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## Material Information- Title:
- Wave transformations by two-dimensional bathymetric anomalies with sloped transitions
- Series Title:
- Wave transformations by two-dimensional bathymetric anomalies with sloped transitions
- Creator:
- Bender, Christopher J.
- Place of Publication:
- Gainesville, Fla.
- Publisher:
- Coastal & Oceanographic Engineering Dept. of Civil & Coastal Engineering, University of Florida
- Language:
- English
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- University of Florida
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- University of Florida
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UFL/COEL-2002/011
WAVE TRANSFORMATION BY TWO-DIMENSIONAL BATHYMETRIC ANOMALIES WITH SLOPED TRANSITIONS by Christopher J. Bender and Robert G. Dean Partial funding by: Bureau of Beaches and Wetland Resources Florida Department of Environmental Protection Tallahassee, Florida 2002 Wave Transformation by Two-Dimensional Bathymetric Anomalies with Sloped Transitions by Christopher J. Bender and Robert G. Dean Partial funding by: Bureau of Beaches and Wetland Resources Florida Department of Environmental Protection Tallahassee, Florida Executive Summary The reflection and transmission of normally incident waves by two-dimensional trenches and shoals of finite width with sloped transitions between the depth changes are studied. Three methods are developed using linearized potential theory. The step method is valid in arbitrary water depth and is an extension of the solution of Kirby and Dalrymple (1983) for asymmetric trenches that allows for sloped transitions to be approximated by a series of steps. The slope method is an extension of Dean (1964) that allows trenches and shoals with a linear transition between changes in depth to be modeled in the shallow water limit. A numerical method is also developed using a backward space stepping routine commencing from the downwave side of the trench or shoal to model the wave field for an arbitrary bathymetry in the shallow water limit. The reflection and transmission coefficients are compared for both symmetric and asymmetric trenches and shoals with abrupt transitions and sloped transitions. The sloped transitions cause a reduction in the reflection coefficient as compared to trenches with abrupt transitions of the same cross-sectional area; a reduction that increases as the waves progress from long waves to shorter period waves. For symmetric bathymetric anomalies complete transmission is found for certain dimensionless wavelengths, a result not found for asymmetric trenches. The wave transformation by domains with Gaussian forms is also investigated with results indicating minimal reflection for waves beyond the shallow water limit. Comparison between the three methods indicates good agreement in the shallow water limit for the cases studied. Several new results were established during the study. The wave field modification is shown to be independent of the incident wave direction for asymmetric changes in depth, a result shown by Kreisel (1949) for a single step. For asymmetrical bathymetric anomalies with the same depth upwave and downwave of the anomaly a zero reflection coefficient occurs only at k1hm = 0. For asymmetrical bathymetric anomalies with unequal depths upwave and downwave of the anomaly the only k1h1 value at which the reflection coefficient equals zero is that approached asymptotically at deep-water conditions. Table of Contents Executive Sum m ary ......................................................................................................... i List of Figures ................................................................................................................ iii 1. Introduction ................................................................................................................ 1 2. Step M ethod: Form ulation and Solution ................................................................... 3 2. 1. Abrupt Transition ................................................................................................ 6 2.2. Gradual Transition ................................................................................................ 8 3. Slope M ethod: Form ulation and Solution ............................................................... 10 3. 1. Single Transition ................................................................................................ 11 3.2. Trench or Shoal .................................................................................................. 13 4. N um erical M ethod: Form ulation and Solution ....................................................... 15 5. Results and Com parisons ......................................................................................... 17 5. 1. Arbitrary W ater D epth ....................................................................................... 17 5. 1. 1. Com parison to Previous Results ................................................................. 18 5.1.2. Sym m etric Trenches and Shoals ................................................................. 19 5.1.3. A sym m etric Trenches ................................................................................. 28 5.2. Long W aves ........................................................................................................ 32 5.2. 1. Sym m etric Trenches and Shoals ................................................................. 33 5.2.2. A sym m etric Trenches ................................................................................. 37 6. Sum m ary and Conclusions ....................................................................................... 39 7. Acknowledgem ents .................................................................................................. 40 8. References ................................................................................................................. 40 List of Figures Figure 1: Definition sketch for trench with vertical transitions ......................................... 6 Figure 2: Definition sketch for trench with stepped transitions ......................................... 8 Figure 3: Definition sketch for linear transition ................................................................ 12 Figure 4: Definition sketch for trench with sloped transitions .......................................... 14 Figure 5: Comparison of reflection coefficients from step method and Kirby and Dalrymple (1983 Table 1) for symmetric trench with abrupt transitions and normal wave incidence: h3 = hi, h2/hl = 3, W/hI = 10 ............................... 18 Figure 6: Comparison of transmission coefficients from step method and Kirby and Dalrymple (1983 Table 1.) for symmetric trench with abrupt transition and normal wave incidence: h3 = hl, h2/h, = 3, W/h, = 10 ...................................... 19 Figure 7: Setup for symmetric trenches with same depth and different bottom widths and transition slopes ........................................................................................ 20 Figure 8: Reflection coefficients versus k1h, for trenches with same depth and different bottom widths and transition slopes. Only one-half of the symmetric trench cross-section is shown ........................................................ 21 Figure 9: Transmission coefficients versus k1h, for trenches with same depth and different bottom widths and transition slopes. Only one-half of the symmetric trench cross-section is shown ........................................................ 21 Figure 10: Reflection coefficient versus the number of evanescent modes used for trenches with same depth and transition slopes of 5000, 1, and 0. 1 ................. 22 Figure 11: Reflection coefficient versus the number of steps for trenches with same depth and transition slopes of 5000, 1, and 0. 1 ............................................... 23 Figure 12: Reflection coefficients versus k1h, for trenches with same bottom width and different depths and transition slopes. Only one-half of the symmetric trench cross-section is shown .......................................................................... 