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Title: 
Motion of a flexible hydrofoil near a free surface 

Physical Description: 
xii, 102 leaves : illus. ; 28 cm. 

Language: 
English 

Creator: 
Reece, Joe Wilson, 1935 

Publication Date: 
1963 

Copyright Date: 
1963 
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Subject: 
Hydrodynamics ( lcsh ) Planing hulls ( lcsh ) Waves ( lcsh ) Engineering Mechanics thesis Ph. D Dissertations, Academic  Engineering Mechanics  UF 

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bibliography ( marcgt ) nonfiction ( marcgt ) 
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Thesis: 
ThesisUniversity of FLorida, 1963. 

Bibliography: 
Bibliography: leaves 99101. 

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Also available on World Wide Web 

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Manuscript copy. 

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Vita. 
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Bibliographic ID: 
UF00097960 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

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University of Florida 

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All rights reserved by the source institution and holding location. 

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alephbibnum  000568378 oclc  13649198 notis  ACZ5109 

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MOTION OF A FLEXIBLE HYDROFOIL
NEAR A FREE SURFACE
By
JOE WILSON REECE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
December, 1963
ACKNOWLEDGMENTS
The author wishes to thank Dr. J. Siekmann, Chairman of his
Supervisory Committee, for suggesting this problem and for his con
tinued encouragement and counsel.
He is grateful to the other committee members, Dr. W. A.
Nash, Head of the Advanced Mechanics Research Section, Dr. S. Y.
Lu, Associate Professor in Engineering Mechanics, Dr. I. K. Ebcioglu,
Assistant Professor in Engineering Mechanics, and Dr. C. B. Smith,
Professor of Mathematics, for their encouragement and helpful criti
cism of the manuscript.
The author is also indebted to Dr. I. C. Statler, Head of the
Applied Mechanics Department, Cornell Aeronautical Laboratory, for
his timely suggestions and advice.
The author thanks Dr. Nash and Mr. B. M. Woodward, Man
ager of the University of Florida Computing Center, for providing
financial assistance for the use of the IBM 709 digital computer.
To the Ford Foundation he expresses profound gratitude for
supporting his graduate studies for the past twentyseven months.
Finally, he is grateful to his wife, Nancy, for her support and
understanding during a difficult time. She has been a "rock" in a sea
of "free surface waves."
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ........ .......... ii
LIST OF TABLES . . . . . . . . ... .. v
LIST OF FIGURES . . . . . . . . . .. vi
LIST OF SYMBOLS ................... vii
ABSTRACT . . . . . . . . . . . . x
INTRODUCTION . . . . . . . . . . 1
Chapter
I. FORMULATION OF THE PROBLEM . . . . 5
1. 1 The Motion of the Plate
1. 2 The Plate Model
1. 3 The Velocity Potential
1. 4 The Integral Equation
II. SOLUTION OF THE INTEGRAL EQUATION . . . 23
2. 1 Form of the Solution
2. 2 Solution for the Infinite Set of Equations
III. THE EFFECT OF THE FLOW ON THE PLATE . . 30
3. 1 Pressure Distribution
3. 2 Lift and Moment
3. 3 Drag and Thrust
IV. NUMERICAL EXAMPLE ............... 46
4. 1 Quadratically Varying Amplitude Function
4. 2 Solution of the Algebraic Equations
4. 3 The Resulting Thrust
TABLE OF CONTENTSContinued
Page
V. SUMMARY AND CONCLUSIONS . . . . .... 58
Appendixes
A. THE FREE SURFACE POTENTIAL . . . .. 63
B. COEFFICIENTS FOR THE INFINITE SET OF
EQUATIONS ... .. . . . .. .. . .. .83
C. A p(e,t) FOR THE CASE IN WHICH Doo . . .. 91
LIST OF REFERENCES .................. 99
BIOGRAPHICAL SKETCH ................. 102
LIST OF TABLES
Table Page
1. Free Surface Flow Conditions . . . . .. 19
2. Real and Imaginary Parts of Bn and Cn . . .. 52
3. Thrust Coefficient . . . . . . . .. .54
4. Thrust Coefficient . . . . . . . . 54
5. Variation of O, j = 1, 2, 3, 4, With kF2 ...... .68
6. G(x,D;) for x oo . ........... ..... 74
7. G(x, D; ) for x  00 . . . . . . . .. .79
8. Variation of Sj, j = 1, 2, 3, 4, With kF2 . . . .. 81
LIST OF FIGURES
Figure Page
1. The Flexible Plate . . . . . ... 6
2. The Plate Model . . . . .. . ... 6
3. Single Vortex Line . . . . . . . .. .12
4. Vortex Sheet With Special Coordinates . . . .. 12
5. Thrust Coefficient Versus Reduced Frequency . . 55
6. Thrust Coefficient Versus Reduced Frequency
for Small Reduced Frequencies . . . . .. 57
/
7. Contours of Integration for the Ij Integrals .. . 69
/
8. Contour of Integration for II . . . . . .. 70
