expresses the proportionality of stress to strain independent of the material. ij ~ 36ij 1 u~r + 2uij (5.4) r where 3 = R2 2. From the discussions following equation (2.18), I can replace the invariant sum with V2( = k2p. rij is symmetric in its indices. Substituting this result into the representation for J in equations (5.3), I can express the pro portionality of the timeindependent part of Ji to a function of the undifferentiated scalar potential, total strain, and displacement. Ji ~ P3 k2 O Ui 2uj uij (5.5) uij separates into shear and longitudinal parts and the longitudinal part is immediately expressible in terms of the scalar potential using 4j)= ,ij (5.6) whereas applying definition (2.2) to u?) and replacing the displacements thereof by curls of the shear vector potential requires the more involved substitution Uij = 2(ia33,arj + EjcrH33,ci) (5.7) Correspondingly, components of the remaining displacement term can be expanded in terms of potentials by Uj = CjabHb,a + (~,j (5.8) With the foregoing substitutions, the ujuij on the right side of equation (5.5) expands to UjUij = (CejabCiapHb,aH/,ac + CjabCja/pHb,aH3,ai) +(CiabHb,aj + CjabHb,ai)L,j +CjabHb,a'P,ij + (,j(,ij (5.9) After a tedious amount of tensor algebra, the first term on the right of equation (5.9) reduces to 2H HI ,, + 2H[a,b]Ha,bi (5.10) The remaining terms do not simplify in useful v, at this level of expression. Collecting the foregoing results, equation (5.5) can now be fully expressed in terms of the scalar potential and components of the vector potential by Ji ~ (P/kf)(CiabHb,a + P,i) 4(H H ,, + H[a,b]Ha,b) (CiabHb,aj + EjabHb,ai)P,j 2CjabHb,aiP,ij 2o,j' ,ij (5.11) The actual physical direction of energy propagation should be invariant under exchange of the transverse coordinates and so, using equation (5.11), I show that in order for this to be true, the y and z components of H cannot be allowed to mix with the x component. Specifically, since the Poynting vector is composed, in part, of curls of H, any exchange of y and z will interact with the handedness of the coordinate system to require an appropriate antisymmetry. To be consistent with this, the wave number in the x direction must also be taken to change sign with any y + z interchange and so all derivatives with respect to x will also be required to change sign. In view of the translational invariance of the modes, this is simply a reflection of the need to change k, the common wave number in the x direction, to k to accommodate inversion of the x direction. The full expansion of the Poynting vector reveals a complex mixing of x versus y, z components of H, but I shall show that failure to separate solutions to avoid this mixing results in failures in the y + z symmetries of some of the terms and therefore implies that solutions must be built distinctly from cases in which H1 = 0 versus those in which, alternatively Hy = Hz = 0. It should also be noted that this will turn out to be consistent with an analysis of the k = 0 case to follow. In the k = 0 case all derivatives with respect to x vanish and the boundary conditions naturally take a form which reflects a decoupling of the x versus y, z components of H. The first additive term on the right hand side of equation (5.11) contributes a term to the x component of the Poynting vector that is proportionate to 6xabHb,a + ,x = 2H[z,y] + ,x (5.12) which is fully antisymmetric under y < z once we incorporate the rule that p,9  yp,x. Moreover, this antisymmetry is preserved even if either one of the terms vanishes. The second term on the right hand side of equation (5.11) contributes a term to the x component of the Poynting vector equal to 4 H [.,,(y](H.,y],. + H[yz],z) +H[x,z](H[x,,],x + H[zy],v) +2H[y,z]H[y,],x,} (5.13) and before sign changes due to differentiation by x, this expression is totally symmetric under y z. Expansion of the antisymmetric parts entails production of the pair ... Hy,H,wx... H,, H,zx ~... (5.14) and thus does not uniformally assemble odd numbers of x differentiations with symmetric yz terms and so the needed antisymmetry is not fully realized unless some terms vanish. If H, is set to zero, then it is easily checked that antisymmetry will be realized, to wit: [Hy,xHyx Hy,H[],z + H,,Hx H ,,H[,,y],y + 8H[y, ]H[y ,]] (5.15) Similarly, if H, does not vanish, but Hy and Hz vanish simultaneously, then, again, the resulting expansion will become antisymmetric under y < z once the xderivatives