24 Figure 13: Reflection coefficients versus k1h, for trenches with same top width and different depths and transition slopes. Only one-half of the symmetric trench cross-section is shown .......................................................................... 25 Figure 14: Reflection coefficients versus k1h, for trenches with same depth and bottom width and different transition slopes. Only one-half of the symmetric trench cross-section is shown.......................................... 26 Figure 15: Reflection coefficients versus k1h, for shoals with same depth and different top widths and transition slopes. Only one-half of the symmetric shoal cross-section is shown........................................................ 26 Figure 16: Reflection coefficients versus k1h, for Gaussian trench with C1 = 2 m and C2 = 12 mn and h,, = 2 m. Only one-half of the symmetric trench crosssection is shown ..................................................................... 27 Figure 17: Reflection coefficients versus k1h, for Gaussian shoal with C, = 1. m and C2 = 8 m and h. = 2 in. Only one-half of the symmetric shoal cross-section is shown.............................................................................. 28 Figure 18: Reflection coefficients versus k1h1 for a symmetric abrupt transition trench, an asymmetric trench with different s, and S2 values and a mirror image of the asymmetric trench.................................................... 29 Figure 19: Reflection coefficients versus k1h1 for an asymmetric abrupt transition trench, an asymmetric trench with different h, and h5 values and s, equal to S2 and a mirror image of the asymmetric trench with S1 = S2 ................... 30 Figure 20: Reflection coefficients versus k1h, for an asymmetric abrupt transition trench, an asymmetric trench with different hi and h5 values and s, not equal to S2 and a mirror image of the asymmetric trench with s, = S2............ 31 Figure 21: Reflection coefficient versus the space step, dx, for trenches with same depth and different bottom width and transition slopes. Only one-half of the symmetric trench cross-section is shown...................................... 32 Figure 22: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with transition slope equal to 5000. Only one-half of the symmetric trench cross-section is shown.......................................... 33 Figure 23: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with transition slope equal to 1. Only one-half of the symmetric trench cross-section is shown.......................................... 34 Figure 24: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with transition slope equal to 0.1. Only one-half of the symmetric trench cross-section is shown.......................................... 35 Figure 25: Conservation of energy parameter versus k3h3 for three solution methods for same depth trench case with transition slope equal to 1. Only one-half of the symmetric trench cross-section is shown................................... 35 Figure 26: Reflection coefficients versus k1h1 for step and numerical methods for Gaussian shoal (h,, = 2 m, C1 = 1 mn, C2 = 8 in). Only one-half of the symmetric shoal cross-section is shown........................................... 36 Figure 27: Reflection coefficients versus k3h3 for step and numerical methods for Gaussian trench in shallow water (h. = 0.25 mn, C1 = 0.2 m, C2 = 3 in). Only one-half of the symmetric trench cross-section is shown ................ 37 Figure 28: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with symmetric abrupt transition trench and asymmetric trench with unequal transition slopes equal to 1 and 0. 1........................ 38 Figure 29: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with asymmetric abrupt transition trench and asymmetric trench with unequal transition slopes equal to I and 0.2.......... 39 1. Introduction There is a long history of the application of analytic methods to investigate wave field modifications by bathymetric changes. Early efforts centered on the effects on normally incident long waves of an infinite step, trench, or sill of uniform depth in an otherwise uniform depth domain. The fluid domain in many of the solutions was divided into two regions: one region comprising the fluid within the change in bathymetry and the other region consisting of the remainder of the fluid domain. This technique has been employed in many studies with a progression from normally incident long waves over a step to solutions that account for oblique incidence and currents along a trench. Lamb (1932) was one of the first to develop a long wave approximation for the reflection and transmission of a normally incident wave at a finite step using the matching technique. Bartholomeauz (1958) performed a more thorough analysis of the finite depth step problem and found that the Lamb solution gave correct results for the reflection and transmission coefficients for lowest order (kh) where k is the wave number and h is the water depth upwave of the step. To solve the case of normal wave incidence and arbitrary relative depth over a sill or a fixed obstacle at the surface, Takano (1960) employed an eigenfunction expansion of the velocity potentials in each constant depth region and matched them at the region boundaries. The set of linear integral equations was solved for a truncated series. Dean (1964) investigated long wave modification by linear transitions in channel depth and/or width. A solution was defined with plane-waves of unknown amplitude and phase for the incident and reflected waves with the transmitted wave specified. In the region of linear variation in depth and/or width, both transmitted and reflected wave forms, were represented by Bessel functions. The solutions were shown to converge to those of Lamb (1932) for the case of an abrupt transition. Newman (1965a) studied wave transformation due to normally incident waves on a single step between regions of finite and infinite water depth with an integral-equation approach. Newman (1965b) examined the propagation of water waves past long obstacles by constructing a domain with two steps placed "back to back" and applying the solutions of Newman (1965a). Complete transmission was found for certain water depth and pit length combinations. The variational approach was used by Mei and Black (1969) to determine the scattering of surface waves by submerged and floating bodies. Black et al. (197 1) applied the variational approach to determine the scattering caused by oscillating or fixed bodies that may be submerged or floating. In Black et al. (197 1) both infinitely long horizontal cylinders of rectangular cross section and vertical cylinders of circular crosssection were studied; rendering the problem two- and three-dimensional, respectively. Lassiter (1972) employed the variational approach to study waves normally incident on a trench where the depths before and after the trench may be different (the asymmetric case). The symmetric 2-D trench problem was studied by Lee and Ayer (1981), who utilized a transform method. In this study the fluid domain is divided into two regions with one uniform depth domain of infinite length overlying a finite rectangular region representing the trench component below the uniform seabed level. The problem was solved by matching the normal derivative of the potential function in each region along the common boundary. Numerical results were compared to those from a boundary integral method with good agreement and to results from a laboratory experiment, which support the general trends seen in the numerical results. Using a boundary integral method, Lee et al. (198 1) studied the interaction of water waves with trenches of irregular shape. The method was compared with good agreement to results from Lee and Ayer (1981) for a rectangular trench. A case with "irregular" bathymetry was demonstrated in a plot of the transmission coefficient for a trapezoidal trench; however the complete dimensions of the trench were not specified. Miles (1982) solved for the diffraction by a 2-D trench for obliquely incident long waves using a procedure developed by Kreisel (1949) that conformally mapped a domain containing certain obstacles of finite dimensions into a rectangular strip. To add the capability of solving for obliquely incident waves, Miles used the variational formulation of Mei and Black (1969). The problem of obliquely