9. Contours of Integration for Ij Integrals . . . .. 76
10. Contour of Integration for Il............. 77
LIST OF SYMBOLS
Bn
Cn
Cmn
Cmn
,0
Cmn
CT
C(k)
Co, C Cz
D
D
Dj
dn
F
G(x, y; )
Coefficient of the Fourier series giving the downwash on
the plate
Coefficient of the series expansion for the pressure distri
bution Ap(x, t)
Coefficient of the Fourier series giving the displacement
function of the plate
Coefficient of the Fourier series giving the slope of the
plate
Complex element of the coefficient matrix of the infinite
set of algebraic equations
Real part of Cmn
Imaginary part of Cmn
Thrust coefficient
Theodorsen function
Coefficients of the quadratic plate amplitude function
Depth of the plate compared to a semichord of 1
Drag
Constants related to the free surface radiation conditions
(j = 1, 2, 3,4)
Auxiliary coefficient used in the series for the pressure
distribution
Froude number
Auxiliary function related to the free surface potential
g Acceleration due to gravity
(z)
Hn () Hankel function of the second kind, nth order, with argu
ment z
h(x, t) Displacement function for the oscillating plate
Jn(s) Bessel function of nth order with argument a
Kn(Z) Modified Bessel function of the second kind, nth order,
with argument a
k Reduced frequency of the plate motion
L Lift per unit span
M Moment per unit span
p Pressure
T Thrust per unit span
T Time average value of the thrust
t Time
U Magnitude of the free stream velocity
u Magnitude of the xcomponent of the perturbation velocity
v Magnitude of the ycomponent of the perturbation velocity
Y(x, t) Displacement in the ydirection of the plate
Yn(s) Neumann function of nth order with argument s
C< Wave number of the plate motion
/8 Auxiliary dissipative constant
r Circulation
< Phase angle of the plate motion
viii
E(x, t)
6(x,t)
(x, t)
(x, y, t)
Re fj
x, yt)
,(x, y, t)
i).
Re~f}
m{f]
Vortex strength distribution for the plate vortex sheet
Vortex strength distribution for the wake vortex sheet
Coordinate of a vortex sheet in the ydirection
Vertical displacement of the free surface relative to its
mean position
Independent variable related to x by x = cos e
Wave length of the plate motion
Dissipative constant
Coordinate of a vortex sheet in the xdirection
Mass density of the fluid
Period of the plate motion
Total velocity potential function
Velocity potential due to the plate vortex sheet and its wake
Velocity potential due to the image system
Spatial part of (x, y, t)
Circular frequency of the plate motion
External force potential
Imaginary unit
Denotes real part of f
Denotes imaginary part of f
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
MOTION OF A FLEXIBLE HYDROFOIL
NEAR A FREE SURFACE
By
Joe Wilson Reece
December, 1963
Chairman: Dr. Julius Siekmann
Major Department: Engineering Mechanics
This study consists of a theoretical investigation of the effect
of free surface waves on the swimming of a thin, flexible, two
dimensional plate immersed in an ideal, incompressible fluid. The
fluid is assumed to be infinitely deep, and the surface waves consid
ered are those generated by the motion of the plate itself. It is as
sumed that the plate moves through the fluid at a constant rate, while
executing lateral motion consisting of waves which travel from front
to rear with increasing amplitude. The amplitude of the plate motion
is made to depend, not only on time, but on the distance which the dis
placement wave has progressed from the leading edge of the plate.
This type motion is used in order to simulate the swimming of a thin
fish near the surface of the water. Of particular interest is the thrust
developed by the swimming action of the plate.
The mathematical theory used is based on velocity potential
methods as employed in unsteady hydrofoil theory. In this theory the
effect of the plate on the flow field is replaced by a continuous vortex
sheet system of oscillating strength. An image system of vortices is
postulated to lie out of the water just above the plate so that the net
effect of the two systems causes no surface disturbance. This allows
the potential of the surface waves to be investigated as a separate func
tion through the free surface boundary conditions. A kinematic condi
tion at the plate between the motion of the fluid and the motion of the
plate leads to an integral equation for the vortex strength distribution.
This distribution is governed by the integral equation in a spatial man
ner only as it has already been assumed oscillatory in the time varia
ble due to the harmonic nature of the plate motion. The wake vortices
shed at the trailing edge of the plate are accounted for in the analysis.
They are assumed to lie in the plane of the plate and to be convected
downstream with the free stream velocity.
The solution of the integral equation for the vortex strength