are considered, to wit: [H,,yH,,yx + Hx,zHx,zx] (5.16) The third term on the right of equation (5.11) contributes the following term to the x component of the Poynting vector: 2 [H[z,y],xu,l + H[z,y],xJ,y + H[z,y],~',z +H[z,y],x45,x + (H[x,z],\,x,y + H[1,y],15, z)] (5.17) Under y + z this is totally antisymmetric when the derivatives of x are consid ered. Moreover, if Hx vanishes or, alternatively Hy and Hz vanish together, the antisymmetry of the result is preserved. Finally, the last two terms on the right of equation (5.11) contribute the following terms to the x component of the Poynting vector: 4 [H ],,i + (H[x,,]p,xy H, .,xz)] 2 [p,xp,,x + (pP,yP,xy + P,zPz)] (5.18) It is easily checked that the desired antisymmetry is preserved and that, again, the vanishing either of Hx alone or Hy and Hz together does not change this result. The conclusion is that, when assembling a propagating normal mode, the shear contribution must be made out of components for which all the Hx's are zero, or for which all the Hy and Hz parts are zero. In considering how to represent the shear superpositions of a propagating normal mode, there is no case in which a superposition for H, will be mixed with ones for Hy and H,. This removes at least one degree of freedom from the problem. 5.3 Propagating Modes Involving Hy, H, Shear 5.3.1 Deriving the Frequency Equations The ease with which the RayleighLamb solution is derived inspires a deriva tion that follows the same pattern. Armed with the conclusion that H, cannot even be accommodated in a normal mode solution that also includes Hy and H, components, I proceed to derive the spectrum of propagating modes with H, = 0. Accordingly, from equations (2.22), the boundary conditions at z = h, with S z, p  x, p + y, and Hx = 0 become 3 k2j p + 2,p,, I,zy  [(Hy,zz Hy,zz) + Hz,zy] + (Hz,zz Hyy) (5.19) As a reminder, under the basis rules devised for this problem (see section 4.2), the representations of potentials will have the following forms: S= Px(k x) di Py(li y) P,(T* z) with T1* i 0 H = P,(k x) E, aP,( y)P(a z) with Ej bjP,(y7 y)P(aj+ z) 2/ _2 ck2 i 2 o( = w2/c2 2 _ 2 a+ S 2Hy,x Substituting into the first boundary condition of equations (5.19) I obtain di (Ok} + 2(l)2)P (riy) P,(Tl h,) 2k {+1 P,(Gj y) Pz (j h,) (5.20) 1 3 1 The iip!." transform devised in section 4.5 is then applied so that Py(rTiy) 6(IA\ ,y) and Py(7j y) ) 6(IA oj). By choosing a value Ao of the transform variable A such that Ao E {(r} i {nj}, the sums on both sides collapse leaving the following equality: do (Ok2 + 2(A )2) P(A h ) 2k { o oA Pz(A, h,) (5.21) 1 I1 This provides one constraint on possible combinations of (w, Ao) at a given value of k. Here and in subsequent steps, Ao = Ao . Repeating this process for the second boundary condition in equations (5.19), the transformed version of the second constraint becomes do k A: Pz(A h,) P P P,(Af h) ao(k2 (A 2) bo Ao(A) (5.22) 1 1 Py P_ Substituting into the third boundary condition in equations (5.19), the trans formed result is do AoA 1 P}(A h) k } bo A ao Ao P (A}+ h,) (5.23) 2 1 1 1 P. Pz Py Equations (5.22) and (5.23) can be reconciled into one constraint by finding a relationship between ao and bo that renders them equivalent. Dividing the two equations will eliminate the transcendental terms and some common factors, leav ing ao ({ k2 (+A)2) + bo A, A+ k2 P (5.24) bo A+ ao Ao Solving this for the required relationship between coefficients yields bo o A[2 (5.25) Substitution of equation (5.25) into either equation (5.22) or (5.23) to eliminate bo will produce the following result: ( 1 1 F(2 A2) A )2 do \A: P,(A h,) a, k ( 2 P (A+ h,) (5.26) 1 2 k2 2 A 0 SP. Having reduced the system to two equations with two unknown coefficients, those coefficients can be eliminated by dividing equation (5.26) into equation (5.21). First, however, inspired by a similar step in the RayleighLamb derivation, the parenthetical on the left side of equation (5.21) can be manipulated and found to have a convenient equivalent representation which coincides with a subexpression in equation (5.26). (/k + 2(A )2) ((k2 A)_ (A+)2) Substituting this and then performing the division followed by the usual rearrange ment produces the following frequency equation: P,(A* h) P,(A+ h,) 4 AA+ (k2 + A2) (5.27) P, (A: h) P (At h) [(k2 + A) (A]+)]2 Comparing with equation (4.20) this is seen to be a RayleighLamb equation with the quantity k2 + A 2 1 i ving the role of the magnitude