incident waves over an asymmetric trench was solved by Kirby and Dalrymple (1983) using a modified form of Takano's method (1960). Differences in the results of Kirby and Dalrymple and those of Lassiter (1972) were found in a direct comparison with a boundary integral method solution verifying the results of Kirby and Dalrymple. Lee and Ayer (198 1, see their Figure 2) also demonstrated differences in their results and those of Lassiter (1972). The Kirby and Dalrymple (1983) solution was later extended to include the effects of currents flowing along the trench in Kirby et al. (1987). The previous studies have all investigated the interaction of water waves with changes in bathymetry. With the exception of Dean (1964) and Lee et al (1980) the study domains have featured an abrupt transition. A more realistic representation of natural trenches and shoals should allow for gradual transitions (sloped sidewalls). The focus of the present study is the propagation of water waves over a 2-D trench or shoal of more realistic geometry. This will extend the study of Dean (1964) that investigated wave modification by a sloped step and Lee et al. (1980), which did not directly address the effect of the transition slope on the reflection and transmission coefficients. Three solution methods are developed for linear water waves: (1) the step method, (2) the slope method and (3) a numerical method. The step method is valid in arbitrary water depth while the slope method and the numerical method are valid only for shallow water conditions. The step method is an extension of the Takano (1960) solution as modified by Kirby and Dalrymple (1983) that allows for a trench or shoal with "stepped" transitions that approximate a specific slope or shape. The slope method is an extension of the Dean (1964) solution that allows for linear transitions between the changes in bathymetry for a trench or shoal creating regular or irregular trapezoids. The numerical method employs a backward space-stepping procedure for arbitrary (but shallow water) bathymetry with the transmitted wave specified. 2. Step Method: Formulation and Solution The two-dimensional motion of monochromatic, small-amplitude water waves in an inviscid and irrotational fluid of arbitrary depth is investigated. The waves are normally incident and propagate in an infinitely long channel containing a twodimensional obstacle (trench or shoal) of finite width. Details of the fluid domain and the formulation of the solution vary depending on the case studied: abrupt transition or gradual transition with the slope approximated by the step method. The step method is an extension of the Takano (1960) formulation for the propagation of waves over a rectangular sill. The method of Takano (1960) was extended in Kirby and Dalrymple (1983) to allow for oblique wave incidence and again in 1987 by Kirby et al. to include the effects of currents along the trench. In the present formulation the method of Takano, as formulated in Kirby and Dalrymple (1983), was generally followed for normal wave incidence. The solution starts with the definition of a velocity potential: Oj(x, z t) = (x, z)e' (j=1 -+ J) (2.1) where j indicates the region, J is the total number of regions (3 for the case of a trench or sill with an abrupt transition), and a is the angular frequency. The velocity potential must satisfy the Laplace Equation: Sa2 + T_ (z0 (2.2) ~2 z)' Z the free-surface boundary condition: aob(x,z) + 2 0=0 (2.3) az 9 23 & g and the condition of no flow normal to any solid boundary: a0f(x,z) = 0 (2.4) an The velocity potential must also satisfy radiation conditions at large Ixi. The boundary value problem defined by Eq. (2.2), the boundary conditions of Eqs. (2.3) and (2.4), and the radiation condition can be solved with a solution in each region of the form: j (x, z) = A: cosh[kj(hj + z)]eik(x) + BJ, cos[K.,(h + z)]e+I'n(xxi) n=1 (j =1-J), (n=1 --) (2.5) In the previous equation A1 is the incident wave amplitude coefficient, A1- is the reflected wave amplitude coefficient and Aj+ is the transmitted wave amplitude coefficient. The coefficient B is an amplitude function for the evanescent modes, (n = 1 oo), at the boundaries, which are standing waves that exponentially decay with distance from the boundary. The values of the wave number for the propagating modes, kj, are determined from the dispersion relation: o-2 =gk tanh(kjhj) (j= J) (2.6) and the wave numbers for the evanescent modes, iKjn are found from: a2 = -gKj, tan(K.,nhj) (j =1 -> J), (n = 1 oo) (2.7) In each region a complete set of orthogonal equations over the depth is formed by Eqs. (2.5) to (2.7). To gain the full solution, matching conditions are applied at each boundary between adjacent regions. The matching conditions ensure continuity of pressure: Oi = j+I (x = xj, (j = 1 --> J -l) (2.8) and continuity of horizontal velocity normal to the vertical boundaries 10i, j( = xj),M j = 1 --> J 1) (2.9) ax ax The matching conditions are applied over the vertical plane between the two regions: (-h < z<0) if hi In order to form a solution, one wave form in the domain must be specified, usually the incident or the transmitted wave. Knowing the value of the incident, reflected and transmitted wave amplitudes, the reflection and transmission coefficients can be calculated from: KR =aR (2.10) a, KT= aT cosh(kjh.) (2.11) a, cosh(k k) where the cosh terms account for the change in depth at the upwave and downwave ends of the trench/shoal for the asymmetric case. A convenient check of the solution is to apply conservation of energy considerations: K __2_, n+ k1 = 1 (2.12) K a, k nlk, where nj is the ratio of the group velocity to the wave celerity, n ={1- sinh2khj) (2.13) 2.1. Abrupt Transition The solution of Takano (1960) for an elevated sill and that of Kirby and Dalrymple (1983) for a trench are valid for abrupt transitions (vertical walls) between the regions of different depth. For these cases the domain is divided into three regions (J = 3) and the matching conditions are applied over the two boundaries between the regions. The definition sketch for the case of a trench with vertical transitions is shown in Figure 1 where W is the width of the trench. z T 1I o- 1 RTI o X XI X2 IL t~ i Region 1 Region 2 Region 3 Figure 1: Definition sketch for trench with vertical transitions. Takano constructed a solution to the elevated sill problem by applying the matching conditions [Eqs. (2.8) and (2.9)] for a truncated series (n = 1 -> N) of eigenfunction expansions of the form in Eq. 2.5. Applying the matching conditions results in a truncated set of independent integral equations each of which is multiplied by the appropriate eigenfunction; cosh[kj(hj+z)] or cos[cj,n(hj+z)]. The proper eigenfunction to use depends on whether the boundary results in a "step down" or a "step up"; thereby making the form of the solution for an elevated sill different than that of a trench. With one wave form specified, the orthogonal properties of the eigenfunctions result in 4N+4 unknown coefficients and a closed problem. By applying the matching conditions at the boundary between Regions 1 and 2 (x = xj), 2N+2 integral equations are constructed. For the case of a trench with vertical transitions (Figure 1) the resulting equations are of the form: 0 0 f 1(x1,z)cosh[k (hi + z)]dz 2 (x,,z)cosh[k (h + z)]dz (2.14) -2, -14 0 0 f (xi, z) cos[@,,, (h, + z)z = 2 (x, z) cos[K1, (h, + z)lz (n = 1 -N) (2.15) -hi -hi So(x, z)cosh[k(h2 + z)]dz = (x,z)cosh[k2(h2 + z)]dz = (, ,)cosh[k2(h2 +z)d z x fx -hl l ax(2.16) -(x,,z)cos[2"(h2 + z)] = z a (x, z)cos[k2 ,(h2 + Z)VZ -hi ax (n= 1 ->N) (2.17) = (X,,Z)cos[K2, (h2+z))iz ax The limits of integration for the right hand side in Eqs. (2.16) and (2.17) are shifted from -hi to -h2 as there is no contribution to the horizontal velocity for (- h2 < z < -h) at x = x, and (-h2 z < -h3) at x = x2, for this case. In Eqs. (2.14) and (2.15) the limits of integration for the pressure considerations are (-h < z <0) at x = x, and (- h3 5 z <0) at X = X2. At the boundary between Regions 2 and 3 the remaining 2N+2 equations are developed. For the case of a trench the downwave boundary is a "step up", which requires different eigenfunctions to be used and changes the limits of integrations from the case of the "step down" at the upwave boundary [Eqs. (2.14) to (2.17)]. o 0 f 02(x2, z) cosh[k3 (h3 + z)]dz = 03 (x, z) cosh[k3 (h3 + z)]dz (2.18) -h3 -h3 0 0 f 02(x2,)cos[K3,,(h3+z)]dz= fq3(x2,Z)cos[CK3,,(h3+z) z (n = 1- N) (2.19) -h23 -h23 J 2~2(x,z)cosh[k2(h2 + z)]dz f 2~(2 ~ohk(2 +z]z(2.20) -h2 ax -h3 00 X1Z o[C, h Iz 00 3x,~o12nh ~z( N (2.21) -k -h3 At each boundary the appropriate evanescent mode contributions from the other boundary must be included in the matching conditions. The resulting set of simultaneous equations may be solved as a linear matrix equation. The value of N (number of nonpropagating modes) must be large enough to ensure convergence of the solution. Kirby and Dalrymple (1983) found that N = 16 provided adequate convergence for most values of kjh1. 