distribution is effected by assuming a Glauert type of series result.
This transforms the integral equation into an infinite system of in
homogeneous algebraic equations. These equations are solved by
truncating the set so as to ascertain as many of the unknowns as are
needed for accuracy in the final solution. The resulting expression for
the vortex strength distribution insures smooth attached flow at the
plate trailing edge, thus satisfying the Kutta condition.
From this vortex strength distribution the hydrodynamic pres
sure acting on the plate is found through the use of the unsteady Euler
equation. The lift and moment on the plate are calculated by integrat
ing the pressure difference between the top and bottom sides of the
plate along the chord of the plate. The thrust is obtained by integrat
ing the product of the pressure difference and the local slope of the
plate along the chord. It is demonstrated that for the case of swim
ming at infinite depth the results of this study and those obtained by
other researchers are in agreement.
A numerical example is given for the case of plate motion with
displacement waves whose amplitudes vary in a quadratic manner along
the length of the plate. The time average value of the thrust over a
period of the plate motion is computed for two different depths and sev
eral frequencies of plate motion. It is found that the influence of the
free surface waves acts to decrease the available thrust. The decrease
is more pronounced at high frequencies than at low frequencies.
INTRODUCTION
Of late, an increasing number of researchers have been inquir
ing into the matter of sea animal locomotion. The manner in which a
fish swims has been viewed both as a biological and an engineering
problem. Attention is called to the papers of Gawn [1, 2], Taylor [3, 4],
Gray [5], Richardson [6], and Dickmann [7].
Two recent papers by Siekmann [8] and Wu [9] have established
identical results for a fish model swimming horizontally with a con
stant velocity in an infinite, ideal, incompressible fluid. Both authors
have used a flat plate of zero thickness, infinite span, and finite chord
to approximate a thin fish. The plate experiences a traveling wave be
ginning at its leading edge and progressing toward its trailing edge with
increasing amplitude. Wu attacks the problem using Prandtl's accel
eration potential theory, while Siekmann approaches the problem through
the airfoil flutter theory developed by Kissner and Schwarz [10], with
particular reference to the paper by Schwarz [11].
Smith and Stone [12] have partially solved the same problem in
an elliptic coordinate system, but they have failed to consider the wake
effects. Their solution has recently been modified and extended to in
clude the wake effects by Pao and Siekmann [13]. Results identical to
those of Siekmann [8] and Wu [9] have been obtained.
In the present work the effect of a free surface on the swimming
of the thin plate model is sought. As the plate approaches the vicinity
of the surface, a disturbance arises on the surface which has an effect
on the motion of the plate. Siekmann, Wu, and Pao have shown that a
positive thrust is developed in case the frequency of the plate oscilla
tion is sufficiently large. Some questions immediately arise when free
surface effects are considered. Will the presence of the free surface
provide an increase or a decrease in the thrust? Just how close to the
surface does the plate have to be in order for the thrust to be influenced
to any great extent?
As a starting point in the analysis, the particularly illuminating
hydrofoil report by Crimi and Statler [14] is used. This report, con
cerning the motion of an oscillating rigid hydrofoil near the free sur
face of a liquid, is based on two papers by Tan [15, 16]. Tan has ob
tained the velocity potential of a point source or vortex line of fluctu
ating strength moving near the free surface of a liquid. In his second
paper Tan investigates the behavior of the free surface and provides
valuable information regarding the nature of the waves produced by a
disturbing point source of fluctuating strength moving at a uniform rate
near the surface. Crimi and Statler use essentially an integrated form
of Tan's results for a moving point vortex of fluctuating strength. This
idea is connected with airfoil flutter theory in that one way of approach
ing the problem is by replacing the thin plate by a vortex sheet of
fluctuating strength [17]. In such flutter theory and in the sea animal
locomotion paper by Siekmann [8], an integral equation results for the
strength of the vortex sheet. The formulation of the problem in an in
finite flow results in a fairly simple integral equation of the form
ftd GQ) e id ,
 