of the propagation vector. This equation is parameterized by Ao and is valid, at a given k, only for certain combinations of Ao and u. Consider, nevertheless, what happens when k > Ao (or, effectively Ao  0 for finite k). A* = /2/c2k2 A k* = k2 = (in equation (4.21)) A, = /w2/c~k2A2 k =i k2 = a (in equation (4.21)) (5.28) Therefore, at large k, it is expected that equation (5.27) becomes more like the classic RayleighLamb equation (4.21) for plane waves in an infinite plate. At this intermediate point, the constraint of boundary conditions at the y = hy surface has not yet been imposed, but the implication nevertheless is that for large k the system looks substantially like a RayleighLamb system. This implies that as k > 0 the effect from the sides tend to decouple and the propagation looks increasingly like simple decoupled RayleighLamb planewave propagation. It now becomes important to focus on a simple observation. The foregoing step produced an equation in terms of the transform variable A whose permissible values (denoted Ao) come from the set of values in {qi} and {jrj} (what mathematicians would call the "support" for p and Hy, H,). There could be more elements of those support sets than possible values of Ao and these additional values may be uniquely connected to the transform variable p applied to boundary conditions at the .,.i i:ent surface. However, connecting the y = hy and z = hz surfaces will be possible only to the extent that some of the values which the two transform variables take on are indeed shared selections from {T1} and {crj}. From equations (2.22), the boundary conditions at y = hy with s  y, p  z, p  x, and H1 = 0 will be 3 k2 P + 29,yy = 2H ,y ,yx i [(Hzxx Hzvy) + Hy,H z Izy (Hy, Hz,zz) (5.29) The transformed first boundary condition from equations (5.29), before appli cation of the delta functions, will be ZdQ 2k + 2T/)P(/i ah,) 6( ) = 2k Ej Py (j ah,) (af+ /) (5.30) 1 3 1 To link the .,11i i:ent surface conditions, I rely upon the fact that each root of equation (5.27) necessarily corresponds to the existence of specific elements of the support sets {T1} and {oaj}. In fact, if Ao is, with some value of w, a root of equation (5.27), it is solely because 3Iro E {ql} and 3o0 E {aj} such that Ao = ro = ao. Now, the left sum in equation (5.30) a will necessarily encounter Ao = To. Suppose, then, that I contemplate the value of p corresponding to To = Ao for which the delta function on the left of equation (5.30) is nonzero. Obviously, it will be A*. If, for that value of p, the right hand side of equations (5.30) is not trivially zero for the same value of p, there must exist some ao such that a = = A . In the alternative, I could first contemplate a value of p for which the argument of the delta function on the right of equation (5.30) is zero. Obviously, it will be A+. If, for that value of p, the left hand side of equations (5.30) is not trivially zero for the same value of p, there must exist some fll such that Tl = p = A+. The upshot of this reasoning is that I can connect the transforms of the two sets of boundary conditions by contemplating simultaneous (w, Ao) roots of both of them. To write the transformed boundary conditions for the .,1i ,i:ent side in terms of Ao, I require either that p > A so that, by operation the delta function, the wave number variables take on values: rli Tlo Ao di do which will be eliminated aoj o i.e. presumed to exist ai al unknown, but to be eliminated J+ A S (A)+ (5.31) or I require that P> A+ so that, by operation the delta function, the wave number variables alternatively take on values: Jj g o = Ao bj bo which will be eliminated + cry  Ao Tli ll i.e. presumed to exist ai al unknown, but to be eliminated i (A)* (5.32) I will name the first of these alternatives (i.e., equations (5.31 )) "LConjugation" since it is premised on equating the longitudinal conjugation of the zsurface solution with the ysurface solution. The second alternative (i.e., equations (5.32)) "SConjugation" since it is premised on equating the shear conjugation of the zsurface solution with the ysurface solution. Because I have stipulated that all members of the support sets are positive and because only positive square roots are used, all of the preceding mappings are guaranteed to be unambiguous. The reader is invited to review the definitions of conjugation denoted with and + superscripts which were introduced in connec tion with equations (4.8) and (4.10). From those definitions, it can be noted that these conjugations have the property (A*) A (A+)+ A A detailed expansion of the final relation in equations (5.31) is (A)+ (A*)2 \A+2 A2 ()2 (5.33) and of the final relation in equations (5.32) is (A) ) 2 (A+)2 )2 + ( 2 (5.34) These expansions illustrate the general rule that LConjugation and SConju gation solutions are related by straightforward substitutions of variables. It will thus be sufficient to complete details of the ongoing derivation for the LConjugate case and then state the analogous results for the SConjugate case. Applying LConjugation to equation (5.30), and after collapsing the sums, the result is do (pk + 2A ) P,(Aoah,) = 2k b1 (A*)+ P1[(A)+ ahz] (5.35) Similarly, the second and third boundary conditions (5.29) under LConjugation can eventually be put into the following forms: 1 1 01k \ \\ }Py(Aoah ) iP,[(A/)+ ah] bi[k2 ((A )+)2]+ a1 A (A)+ (5.36) 1 1 2 ~L\/o \'\ do0AAo{ }{ Py(Aoahz) 1t 1 [y bl P Sk at (A *) b1 A:* P[(A*)+ ahz] (5.37) 1 1 1 Dividing equations (5.36) and (5.37) to put one al in terms of b1, I obtain 1 1 t (A)+Ao* a = bi )(5.38) k2 + 2 ht t 0iM^ It may be noted that equation (5.38) is not identical to (5.25). This highlights the fact that al and bl are expected to be distinct from ao and bo. If equation (5.38) is substituted into either equation (5.36) or equation (5.36) to eliminate al, the result is identical, to wit: 1} 1 t (k2 + (A*)2) ((A*)+)2 do Ao Py(Aoaho ) b k k2 Pm[(A) ah (5.39) 12 k 2 + (A* )20 P. In what is by now a ritual, the parenthetical on the left of equation (5.35) can be manipulated to show its relation to the numerator in the fraction on the right hand side of (5.39). (p3k + 2A ) = [(k 2 + (A)2) ((A )2] (5.40) Having reduced the unknown coefficients to only do and bl, equation (5.39) can be divided into equation (5.35) to eliminate them and produce the frequency equation associated with the y = hy side. Py(Ao ah,) P,[(A:)+ ah,] 4 A, (A)+ ( (k2 + (A)2) (5.4t) Py(Ao ah,) Py[(A\)+ ah,] [(k2 + (A)2) ((+A)22 (2 Once again, this is a RayleighLamb equation. Here, the quantity (k2 + (A2)2) 1p i'l the role of the square of the magnitude of propagation. Equation (5.41) is manifested in the classic RayleighLamb structure which shows the consistency of the result with preceding derivations. However, in this case, that consistent manifestation masks an interesting feature. Namely, all k dependency in equation (5.41) cancels internally. To see this, expand the terms which ostensibly appear to show a k dependency and observe that k is eliminated. S 2 [2 2 k2 A] ( A ( 2 2 c2 2 C2 R 2 (k + ) k2 + k2 a A2 Equations (5.27) and (5.41) define (w, Ao) root systems for independent equa tions. The coincidences of these simultaneous constraints define values of w that constitute the eigenspectrum with respect to the LConjugation case. There is a set of common roots for each distinct value of k. Plotting the k versus w dispersions requires methodical plotting at different values of k, but, in principle, the spectrum (for LConjugation) has been analytically and precisely specified. The entire derivation is also invariant under y z exchangethough this might not be immediately apparent. The key to realizing that it must be so is to realize that relabeling the directions must also be applied to the representations of the potentials as superpositions and that with directions relabeled the aspect ratio is defined by h, = ahy. The reader can easily verify that the results of derivation will be equivalent to the foregoing. Repeating the same sequence of steps from equation (5.30) to the present point, using SConjugation defined by equations (5.32), the SConjugate analog to equation (5.41) is P,((A )* ah ) P [A, ah 4 Ao (A,+)* (k2 + (A+)2) P,((A)* ah,) Py[Ao ah] [(k2 + ()2) ]2 5.3.2 Interpretation Had anyone been insightful enough to anticipate that coupled modes of a rectangular waveguide could be characterized by a coincidence of RayleighLamb solutions, proving that it was so would have remained as elusive as history shows the main problem to have been. Moreover, there are aspects of the result which, had it been somehow forseen as possible, would have argued against believing it. The main impediment would have been that there is an intrinsic interference built into the result which precludes boundary satisfaction at both surfaces without the rest of the superposition. Although the elegantlooking result involves a coincidence with the full superpositionit is still not the full superposition. What is intriguing is that I do not need to have a full description of the superposition in order to find the eigenspectrum. The derivation tells us that any superposition of shear and longitudinal compo nents that satisfy boundary conditions at .,.i i,:ent sides must include a particular combination of components that, in a partial sense, mimic a RayleighLamb wave system. The LConjugation and SConjugation cases are merely two different vi, of realizing this. Figures 5.1 and 5.2 illustrate the essence of these alternatives. They imply two recipes for building the superpositions. 