2.2. Gradual Transition The step method is an extension of the work by Takano (1960) and Kirby and Dalrymple (1983) that allows for a domain with a trench or sill with gradual transitions (sloped sidewalls) between regions. Instead of having a "step down" and then "step up" as in the Kirby and Dalrymple solution for a trench, or the reverse for Takano's solution for an elevated sill, in the step method a series of steps either up or down are connected by a constant depth region followed by a series of steps in the other direction. A sketch of a domain with a stepped trench is shown in Figure 2. In this method, as in the case of a trench or a sill, a domain with J regions will contain J-1 steps and boundaries. z T1 11 TI --! .X1 X2 X3 X4 XS h1 h 3 5 h6 Lw .1 II I SI I II I I Region I (R1)I R2 R3I Region 4 jRS RI Figure 2: Definition sketch for trench with stepped transitions. Each region will have a specified depth and each boundary between regions will have a specified x location where the matching conditions must be applied. At each boundary the matching conditions are applied and depend on whether the boundary is a "step up" or a "step down." With the incident wave specified, a set of equations with 2(J-1)N+2(J-1) unknown coefficients is formed. The resulting integral equations are of the form: for (j = 1 -> J 1) if (- h, > -h+l ) at x = xj then the boundary is a "step down"; 0 0 f O (xi, z) cosh [kj (hj + z z= fj+1 (xj, z) cosh [k (hj + z)}z (2.22) -h -h1 o 0 f Oi(xj,z)cosr[,C,,(hi + z) z fO #+(x,,z)cos [my.,,(h z)z (n=1-+N) (2.23) -h -hj J-0-'(xy,z)cosh[kj+, (h +z z = f -- (xJ,z)cosh[kj+,(hj+1 +zz (2.24) -hiax -hj+l J (x, z)cos[y.1,, n(h+ +z z)}z= (x~)(XJZcos[J+,(h+, z+z)z -hi -hJ+I (n = 1 -> N) (2.25) if (- h, <-hi+,) at x = xj then the boundary is a "step up"; 0 0 j(xjz)cosh[kj+(hj+I+z z= ++(Xi,z)cosh[kj+(hj++ z (2.26) -hj+1 -h]+1 o 0 Si (x, z)cos [yK+I,,(h+, + z= J A+,(xy,z)cos [Kj+l,,(hy+l + z)}z (n =1 -> N)(2.27) -hJ+I -hj+1 (x, z)coshak(h;+zJz = (x ,z)cosh [kj(hj + z)z (2.28) aoi jzcoh~ +ax a x z -hI -hJ+ 0 0(\1 S (x,,z)cos ,, (h +z)z= L (xj z)cos [,, (h + z)z (n=1-+ N)(2.29) -h -hj+l At each boundary (xj) the appropriate evanescent mode contributions from the adjacent boundaries (x.j-1, xj+i) must be included in the matching conditions. The resulting set of simultaneous equations is solved as a linear matrix equation with the value of N large enough to ensure convergence of the solution. 3. Slope Method: Formulation and Solution The slope method is an extension of the analytic solution by Dean (1964) for long wave modification by linear transitions. Linear transitions in the channel width, depth, and both width and depth were studied. The solution of Dean (1964) is valid for one linear transition in depth and/or width, which in the case of a change in depth allowed for an infinite step, either up or down, to be studied. In the slope method a domain with two linear transitions allows the study of obstacles of finite width with sloped transitions. The long wave formulation of Dean (1964) for a linear transition in depth was followed. By combining the equations of continuity and motion the governing equation of the water surface for long wave motion in a channel of variable cross-section can be developed. The continuity equation is a conservation of mass statement requiring that the net influx of fluid into a region during a time, At, must be equal to a related rise in the water surface, il. For a channel of uniform width, b, this can be expressed as: [Q(x) Q(x + Ax)]At = bAx[)7(t + At)- )(t)] (3.1) where Q(x) and Q(x+Ax) are the volume rates of flow into and out of the control volume, respectively. The volume flow rate for the uniform channel can be expressed as the product of the cross sectional width, A, and the horizontal velocity, u, in the channel: Q=Au (3.2) By substituting Eq. (3.2) into Eq. (3.1) and expanding the appropriate terms in their Taylor series while neglecting higher order terms, Eq (3.1) can be rewritten as: - a--(Au) =barL (3.3) ax at The hydrostatic pressure equation is combined with the linearized form of Euler's equation of motion to develop the equation of motion for small amplitude, long waves. The pressure field, p(x,y,t), for the hydrostatic conditions under long waves is: p(x,z,t) = ,g [q7(x,t) z] (3.4) Euler's equation of motion in the x direction for no body forces and linearized motion is: lap = aU (3.5) p ax at The equation of motion for small amplitude, long waves follows from combining Eqs. (3.4) and (3.5): -g -- (3.6) ax at The governing equation is developed by differentiating the continuity equation [Eq. (3.3)] with respect to t: a [ la alAl al a3.7 &[-.x(Au)=b yt --L a (3.7) and inserting the equation of motion [Eq. (3.6)] into the resulting equation, Eq. (3.7) yields the result: - Al l = b a2r1 (3.8) ax L ax at2 Eq. (3.8) is valid for any small amplitude, long wave form and expresses rj as a function of distance and time. Eq. (3.8) can be further simplified under the assumption of simple harmonic motion: q(x, t) = q, (x)e(at+a) (3.9) where a is the phase angle. Eq. (3.8) can now be written as: ga [bh + U270 (3.10) baxL ax ] where the subscript i(x) has been dropped and the substitution, A = bh, was made. 3.1. Single Transition The case of a channel of uniform width with an infinitely long step either up or down was a specific case solved in Dean (1964). The definition sketch for a "step down" is shown in Figure 3. The three regions in Figure 3 have the following depths: Region 1, x < x; h = h (3.11) Region2, xI Region 3, x > x2; h=h3 (3.13) Region 1 Region 2 Region 3 X, X h. .. ,. . h 2 h 3 S Figure 3: Definition sketch for linear transition. For the regions of uniform depth, Eq. (3.10) simplifies to: 2,2 which has the solution for il of cos(kx) and sin(kx) where k = 21_ and X is the wave length. The most general solution of il(x,t) from Eq. (3.14) is: r/(x, t) = B, cos(kx ot + a1) + B2cos(kx + Ot + a2) (3.15) The wave form of Eq. (3.15) consists of two progressive waves of unknown amplitude and phase: an incident wave traveling in the positive x direction and a reflected wave traveling in the negative x direction. For the region of linearly varying depth, Eq. (3.12) is inserted in Eq. (3.10) resulting in a Bessel equation of zero order: xa27 + L+/8q=0 (3.16) ax x where )6 = a 2 X2 (3.17) Ah The solutions of rI(x) for Eq. (3.16) are: i(x)= Jo(2fl1/2x112) and Yo(2/3112x"2) (3.18) where Jo and YO are zero-order Bessel functions of the first and second kind, respectively. From Eq. (3.18) the solutions for rl(x,t) in Region 2 follow: q(x, t) = B3 JO (2fl"/2x1/2 ) cos(ut + a3) + B3Y (2f1/2x112) sin(ut + a3) (3.19) + B4Jo (2p 1/2xl/2) cos(at + a4) B4Yo (2fl/2x'/2) sin(at + a4) The wave system of Eq (3.19) consists of two waves of unknown amplitude and phase; one wave propagating in the positive x direction (B3) and the other in the negative x direction (B4). The problem described by Figure 3 and Eqs. (3.15) and (3.19) contains eight unknowns: B1,4 and al,4. Solution to the problem is obtained by applying matching conditions at the two boundaries between the three regions. The conditions match the water surface and the gradient of the water surface: 77j -= 'j+, atx = xj (j = 1,2) (3.20) - atx=xj (j = 1,2) (3.21) ax ax Eqs. (3.20) and (3.21) result in eight equations (four complex equations), four from setting o- t = 0 and four from setting at = -,which can be solved for the eight unknowns as a linear matrix equation. 