where H( ) is the sought for function related to the vortex distribution
strength, and \ and CT are constants. This equation has been dis
cussed by SBhngen [18], and the solution has been modified and used by
Schwarz [11]. The presence of the free surface alters the form of the
analogous integral equation and makes it much more complicated.
Crimi and Statler have employed a method of solution giving
results in approximate but extremely accurate form. In their solution
none of the effects of the free surface are lost. A similar equation will
result in the following analysis, and the same method will be used to
solve it.
Other investigators have studied the effects of the free surface
on an oscillating rigid hydrofoil, but several simplifying assumptions
have been made in their work. Chu and Abramson [19] assumed an in
finite Froude number allowing for no surface waves. Their solution is
in the form of a chordtodepth ratio expansion and therefore holds
only for relatively high velocities and large submergence depths. Kap
lan [20] has analyzed the same problem considering the primary effects
4
of a finite Froude number, but his model consists of a single vortex
and a doublet and is not as precise as the model of Crimi and Statler.
It should be mentioned that the problem is formulated here and
solved within the bounds of linear theory. The results obtained are
perhaps as pertinent to the field of hydrofoil hydrodynamics as they
are to sea animal locomotion.
CHAPTER I
FORMULATION OF THE PROBLEM
Consider the twodimensional flow of an incompressible, invis
cid fluid around a flexible solid plate of zero thickness. The flow is
influenced by a free surface of the fluid. We allow that the plate spans
from x = 1 to x = +1 and lies at a distance D beneath the mean free
surface as shown in Figure 1.
An axes system is taken fixed in the plate with the xaxis in the
direction of the flow. The flow, with constant velocity U, is allowed to
stream over the undulating plate in the positive xdirection or in a left
toright sense.
1. 1 The Motion of the Plate
The deflection of the solid deformable plate is taken as
Y = h(6, ) 4 X + 1 (1. 1)
and it is assumed that these deflections are small with respect to the
chord of the plate, i. e.,
<./ < I < << (1.2)
If we assume that the motion of the plate consists of a traveling
wave of small amplitude, we may write
1
Y = h(x,t)
Mean Free Surface
Figure 1.
The Flexible Plate
Image
Plate Vortex Sheet of
Strength Y(x, t)
Figure 2. The Plate Model
U
U
7
I
I
W r
m I
Y= h(x,t) = (x) cos (x it + 9S 1 x 1 (1.3)
in which f(x) is the amplitude function of the wave, oc is the wave num
ber, W is the circular frequency, and 4 is the phase angle. Upon
insertion of the wave length X
2 = T (1.4)
the period '
S (1.5)
and the propagation velocity V
*V ^= W (1.6)
cx.
the plate displacement can be written as
y = t)= (x) cos t + (1. 7)
Such motion can also be described generally by
y H (x,) e. t (1.8)
if the real part is used for a physical interpretation. It should be noted
that H(x) may be complex.
1. 2 The Plate Model
The deformable solid plate is replaced by a vortex sheet of os
cillating strength. As the circulation is time varying, vorticity must
be shed into the wake in order to satisfy Kelvin's circulation theorem
[21]. These shed vortices are carried downstream with the free
stream velocity U. Hence, the model of the plate and its total effect
on the flow is represented by the plate vortex sheet and its trailing
vortex wake. It is also assumed that the shed vortex trail lies in the
plane y = 0. This is not precisely the case, but the assumption greatly
simplifies the analysis.
In order to satisfy the free surface conditions, an image sys
tem is located 2D units above the plane y = 0. This image system con
sists of a vortex sheet along with its trailing shed vortices identical to
the plate system located in the plane y = 0 (see Fig. 2). Thus, the
combined effect of the plate system of vortices and its image is to
make the free surface appear as undisturbed. As the surface will ob
viously be disturbed, a means must be found which will allow the de
termination of the effect of the disturbed surface on the flow.
1. 3 The Velocity Potential
An obvious method of attack is seen in the use of potential the
ory in the case of nonaccelerated flow. The potential due to the vortex
sheet of oscillating strength over the plate as well as that of its image
may be written down. The same holds true for the vortices in the wake
and their images. Then, a potential function for the effect of the free