88 z=h r7 =r 17 long shear ..+ II r + C b II S on,= (7*)+ shear LConj II Figure 5.1: Illustrating the LConjugation Case for Modal Solutions. Shown are the relationships among longitudinal and shear wave vector components of the defining physical waves which must be among those making up the total superposition needed for a solution. The common x directional component (k) is normal to the page. The recipe implied by LConjugation begins with a longitudinal wave at a desired k value. Conceptually, one could imagine starting out with some r] and some u close to a modal solution. Add a shear wave with the same fixed k. The polarization of this initial shear contribution is such that shear displacement is not parallel to the sides. Now adjust u until the RayleighLamb equation is satisfied with respect to the z = h, sides. There will be a range of o's for which RayleighLamb can be satisfied at these parallel sides. For each possible U the bulk dispersion relations will fix if* and a+ wave vector components pointing along the z directions. Now add a second ("conj i, i. ) shear wave at the same k and close 89 z= h cr I rj = cr long shear * b II + b II 7conj= ("+)* S SConj long b II 11F 0 Figure 5.2: Illustrating the SConjugation Case for Modal Solutions. Shown are the relationships among longitudinal and shear wave vector components of the defining physical waves which must be among those making up the total superposition needed for a solution. The common x directional component (k) is normal to the page. to the w implied by the process so far. The polarization of this shear wave should also result in a displacement not parallel to the sides. The initial a+cj is set equal to *. Just as the first shear wave had a wave vector component common with the foundation longitudinal wave along the y direction, this conjugate shear wave has a wave vector component common with the foundation longitudinal wave along the z direction. Bulk dispersion will fix the value of ocomj. Now adjust w over the range of values that continue to satisfy RayleighLamb at the z = h, surfaces until one is found for which RayleighLamb is also satisfied for the foundation longitudinal wave and the justadded conjugate shear wave at the y = hy surfaces. When such an u is found, it defines an eigenfrequency of a propagating system at the set value of k. There will be as irn Ii such o/s as there are subbands. The recipe implied by SConjugation is procedurally the same as the recipe for LConjugation except that the foundation wave is a shear wave instead of a longitudinal one and two longitudinal ones are added instead of two shear waves. The recipes each focus on combining waves in total disregard of their reflections at .Ii] ,' ent surfaces. Satisfaction of RayleighLamb incorporates reflections only at parallel surfaces. The reflections from .ili ,i'ent surfaces being ignored constitute the rest of the superpositions. What the derivation shows is that to find the eigenspectrum, one can ignore these reflections. To build a complete description of the wave function, however, these reflections must be included. Incorporating the ignored reflections is not trivial. There are two principle complications. The first is that shear and longitudinal waves scatter into each other upon reflection and the scattering amplitude ratios are non trivial transcendental relations even when the surface environment is free of other interactionswhich they are not. Secondly, from plotting the roots of the equation (5.27) paired alternately with equation (5.41) or equation (5.42) there appear to be cases where, for a given eigenfrequency, there are multiple root coincidences and it is not clear whether each is an independent foundation from which the reflections can be taken as El i i, i lii. or whether the