3.2. Trench or Shoal The slope method is an extension of the Dean (1964) solution that allows for a domain with a trench or a sill with sloped transitions. Two linear transitions are connected by a constant depth region by placing two solutions from Dean (1964) "back to back." A trench/sill with sloped side walls can be formed by placing a "step down" upwave/downwave of a "step up." The definition sketch for the case of a trench is shown in Figure 4. In the slope method the depths are defined as follows: Region 1, x < x,; h = h (3.22) Region2, xI Region 4, x3 where hl, h3, hs, Sl, s2, and W are specified. With the new definition for the depth in o-2 regions 2 and 4, the definition of the coefficient 3 in Eq. (3.16) changes to ,8 =- and gs1 .2 18= --in regions 2 and 4, respectively. gs2 ~- w -~. Figure 4: Definition sketch for trench with sloped transitions. The matching conditions of Eqs. (3.20) and (3.21) are applied at the four boundaries between the regions. With the transmitted wave specified and by setting a-t = 0 and crt = 'r for each matching condition a set of 16 independent equations is developed. 2 Using standard matrix techniques the eight unknown amplitudes and eight unknown phases can then be determined. The reflection and transmission coefficients can be determined from KR =aR and K = al (3.27), (3.28) a, a, Conservation of energy arguments in the shallow water region require: K 2 + K2F ] =1(3.29) This method can be extended to the representation of long wave interaction with any depth transition form represented by a series of line segments. 4. Numerical Method: Formulation and Solution A numerical method was developed to determine the long wave transformation caused by a trench or shoal of arbitrary, but shallow water bathymetry. A transmitted wave form in a region of constant depth downwave of the depth anomaly is the specified input to the problem. Numerical methods are used to space step the wave form backwards over the trench or shoal and then into a region of constant depth upwave of the depth anomaly where two wave forms exist; an incident wave and a reflected wave. As in the long wave solution of Section 3, the continuity equation and the equation of motion are employed to develop the governing equation for the problem. The continuity equation and the equation of motion in the x direction are written in a slightly different form than in Eqs. (3.3) and (3.6) of Section 3: ar = aq (4.1) at ax gha7 = q- (4.2) ax at Taking the derivative of Eq. (4.1) with respect to t and the derivative of Eq. (4.2) with respect to x results in the governing equation for this method: 2rl gh r dhatu=O gha27 g (4.3) 0t2 dx ax where the depth, h, is a function of x and 1i may be written as a function of x and t: r7 = i(x)ei'r (4.4) Inserting the form of i in Eq. (4.4) into the governing equation of Eq. (4.3) casts the equation in a different form: gh a217(x) g dh aq(x) = 0 (4.5) o' Oxax2 2 dx ax Central differences are used to perform the backward space stepping of the numerical method. F"(x) = F(x + Ax) 2F(x) + F(x Ax) (4.6) Ax2 F'(x) = F(x + Ax) F(x-Ax) (4.7) 2Ax Inserting the forms of the central differences into Eq. (4.5) for 1 results in: 7(x) +-gh [(x+Ax)-2r(x)+ i(x-Ax)]+ g dh [7(x+Ax)-r(x-Ax)]=0 (4.8) a 2 Ax2 I d 2Ax For the backward space stepping calculation, Eq. (4.8) can be rearranged: (x + Ax) gh g dh 1 2(x)1 2gh ?zAx' '. dx 2Ax/x (x Ax) 2 dx 2Ax 22 (4.9) Sgh g dh 1 ] o2Ax2 A.2 dx 2Ax To initiate the calculation, values of l(x) and l(x+Ax) must be specified in the constant depth region downwave from the depth anomaly. If the starting point of the calculation is taken as x = 0 then the initial values may be written as: H q(0) = H (4.10) 2 H q(Ax) = H [cos(kAx) i sin(kAx)] (4.11) 2 The solution upwave of the depth anomaly comprises of an incident and reflected wave. The form of the incident and reflected waves are specified as: = H-cos(kx- ot- s) (4.12) 2 7R =R cos(kx + ot R) (4.13) 2 where the s's are arbitrary phases. At each location upwave of the depth anomaly the total water surface elevation will be the sum of the two individual components: q = rh + R= HI cos(kx e,) cos(ot) + sin(kx e,) sin(ot) 2 HR + R cos(kx eR) cos(ot) sin(kx 6R) sin(at) 2 =cos(ot) -Hcos(kx + H cos(kx- ER 2 2 I + sin(ot) sin(kx )- H sin(kx 6R )[HH 2 2 II = I2+II 2 COS(t = tan-I(L (4.14) Using several trigonometric identities, Eq. (4.14) can be reduced further to the form: W =1 VH +H +2HIHRcos(2kx-e S-R)cos(t-8) (4.15) 2 1 R which is found to have maximum and minimum values of: Tmax = (HI +HR) (4.16) 2 r/r.n =-' H -HR) (4.17) Eqs. (4.16) and (4.17) are used to determine the values of HI and HR upwave of the trench/shoal, and allow calculation of the reflection and transmission coefficients. 5. Results and Comparisons 5.1. Arbitrary Water Depth The step method was used to study wave transformation by 2-D trenches and shoals in arbitrary water depth. The focus of the study was trenches or shoals with sloped transitions, which for the step method was performed by approximating a uniform slope as a series of steps of equal size. For shallow water conditions the results of the step method were compared to the results of the slope method and the numerical method, which are long-wave models. 5. 1.1. Comparison to Previous Results A comparison to results from Kirby and Dalrymple (1983) was made to verify the step method (Figures 5 and 6). In these figures the magnitudes (i.e. the phases are not reported) of the reflection coefficient, KR, and magnitudes of the transmission coefficient, KT, are plotted versus the dimensionless wave number, kjh1, using the step method for the case of a symmetric trench with vertical transitions and normal wave incidence. Also plotted are the values (shown as "'*") of Kr and Kt for three different kjhj, values taken from Table 1 in Kirby and Dalrymple (1983) with good agreement between the results. 0.45 (* Kirby and Dalrym pie (1983) Table 1 0.4 0.35 0.3 S0.250.2 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1l 1.2 1.3 1.4 1.5 k~hI Figure 5: Comparison of reflection coefficients from step method and Kirby and Dalrymple (1983 Table 1) for symmetric trench with abrupt transitions and normal wave incidence: h3 = hl, h2/h1 = 3, W/h1 = 10. The results in the figure include 16 evanescent modes, an amount which was found by Kirby and Dalrymple (1983) to provide adequate convergence for most values of kjhj. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 kIhI Figure 6: Comparison of transmission coefficients from step method and Kirby and Dalrymple (1983 Table 1) for symmetric trench with abrupt transition and normal wave incidence: h3 = hl, h2/h, = 3, W/h, = 10. A comparison to the results of Lee et al. (1981) was also carried out. Good agreement was found between the results of the step method and those from Lee et al. for several cases of a trench with abrupt transitions. A direct comparison between the results of the step method and data from the one case in Lee et al. (198 1) for a trench with gradual transitions was not made as the complete dimensions of the trench were not specified in the article and were not able to be obtained at present. 5.1.2. Symmetric Trenches and Shoals To study the effects of sloped transitions on the wave transformation, a number of trench and shoal shapes were examined. For the first component of the analysis, the cross-sectional area was kept constant for several different symmetric trench configurations. The objective was to investigate wave transformation for various slopes with a fixed cross-sectional trench area in a two-dimensional domain. The first set of trench shapes has the same cross-sectional area and depth with different bottom widths and transition slopes as shown in Figure 7. The transition slopes range from an extreme value of 5000 that represents an abrupt transition, to a gradual transition with a slope of 0. 