surface may be postulated and a partial differential equation found for
its determination. Thus, a total perturbation velocity potential ((x, y, t)
is postulated which takes into account the effect of the plate system of
vortices and its wake, the image system of vortices and its wake in
addition to the effect of the disturbed free surface.
We assume that the velocity vector q of the flow is given by
S= tV +ut, V} I, (1.9)
where u and v are the x and y perturbation components of the velocity
LL = (1. 10)
This means that we have assumed irrotational flow. Upon substituting
(1. 10) into the continuity equation
d;v t O (1. 11)
for incompressible flow, we see that the velocity potential, < must
satisfy the Laplace equation
2 = 0. (1. 12)
Hence, is a harmonic function of the space variables in the field of
the flow.
The potential, ( must also satisfy certain boundary condi
tions. If the free surface disturbance given by
j = (X, t)
is to be a stream line, then
) ) ' + t .
d t ^t Y. d1
If the total velocity vector is taken in the form give
(1. 13) becomes
w n U +on) oy +
which, in the bounds of linear theory, is altered to
, D.
(1. 13)
n by (1.9), equation
0 )
(1. 14)
4 V j 0. (1.15)
The unsteady Bernoulli equation
['. i 4. s wct) y+ 0, (.1t6)
must also be satisfied at the free surface. In equation (1. 16), p is the
pressure at the free surface, P is the density of the fluid, w(t) is an
arbitrary function of time which can be specified by the flow conditions
at infinity, and A. is the external force potential given by
An. .
If the velocity vector q and the force potential .1 are inserted into
equation (1. 16), and if the pressure is required to equal that at infinity,
i. e., a constant, the equation becomes
uU .U1^ ) 1 = ,y D (1. 17)
where g is the acceleration due to gravity. Again, due to the linear
theory, this boundary condition may be simplified to read
+ =  0 (1. 18)
For the third boundary condition the face that the plate is a solid im
penetrable body is used. The corresponding kinematic condition
'()Y V + (1. 19)
must be met. This relation is also linearized to
V" P O + U3 0 (1. 20)
where Vp is the fluid velocity in the vertical direction of a particle ad
jacent to the plate.
It should also be noted that the Kutta condition should be satis
fied at the trailing edge of the plate.
Having set forth the fundamental equations governing the be
havior of the velocity potential, we are now in a position to proceed to
the development of the various potential functions.
Consider the single vortex line of fluctuating strength located
at the origin as in Figure 3. The velocity potential of this vortex line
as given in polar coordinates is
Sr,et) (1. 21)
or in rectangular coordinates
XP(.) Y LT 3 (1.22)
where P(t) is the total time varying circulation.
For a vortex sheet of strength Y(, t) located at a 4 < b, y =
as in Figure 4, the contribution to the infinitesimal element of the ve
locity potential di at the point (x, y) due to the infinitesimal element of
the sheet dE is given by
(d t +t' dde (1.23)
and the potential due to the whole sheet is
(I= C(,t) ta ' (1.24)
V J Q
I 1
(r, e)
lx, Y)
eo
I
rct)
Figure 3.
Single Vortex Line
(x, )
^i^
f=Q. .
4
Figure 4. Vortex Sheet With Special Coordinates
j1
.I
I
X
The circulation around the vortex sheet is given by
(t) / x,t) dX (1.25)
and is taken positive in a clockwise direction.
Since the circulation is time varying, it is obvious that vorticity
must be shed into the wake to satisfy Kelvin's theorem. Denoting the
strength of these vortices shed at the trailing edge of the plate by E(x, t)
as in Figure 2, and observing that they are carried away by the free
stream velocity U, we write
E(,t) = (x to t to) o= A (1.26)
or
E(XX,t)= E 1 1.) ; > 1 (1.27)
UV
The change in circulation around the plate in time dt is
dr dt (1. 28)
dt
and is of equal magnitude and opposite in sign to the vortex
(1,tt)dx d = Udt (1.29)
shed at the trailing edge. Thus,
(rt)d =U (1,t) dt
d( t) d dt (1.30)
f
Using (1. 27), the vortex strength in the wake becomes
r1
/1
Now it seems reasonable to assume that
x,t) = Y(X) e (1. 32)
since the motion of the plate is harmonic. Hence, it follows from
(1. 31) that
E (X, t) = v(Ce) d
U
f
i1
If we recall (1. 25) defining the total circulation around the plate and
use the standard form for the reduced frequency
K = (1.34)
ET
the wake vortex strength becomes
E(Xt) = / e ( e (1. 35)
where
1
r . Y ,) dx (1.36)
so that
re r. (1. 37)
Upon applying equation (1. 24) for the plate vortex sheet and again for
its wake and adding, there results the potential due to the entire vor