multiple root coincidences are, in some sense, resonances of each other and building reflections from only one of them is sufficient. Since the amplitude of reflection into either shear or longitudinal is bounded by unity, successive reflections must progressively dissipate in amplitude and the implication is that each foundation set constitutes a firstorder characterization of the entire wave function. In spite of the practical difficulties involved in building a complete description of the wave function that fully describes the superposition, the implication of the derivation remains a strong one. Namely, no matter how complicated the details of the wave function are, the dispersion is precisely defined by the behavior of only one overlapping set of components which must exist as a dominant part of the full superposition. 5.3.3 Mode Dispersions The most important physical feature which can now be exhibited are the mode bands and particularly the dispersions. As the terms LConjugation and SConjugation will now appear more frequently and together, the contractions "LConj" and "SConj" will begin to be used routinely. Equation (5.27) paired alternately with equation (5.41) or equation (5.42) defines simultaneous transcendental relationships which determine the propagat ing normal modes of an elastic isotropic rectangular waveguide. Extracting the actual subband dispersions is accomplished by substituting successive K values into a dimensionless form of equation (5.27), contour plotting its root system, and superimposing that root system on top of the root system plotted for equation (5.41) or (5.42). It is of some help that the latter equations are Kindependent and need only be plotted once in dimensionless form. The subbands resulting from LConjugation (viz. equations (5.27) and (5.41) combined) have a strong similarity to the standard RayleighLamb bands. This can be seen by methodically plotting the lowest subbands over a range of K values and superimposing them on the lowest subbands of the standard RayleighLamb solution. This is shown for dilatational modes in Fig. 5.3 where the dispersion for the first three subbands of these propagating waveguide modes are shown against 3.  4. PY = E R2 = 3 PZ = E 0 0 1 2 3 4 K [hkx] Figure 5.3: Plotted KDispersion for Propagating Dilatational Modes L Conjugate Case. Heavy lines correspond to waveguide modes (solid lines are square cross section, dashed lines are 1x2 cross section) and thin reference lines are RayleighLamb infinite plate "symmetric" solutions. the background of the first three RayleighLamb branches. Waveguide modes for the square cross section (solid lines) are shown together with modes corresponding to a 1:2 crosssectional aspect ratio (dashed lines). (LConj and SConj dilatational modes in combination are compared with numerical mode computations in Fig. 6.4.) The dilatational subbands resulting from SConjugation are not intrinsically similar to infinite plate modes of the same parity pattern. Figure 5.4 shows the dispersion of the lower SConj subbands for the dilatational family against the background of the first three RayleighLamb branches. Waveguide modes for the square cross section (solid lines) are shown together with modes corresponding to a 1:2 crosssectional aspect ratio (dashed lines). 4. '44 P =E 3. 0 0 1 2 3 4 K [hkx] Figure 5.4: Plotted KDispersion for Propagating Dilatational Modes S Conjugate Case. Heavy lines correspond to waveguide modes (solid lines are square cross section, dashed lines are 1x2 cross section) and thin reference lines are RayleighLamb infinite plate "symmetric" solutions. For combined LConj and SConj dilatational modes compared to numerical results, please see Fig. 6.4. Figure 5.5 shows the dispersion for LConjugated flexural solutions against a background of RayleighLamb flexural modes. The dashed lines correspond to a 1:2 crosssectional aspect ratio where the flexing motion of the fundamental modes is of the same plane (y x) in both the RayleighLamb infinite plate and rectangular waveguide scenarios. Although no torsion of an infinite plate is possible, the displacement pattern of loworder torsion modes (viz. dominantly parallel to the surfaces) is analogous to the uncoupled SH modes (viz. also parallel to the surfaces, but vanishing there) of the infinite plate derived at the end of section 4.4. Figure 5.6 shows torsional modes of a square waveguide against a background of infinite plate SH modes. 