1. In this figure of the bathymetry, and those that follow, the still water level is at elevation = 0 mn. The trench with the abrupt transitions contains one step while ten equally sized steps approximate the slopes in the other three trenches. The reflection and transmission coefficients versus the dimensionless wave number, kjhj, are shown in Figures 8 and 9, respectively for the trenches with the same depth. An inset figure of the trench through the centerline is included in each figure with four different line types indicating the corresponding trench configurations. The reflection coefficients oscillate with decreasing peaks as k1h, increases and with k1h, values of complete transmission (KR = 0). Decreasing the transition slope is seen to reduce the reflection caused by the trench, especially at larger values of kjhj. As the slope is reduced, the location of the maximum value of KR is shifted and the number of instances where complete transmission occurs is also seen to increase greatly. The plot of the transmission coefficients shows the same features as the reflection coefficients with the effect on the wave field reduced as the slope is decreased. For all four trench configurations and all values of kjh1, conservation of energy requirements were satisfied. -1.5 -2 ()(3)1 (2)1 (1) -2.5-1I C -3.5-I STrench Slope (1) 5000 ___ ___ __ ___ ___ __ ___ __ (2) 1 -4 (3) 0.2 (4) 0.1 0 10 20 30 40 50 X Distance (in) Figure 7: Setup for symmetric trenches with same depth and different bottom widths and transition slopes. 0 0.5 1 1.5 k1h1 Figure 8: Reflection coefficients versus k1hl for trenches with same depth and different bottom widths and transition slopes. Only one-half of the symmetric trench cross-section is shown. Figure 9: Transmission coefficients versus k1h1 for trenches with same depth and different bottom widths and transition slopes. Only one-half of the symmetric trench cross-section is shown. The results shown in Figures 8 and 9 are for 16 evanescent modes taken in the summation and trenches with 10 steps approximating the sloped transitions. The influence of the number of evanescent modes taken in the summation on the reflection coefficient is shown in Figure 10 for the same depth trenches with slopes of 5000, 1, and 0. 1 at a specified value of kjhj. 0.195 0.19 -Trench (1): Slope =5000 ~~ III9, =h 0.95I 0. 5 5 1 .0 1 .5 2 .0 2 .5 3 .0 35 40 0.242 0.241 Trench (2): Slope I YO:0.24 k1h1 0.7e 0.2390. 5 L 10 15 20 25 30 35 40 0.2821 0.2815- Trench (4): Slope = 0.1 w 0.281 k1 . 0.280500 5 10 15 20 25 30 35 40 Number of Evanescent Modes in Summation Figure 10: Reflection coefficient versus the number of evanescent modes used for trenches with same depth and transition slopes of 5000, 1, and 0. 1. The reflection coefficient is seen to converge to a near constant value with increasing number of evanescent modes. A value of 16 modes is found to produce adequate results and was used for all the step method calculations. The influence of the number of steps used in the slope approximation is shown in Figure 11 for the same depth trench configurations. The reflection coefficient is seen to converge to a steady value with increasing number of steps used for the trenches with slopes of 1 and 0. 1; however for the abrupt transition trench (slope = 5000) convergence with an increasing number of steps was not found. Convergence for the abrupt transition was found as the step number decreased to 1, which represents an almost vertical wall. This could be due to the small distance between points that results when dividing the nearly vertical wall into an increasing number of segments. For this reason the abrupt transition trenches were configured with 1 step, resulting in an almost vertical wall and for all other slopes 10 steps were used. This convention was followed for all the step method calculations. 0.27. 0.26- Trench (1): Slope = 5000 W, 0.25 klh1 0.95 0.24- 1 0.23' 0 5 10 15 20 25 30 Trench (2): Slope 1 p 0.295 k1h = 0.7 0.29 5 10 15 20 25 30 0.285 1 1 1 : 0.28 ) Trench (4): Slope 01 k1ht = 0.1 0. 275L L__I 0 5 10 15 20 25 30 Number of Steps Figure 11: Reflection coefficient versus the number of steps for trenches with same depth and transition slopes of 5000, 1, and 0.1. A second way to maintain a constant cross-sectional area is to fix the bottom width of the trench and allow the trench depth and transition slopes to vary. The reflection coefficients for four trenches developed in this manner with the same bottom width are shown in Figure 12. An inset figure is included to show the trench dimensions through the centerline with slopes of 5000, 1, 0.2 and 0.05 being used. For this case, as the slope decreases the depth of the trench must also decrease to maintain the fixed value of the cross-sectional area. The maximum reflection coefficient is seen to decrease and shift towards a smaller value of k1h, as the slope decreases, due to the associated decrease in the depth of the trench. Generally the depth of the trench determines the magnitude of KR and the trench width determines the location of the maximum value of KR. - 0 0.5 1 1.5 k1hi Figure 12: Reflection coefficients versus k1h, for trenches with same bottom width and different depths and transition slopes. Only one-half of the symmetric trench cross-section is shown. Keeping the trench cross-sectional area and top width fixed while allowing the transition slopes and depth to change, results in another series of trench configurations. The reflection coefficients for four trenches with the same cross-sectional area and top width are shown in Figure 13. The slopes of the transitions are 5000, 5, 2, and 1. The trench with the smallest slope is found to produce the largest value of Kr due to the large depth associated with that slope; however the abrupt transition results in the largest KR values after the first maximum. For this case decreasing the slope does not shift the locations of the maximum values of KR as much as in the same depth and same bottom width cases, although no slopes less than one were used as they do not satisfy the constraints of the domain. These three cases have demonstrated that trenches with the same cross-sectional area can have very different reflective properties depending on the trench configuration. Figure 13: Reflection coefficients versus k1h, for trenches with same top width and different depths and transition slopes. Only one-half of the symmetric trench cross-section is shown. The effect of the transition slope can be viewed in another manner when the cross-sectional area of the trench is not fixed. For a fixed depth and bottom width, decreasing the transition slope results in a trench with a larger cross-sectional area. Plots of the reflection coefficients versus k1h, for four trenches with the same depth and bottom width, but slopes of 5000, 0.2, 0. 1, and 0.05 are shown in Figure 14. Decreasing the transition slope is seen to reduce KR, even though the trench cross-sectional area may be much larger. The step method is also valid for the case of a submerged shoal with sloped transitions approximated by a series of steps. Figure 15 shows the reflection coefficients versus k1h, for four different shoal configurations with transition slopes of 5000, 0.5, 0.2 and 0.05. Decreasing the transition slope is found to reduce the, value of KR as was the case for the trench, however the reduction in KR is not as significant for large k1h, as in the trench cases. p0.15 0 0.5 1 1.5 k~hI Figure 14: Reflection coefficients versus k1h, for trenches with same depth and bottom width and different transition slopes. Only one-half of the symmetric trench cross-section is shown. Figure 15: Reflection coefficients versus k1h, for shoals with same depth and different top widths and transition slopes. Only one-half of the symmetric shoal cross-section is shown. A uniform transition slope, approximated by a series of uniform steps, was used in the previous cases for the wave transformation caused by symmetric trenches and shoals. In the following cases the wave transformation by depth anomalies with variable transition slopes is investigated. The domain was created by inserting a Gaussian form for the bottom depth into an otherwise uniform depth region. The equation for the Gaussian shape centered at x0 was: h(x)=k + Cle 2C2 (5.1) where ho is the water depth in the uniform depth region and C1 and C2 are shape parameters with dimensions of length. To implement the step method, steps were placed at a fixed value for the change in depth to approximate the Gaussian shape. Two extra steps were placed near ho where the slope is very gradual and one extra step was placed near the peak of the 'bump' to better simulate the Gaussian form in these regions. The reflection coefficients versus k1h1 for a Gaussian trench with C1 and C2 equal to 2 m and 12 m, respectively, and a uniform water depth, ho, equal to 2 m is shown in Figure 16. 