tex sheet which is denoted by
+ K eK ta~ t1 
For the image system we write correspondingly
For the image system we write correspondingly
, 2"I
& 2W
1 X
+iKr I K
JI
tat" D Ii
The two potential functions and az are clearly harmonic.
It is not unreasonable to expect the third potential function 3, due to
the disturbed surface, to be harmonic also. Hence, we write
c, ,1,) Re P + a v e93 (1.40)
where
^( 't) ) e L= U 1 ,,
and
S(x, y, t). To investigate the nature of Ip we combine the boundary
conditions (1. 15) and (1. 18) by eliminating 'y (x, t) to get
+ 8 a ( a =D. (1.41)
If equations (1. 38) and (1. 39) are rewritten using
L e
~, = ~r~ e ,
(1. 38)
(1. 39)
Lwt
gE= (PX, )e
16
and substituted into equation (1. 41), it is not difficult to obtain a par
tial differential equation for cq3 in the form
93 + z' P 3L _9_ 9_3
S7Tf F
LFK J c ) + o
1
where the Froude number F is given by
j
F = (1.43)
Tan [15] has solved a partial differential equation similar to
(1.42) in the case of a single vortex of oscillating strength. Hence,
according to Crimi and Statler [14], the solution to (1.42) can be ob
tained by taking an integral form of Tan's results as
3(?,) = __ f (x) X +
I
00
+ K r G(; x, ) d (1. 44a)
tT Fz
e
+I
where
S S(? D) ee L S( P) r s_
0
4 D e + Dz e +
T e.s[jZD LX + D) eS4. zo J(x (1.44b)
and
(ss )(s ZK)S K'
(Cs53)( S) = S + Zk)S + (I.44c)
We refer to Appendix A for a more detailed explanation.
The terms of (1. 44) involving the double integration give a par
ticular solution to (1.43), and the remaining terms containing the D1,
D2, D3, and D4 are the complementary solution. Obviously, the solu
tion is incomplete in this form since we do not yet know the Dj. The
constants Dj will have to be determined by the free surface flow condi
tions at infinity.
A detailed analysis of the free surface flow conditions is pro
vided in Tan's paper [16] for the case of a single source of fluctuating
strength, and a similar analysis is outlined in the paper of Crimi and
Statler [14] for a vortex sheet of fluctuating strength. The method em
ployed was first used by Rayleigh [22] and constitutes the introduction
of a small fictitious dissipative force / grad I >/O. The results
are then examined as 7 tends to zero in the limit.
Crimi and Statler have shown that the constants Dj may be as
signed to provide the correct free surface flow conditions depending
upon the following combination of the Froude number and the reduced
frequency: kF2
The values of the Dj as so determined are reproduced in tabu
lar form as a convenience in Table 1. Appendix A offers an analysis
in some detail leading to these constants.
In examining the table we note that for kF2 > 0, S3 and S4 will
be real and positive so that at least two harmonic wave trains always
exist on the surface. Their wave lengths are 27T/S3 and 27T/S4. They
propagate downstream from the plate and cannot exist physically up
stream of the plate.
For kFZ > 1/4, the roots S1 and S2 are complex conjugates and
therefore produce damped waves which do not influence the flow at
great distances from the plate. In case kF2 < 1/4, S1 and S2 are real
and positive. The group velocity of the wave arising from S1 is less
than U, requiring that this wave trail the plate. The S2 wave has a
group velocity greater than U and will lead the plate. Thus, there are
two harmonic waves propagating upstream for 0 < kF2 < 1/4, one
leading and one trailing the plate.
In the special case kF2 = 0 (steady motion), the only roots
present are S1 and S3. The solutions combine to give a single standing
wave of wave length 27TU2/g downstream of the plate. Let us refer to
" o i '*
9')
a a NI O
( I
m 0 m
z*4
H 0
X04
0)
>
0 N
*4
a0
4 W
W >o
ua
^! <
i<
"4
rl
0
0
H
a) a)
> O
u.. O $ 
LL0 0 0
0 01
0U
_ _ _ p4 _
Y Y N
N~ f.v lt
bO
0
0
0
d
0
a
bD
4.
0
0
CO
4.
o
04
0
nl
4
0,
,jI"
C
n
= Ito
wi
the paper by Isay [23] for a solution to the problem of a flat plate in
steady motion moving near the free surface.
When kF2 = 1/4, the integral associated with G( ; x, y) does
not exist. The roots Sl and S2 do exist, however, but the group ve
locities of the associated waves are the same, creating a resonance
condition. Thus, for a given flow velocity U and, hence, for a given
Froude number F, there exists a critical frequency, critical, for
which (kcritical)F2 = 1/4, and for which no solution exists within the
framework of the present formulation.
1. 4 The Integral Equation
The last step in the actual formulation of the problem requires
that the boundary condition on the plate be used. From the last section