0.25- -2 -2.2 -2.4 0.2- -2.6Ef-2.8 0. -3 0.15 "-3.2 -3.4 0.1- -3.6-3.8 -4 -40 -30 -20 -10 0 0.05 X Distance (in) 0 0 0.5 1 1.5 k h I Figure 16: Reflection coefficients versus k1h, for Gaussian trench with C1 2 m and C2 = 12 m and ho = 2 m. Only one-half of the symmetric trench cross-section is shown. An inset figure is included to show the configuration of the stepped trench and the form of the Gaussian bump. The Gaussian form is approximated by the step method with a step at every 0.05 m change in depth. The reflection caused by the Gaussian trench is seen to differ considerably from that for the trenches with constant transition slopes shown previously where KR showed significant oscillations that diminished as k1hl increased, even with gradual transition slopes. The reflection coefficient is seen to have one peak near k1hl equal to 0.1 with minimal reflection for larger values of klhj. The reflection caused by a Gaussian shoal is shown in Figure 17 for C1 and C2 values of 1 m and 8 m, respectively. The steps approximating the Gaussian form are placed a 0.05 m depth intervals. As for the case of the Gaussian trench, a single peak in KR occurs followed by minimal reflection at larger values of kjhl. 0.16 0.12 Figure 17: Reflection coefficients versus k1h for Gaussian shoal with C1 = 1 m and C2 = 8 m and ho = 2 m. Only one-half of the symmetric shoal crosssection is shown. 5.1.3. Asymmetric Trenches The trenches considered previously have been symmetric with the upwave and downwave water depths equal and the same slope for the upwave and downwave transitions. Figure 18 shows the reflection coefficients versus k1h, for 3 trenches with the same depth upwave and downwave of the trench: a symmetric trench with an abrupt transition, an asymmetric trench with a steep (s, = 1) upwave transition and gradual (S2 = 0.1) transition and a mirror image of the asymmetric trench with a gradual (sI = 0.1) upwave transition and steep (S2 = 1) downwave transition. The trench configurations are shown in the inset diagram with dashed and dotted lines. The order of the transition slopes (steep slope first versus gradual slope first) is seen to have no effect on the value of KR; therefore only one dotted line is plotted in the KR versus k1h1 plot. Stated another way, the direction of the incident wave (positive x direction versus negative x direction) does not matter for this case. For the asymmetric trenches with the different transition slopes, complete transmission only occurs at k1h, = 0, which is different from the case for the abrupt transitions and the previous cases where s, equals S2. 0.35 -1.5 1 --Im Wave -2 0.3 % C I o -3 '. 0.25 -; -3.5 -. 1 -4 0.2- 145-0 20 30 40 50 X Distance (m) 0.15 0.1 % 0.05 0 0 0.5 1 1.5 k h1 Figure 18: Reflection coefficients versus k1hl for a symmetric abrupt transition trench, an asymmetric trench with different S, and S2 values and a mirror image of the asymmetric trench. Asymmetric trenches can also be studied with different depths upwave and downwave of the trench. Figure 19 plots KR versus k1h1 for domains with different depths upwave (hj) and downwave (hj) of the trench. The case of an abrupt transition (solid line) with the downwave depth greater than the upwave depth can be compared to the case with transition slopes of 0.2 (dashed line) with the same cross-sectional area. Complete transmission does not occur for either the trench with the abrupt transition or the trench with sloped transitions. The trench with sloped transitions is found to cause less reflection with an irregular oscillation in KR with increasing k1hl as compared to the abrupt transition. 0.3 -1.5 -2-Wave -2 0.25 9-2.5. 1 -3.5 .... " 1 0.2-t -4 -4.5 10 20 30 40 50 60 p2'0.15 X Distance (m) oI iI I 0.1 1 I Ih Ik h Figure 19: Reflection coefficients versus kh, for an asymmetric abrupt transition trench, an asymmetric trench with different h, and h5 values and s, equal to S2 and a mirror image of the asymmetric trench with Sl = Sz. For this asymmetric case KR does not approach zero as kh] approaches zero as was found when h, was equal to hj, but approaches the long wave value for a single abrupt step (either up or down) of elevation 1h, hjI. Also plotted (dotted line) are the results for an asymmetric trench with the depth upwave of the trench greater than the depth downwave of the trench with transition slopes of 0.2, which is a mirror image of the trench shown with the dashed line. The reflection coefficients for this case are identical in magnitude to (but differ in phase from) the miror image trench if KR is plotted versus the shallower constant depth, kjhj; therefore only one dashed line for KR is plotted. This result is another case of the incident wave direction not affecting the magnitude of the reflection coefficients for an asymmetric trench as shown in Figure 18. Another asymmetric trench case is shown in Figure 20 where the upwave and downwave depths and transition slopes are different with the upwave depth being larger. In this figure two asymmetric trenches with the same cross-sectional area are compared: a trench with an abrupt transition (solid line) and a trench with an upwave transition slope (si) equal to 1 and a downwave slope (2) of 0.2 (dashed line). The sloped transitions are seen to reduce significantly the value of KR, especially as k1hl increases. 0.4 1 -2 Wave 0.35 -" I. ' r 0.3 t--, -5 0.25- 20 40 60 X Distance (m) 0o.2 0.15 %II 0.1 I I 0 0.5 1 1.5 klhi Figure 20: Reflection coefficients versus khl for an asymmetric abrupt transition trench, an asymmetric trench with different h, and h5 values and s, not equal to S2 and a mirror image of the asymmetric trench with sI = S2. As shown in the previous figure, KR does not approach zero as kh, approaches zero for this case where the upwave and downwave depths are not equal. Also plotted is a mirror image of the asymmetric trench with the sloped transitions (dotted line).Only one dashed line for KR is plotted, as the reflection coefficients for this case are identical to that for the mirror image trench if KR is plotted versus the deeper constant depth, kh5. This result is another example of the independence of KR to the incident wave direction as shown in Figures 18 and 19. 5.2. Long Waves Direct comparison of the step method to the slope and numerical methods can only be made in the long wave region. While the slope method and numerical method are limited by the long wave restrictions, it is in this region that changes in bathymetry are most effective at modifying the wave field. The slope method does not contain any variables that will affect the accuracy of the results such as the number of evanescent modes or the number of steps as in the last section. The numerical method however is sensitive to the spacing between points used in the backward space stepping procedure described in Section 4. Figure 21 shows the value of KR versus the step spacing, dx, using the numerical method for three trench configurations at a single value of k1hl for the three of the same depth cases (Figure 7) presented earlier. The reflection coefficient is seen to converge for dx < 1 m and a value of dx = 0.5 m was found to be adequate for the numerical model. 0.36 _1.5 I I 0.355 -g, -2.5 _k1hI =0.142 0.35 -3 ,0........................ .... ...................... S0.345- ..... ............... 0.34 010 0 0.1355.... 0.33 0.325 klh1 = 0.1293 0.32 .. 0.315 I I I I 0 1 2 4 5 6 7 8 9 10 dx (in) Figure 21: Reflection coefficient versus the space step, dx, for trenches with same depth and different bottom width and transition slopes. Only one-half of the symmetric trench cross-section is shown. 5.2. 1. Symmetric Trenches and Shoals To compare the three solution methods, the same depth trench configurations shown in Figure 7 were used. In order to maintain long wave conditions throughout the domain the limit of k3h3 n/10 was taken with the subscript 3 indicating the constant depth region inside the trench between the transitions. Figure 22 shows KR versus k3h3 for the three solution methods for the same depth trench case with a slope of 5000. The slope method and the numerical method are found to yield nearly the same value of KR at each point with the peak value occurring at a slightly smaller k3h3 than for the step method. Overall, the agreement between the three models is found to be very good for this case. 0.35 0.3 0.25 (dash) = slope method 0.2- (o) =numerical method -2 0.15-925 C 0 -3. (U .4 0.05- -.51 0 10 20 X Distance (in) 0 0 0.05 0.1 0.15 0.2 0.25 0.3 k3h3 Figure 22: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with transition slope equal to 5000. Only one-half of the symmetric trench cross-section is shown. Figures 23 and 24 show KR versus k3h3 for a slope of 1 and 0. 