we see that with the use of equations (1. 38), (1. 39), and (1. 44) the total
potential given by
I (x',) = Rei. e LWt
is known except for the vortex strength distribution Y( ). If the
boundary condition (1. 20) relating the velocity of a fluid particle near
the plate to the motion of the plate is used, there results an integral
equation for the determination of the vortex strength distribution. If
we keep in mind the type of motion which is to be imposed on the plate,
namely that given by equation (1. 8), then the boundary condition (1. 20)
becomes
= U H'x + L K H() (1.45)
where
H() dH
dX
Upon substituting the determined value of in (1.45), there results
the integral equation
1 1 f
so r) xf ii r a o
2z J1 xl (.x)+ ,+o F poJ +
+ vl 'x + LKH(~ o= IX $ (1.46)
where the Cauchy principal value is to be taken for the integral on the
lefthand side, and the quantity is found to be from (1.44b):
00 Z s scx 
,lzo D(x ) 3 s,z0 ,(x S)L( ]
4 SD, e 1 D, eQ
In addition to the requirement that Y(F) satisfy the integral equation
(1. 46), X ( ) must produce a velocity field which allows a finite ve
locity at the trailing edge of the plate.
If D is allowed to approach infinity in the integral equation,
another integral equation results, governing the value of (r ) for the
case of infinite flow about the oscillating plate. In this case the equa
tion becomes
ifT ) Z xKP e d 4 G/ x)
which is seen to be equivalent to the equation investigated by Sbhngen
[18] and applied by Siekmann [8] to the study of sea animal locomotion.
With the derivation of the integral equation (1. 46), the formu
lation of the problem is complete.
CHAPTER II
SOLUTION OF THE INTEGRAL EQUATION
As mentioned in the Introduction and demonstrated at the end of
Chapter I, the formulation of the problem of an oscillating hydrofoil or
thin plate in infinite flow (Doo) leads to a wellknown integral equa
tion. The presence of a free surface obviously has complicated the
integral equation considerably, as evidenced by equation (1.46). An
approximate method of attack seems to be the most expedient way of
solving this equation. It is possible to make the final results as accu
rate as desired through the use of highspeed computing techniques.
2. 1 Form of the Solution
Crimi and Statler [14] have been successful in solving an inte
gral equation much like (1. 46). Equation (1. 46) differs from the one
solved by Crimi and Statler in that an infinite number of terms enter
due to the plate boundary condition.
We first assume that the vortex strength distribution X(e) is
given by the Glauert trigonometric series
Y (e) = 2SU aI co+. CLsix eX ,9 (2.1i)
'V's f
where x = coso 1 \ x v+1, O ,8 27T, and the c2 may be
complex.
It is easy to find the total circulation around the plate with this
form of the vortex strength distribution. From (1. 36) and (2. 1), f is
calculated as
1 0
Sy (,X,)dX, = o ,e) sie de
S2 U [ o co e 4 Qc stV e siw e de
z=7rvc U) (2.2)
Substituting (2. 1) into the left side of the integral equation
(1.46), there results after the integration
X C CO Inc) (2.3)
1
Now, if the righthand side of (1. 46) can be expressed as a Fourier
cosine series, like coefficients of cos ne can be equated to obtain an
infinite set of inhomogeneous algebraic equations involving an infinite
number of unknown Q, The expansion of the righthand side of (1.46)
involves much manipulative work. Fortunately, the results of Crimi
and Statler can be used, as the only differing terms are incorporated
in the inhomogeneous terms of the infinite set of equations:
^ C= Z' C n + a o0, 1,2... (2.4)
Crimi and Statler have considered the motion of a hydrofoil with two
degrees of freedom, pitching and plunging, whereas, the inhomogene
ous terms of (2.4) reduce to only two, r. and r,. The oscillating plate
dealt with in this work has an infinite number of degrees of freedom.
Thus, as might be expected, an infinite number of inhomogeneous
terms are involved.
The technique used in expanding the integral expressions in the
righthand side of (1. 46) deserves mention. Consider the integral
00
I1 eJ (ix4) r (2. 5)
2 (Tr f + 4 D"
where x = cos 9. Now in transform theory, the Laplace transform
of a function F(s) is
M) = 3W(s)} f e~ FCs) ds ,
and in particular
a IVs Qs (2.6)
CLI + CLL
In the fraction of the integrand in (2. 5)
(cos e +
(zD), + (cose +
let a = (cos e ) + and q = 2D when applying (2. 6). Thus,
00
Cos f ZDS Ls(J+Cose) +js(t+cos e)
cos( e = e e (2.7)
(Zo) +t (f+cose)z 2 1 7
0
Letting
_cose+ . b cos ie
(2 D)' + (+ +cos e) 2
n=1
and employing (2. 7), the bn Fourier coefficient is given by
o S 7 S r Scos
b e e Cos >e de