1, respectively for the same depth trench case with all three models results plotted. Good agreement is found for the results of the three models. In Figure 24 the numerical model results match those of the slope method up to the peak value for KR, but as k3h3 increases after the peak the numerical results match those of the step method and the slope method results diverge slightly. The similarity in the results of Figures 22 through 24 provides reasonable verification for the three solution methods. While the conservation of energy requirement is satisfied exactly in the step method and the slope method, for the cases studied, the numerical method is found to have some variance in the conservation of energy requirement for the space step used in Figures 22 through 24. Figure 25 shows the conservation of energy parameter versus k3h3 for the same depth trench with a slope of 1. As noted, the step method and the slope method have values exactly equal to 1, while the numerical method results are equal to one for small k3h3 with increasing deviation from one, although very small, as k3h3 increases. 0.3 0.25 (solid) =step method (dash) = slope method 0.2- (o) =numerical method -1.5 -2 0.15 .0 -3 0.1 -4. 0.05- -4.51 0 10 20 X Distance (in) 0 0.05 0.1 0.15 0.2 0.25 0.3 k3h3 Figure 23: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with transition slope equal to 1. Only one-half of the symmetric trench cross-section is shown. Figure 24: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with transition slope equal to 0.1. Only one-half of the symmetric trench cross-section is shown. 1.0001 E a. 0) -1.5 W, 0.9999 -2 C! 16C /(solid) =step method 0 0).35 (--x--) = slope method 0.9998 (--o--) = numerical method -4 -4.5 0 10 20 X Distance (m) 0.99971 0 0.05 0.1 0.15 0.2 0.25 0.3 k3h 3 Figure 25: Conservation of energy parameter versus k3h3 for three solution methods for same depth trench case with transition slope equal to 1. Only one-half of the symmetric trench cross-section is shown. 40: 0. 1! Using the step method the reflection coefficients for Gaussian trenches and shoals were shown to have a single peak followed by minimal reflection as k1h, increased. Figure 26 shows a comparison of the step method and numerical method results for a shoal with h. = 2 m, C1 = 1 m, and C2 = 8 m. The results for the step method were shown previously in Figure 17. The slope method can only contain two slopes and therefore cannot be employed to model a Gaussian form and is not included in the following comparisons. The numerical results show good agreement with those of the step method with a slight divergence at higher k1h values near the shallow water limit. 0.28 ... 0.24 (solid) step method, dh 0.1 m (o) numerical method, dx = 0.5 m 0.2 0.16 0.12 ,-1.2 '-1.4 0.1 0.08 -1.6 10-1.8 t i -20 -10 0 X Distance (m) 0.05 0.1 0.15 0.2 0.25 0.3 klh1 Figure 26: Reflection coefficients versus k1hl for step and numerical methods for Gaussian shoal (h. = 2 m, C1 = I m, C2 = 8 in). Only one-half of the symmetric shoal cross-section is shown. Figure 27 is a comparison of the results of the step method and the numerical method for a Gaussian trench in shallow water. The step method has a step for every 0.01 m in elevation and the numerical method has a space step of 0.1 m. The domain is in very shallow water (ho = 0.25 m, CI = 0.2 m, C2 = 3 m) to extend the numerical method results into the region of minimal reflection following the initial peak in KR. The numerical method results agree closely to those of the step method and verify the small KR values of the step method that occur as k3h3 increases after the initial peak in KR. 0.25 -0.25 .. . -0.3 0.2 0 -0.35 W 0.15 -0.4 v -0 .4 5 '. . -10 -8 -6 -4 -2 0 0.1 X Distance (m) (solid) step method 0.05 (o) numerical method C 1 0 10 0 0.05 0.1 0.15 0.2 0.25 0.3 k3h3 Figure 27: Reflection coefficients versus k3h3 for step and numerical methods for Gaussian trench in shallow water (ho = 0.25 m, C1 = 0.2 m, C2 = 3 m). Only one-half of the symmetric trench cross-section is shown. 5.2.2. Asymmetric Trenches The case of a trench with unequal upwave and downwave slopes as shown in Figure 18 is now compared for the three solution methods. In order to maintain shallow water conditions throughout the domain a limit of k3h3 < r/10 was taken for the slope and numerical methods. Figure 28 shows the value of KR versus k3h3 for the same depth trench with an abrupt transition (solid line) and for s, and S2 equal to 1 and 0.1, respectively. The bottom width for all three cases was maintained the same. The slope and numerical method results show good agreement with the step method solution for both cases. Due to the location of the shallow water limit, the increase of KR for the step method at k3h3 equal to 0.4 cannot be verified directly. \ / \ 0.15 I 0. I" \ / 0.1 -solid) step method, abrupt transition ( --) step method, si notequal to s2 0.05(+) numerical method (o) slope method 0!/ . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k3h3 Figure 28: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with symmetric abrupt transition trench and asymmetric trench with unequal transition slopes equal to 1 and 0.1. The case of an asymmetric trench with unequal upwave and downwave slopes, and depths as shown in Figure 20 is now compared for the three solution methods. For the slope and numerical methods a limit of k3h3 < 7r/10 was taken to maintain shallow water conditions. Figure 29 shows the value of KR versus k3h3 for an asymmetric trench with an abrupt transition (solid line) and for s, and s2 equal to 1 and 0.2, respectively (dashed line). The bottom width was changed to maintain a constant cross-sectional area. The slope and numerical method results show good agreement with the step method solution for both cases with less divergence in the KR values at larger k3h3 for the sloped transition trench. 0.1 4I soid ste mehd arptanito 0.05 M- )se ehd s, not equal tos2 ()numerical method 0 20 40 60 (o) slope method X Distance (in) 01 1 0 0.1 0.2 0.3 0.4 0.5 0.6 k3h 3 Figure 29: Reflection coefficients versus k3h3 for three solution methods for same depth trench case with asymmetric abrupt transition trench and asymmetric trench with unequal transition slopes equal to 1 and 0.2. 6. Summary and Conclusions The interaction of linear water waves with two-dimensional trenches and shoals has been demonstrated using three methods. The three methods show good agreement for the cases presented for various bathymetric changes with abrupt and gradual transitions between changes in depth. Gradual transitions in the depth for both symmetric and asymmetric bathymetric changes are seen to reduce the reflection coefficient, especially for non-shallow water waves. Linear transitions are shown to result in instances of complete wave transmission for symmetric trenches and shoals, while for asymmetric trenches and shoals complete transmission does not occur. Changes in depth that are Gaussian in form are demonstrated to result in a single peak of the reflection coefficient in the long wave region, followed by minimal reflection for shorter wavelengths. Several new results have been found in this study: (1) The wave field modifications are shown to be independent of the incident wave direction for asymmetric changes in depth; a result shown by Kreisel (1949) for a single step, (2) For asymmetrical bathymetric anomalies with hi = hj, a zero reflection coefficient occurs only at k1h = 0, and (3) For asymmetrical bathymetric anomalies with h, # hj, the only k1h, value at which KR = 0 is that approached asymptotically at deep-water conditions. 7. Acknowledgements An Alumni Fellowship granted by the University of Florida sponsored this study with partial support from the Bureau of Beaches and Wetland Resources of the State of Florida. 8. References Bartholomeusz, E.F. 1958. The reflexion of long waves at a step. Proc. Camb. Phil. Soc. 54, pp. 106-118. Black, J.L., Mei, C.C., and Bray, M.C.G. 1971. Radiation and scattering of water waves by rigid bodies. Journal of Fluid Mechanics. 46, pp. 151-164. Dean, R.G. 1964. 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