TrT
f0 0
7T
Ls, r s cos e
0
00 DS +Ls"
= e T (S) e
In the last step the integral formula
J(S) L e cos 7
7r
7TT
iS cos 8
7 f
77"
0
CoS 'r
os v e de I cIS
 (i dS S
9e de
ae de
for the Bessel function Jn of the first kind and order n has been used.
Inserting the total circulation T from (2. 2), the integral I1 can now
be written as
(2. 8)
(2. 9)
27
00
I LKU ( +. e b cos n d0 (2. 10)
n=1
Finally, if bn from (2. 9) is placed in (2. 10), and the ) is integrated
out, Il can be written as
z,
4 b h cos vIt 9
1A I
(2. 11)
where
00
f 2DS
0
',(s, [
kS
S
8+S
ds 
Other of the integrals are less difficult to expand.
Consider
the integral
2 2= T
Z, Tr
S00 ik 1)
e
IT
where x = cos 9 Assume that
1 
2 ~ 2
Then,
C = _ U
L7
7r
( e COS n e
ff Cos q 8
o + ( e CoSne
A"0" cose
of
(2. 13)
dyde .
Interchanging the order of integration and integrating with respect to ,
we obtain
00
C Z1aU (a ) i)
I
e ( 1)
(2. 14)
(2. 12)
I, = U(a
When expansions for all the integrals on the righthand side of (1. 46)
are found, the like coefficients of cos n9 are equated, giving expres
sions for the Cmn terms of (2. 4). These are listed by Crimi and
Statler and are reproduced in Appendix B for convenience.
Return now to the inhomogeneous terms of the infinite set of
equations. If the "downwash"
 [H'(x) +. i\ H (A ] eP
is assumed equal to the Fourier expansion
cU e 4 +Zcosn e (2. 15)
then it is easy to see that the inhomogeneous terms of (2. 4) are
Ar
r = n I, 3, ... (2.16)
Note that the An are assumed known, as are the Cmn. The integrals
involved in the Cmn cannot generally be expressed in closed form, but
they may be calculated for various flow conditions by utilizing a high
speed computer.
We are now in a position to consider the infinite set of equations
oo
s.a = C aN + r f (2.17)
seeking a solution for the 0. 's.
2. 2 Solution for the Infinite Set of Equations
Some of the fundamental theorems dealing with a system of
equations like (2. 4) are presented in the book by Kantorovich and
Krylov [24]. In certain cases it is possible to show that a solution
exists and is unique. Such is not the case here, but we expect that a
solution exists due to the behavior of the Cmn and the r for increas
ing m and n. The Cmn decrease rapidly when either m or n increases
or when both m and n increase. In the particular example to be con
sidered in Chapter IV, it will be shown that the t, decrease also with
increasing m.
Assuming that a solution does exist, the procedure leading to
an approximate solution is as follows [24]. We first find the solution
to a reduced finite system of equations
m C + r 0 2,..,N (2. 18)
7n=o
and then repeat the process for N + 1. This, of course, introduces an
additional unknown, ON but at the same time limiting values for Oo ,
., CL . ,MN become apparent. It develops that only a finite num
ber of the am will be needed to define the lift and moment on the plate,
but the thrust involves an infinite sum of the m. which converge very
rapidly. Also, it becomes apparent that approximately N + 4 complex
equations of the set (2. 4) need be solved to ascertain N of the am to a
reasonable degree of accuracy.
CHAPTER III
THE EFFECT OF THE FLOW ON THE PLATE
In this chapter the forces and moments acting on the oscillating
plate are discussed. Due to the unsteadiness of the motion, a pres
sure difference results between the top and bottom of the plate. This
pressure difference is not constant, but varies with time and the x
coordinate along the plate in the chordwise direction. The first task
is to derive an expression for this pressure difference distribution
along the plate.
3. 1 Pressure Distribution
If the pressures at the top and bottom sides of the plate are de
noted by p+ and p respectively, the pressure difference is given by
a p = p P ,p (3.1)
where it is assumed that the Ap will be positive in a sense implied by
equation (3. 1). This is in conformity with the negative direction for
the downwash assumed in deriving the integral equation (1.46) govern
ing the vortex distribution along the plate.
Considering the velocity vector q to be given as in (1. 9), the
unsteady Euler equation
___ u\ iL vI p
t + V  (3. 2)
may be linearized to read
S+ U (3.3)
^t T < {\
Writing the last equation for the top side of the plate,
U + :1 P'
and for the bottom side,
M _j = P_
Yt >KC X
and subtracting one from the other, we obtain
+(') 4 j C .) 1 (3.4)
^t ^X C X
Now u+ u~ is exactly the strength, X(x, t), of an element of the vor
tex sheet which we have allowed to replace the plate. Hence, we can
rewrite (3. 4) as
a + L 1 A (3.5)
S^x e x
If the total vortex sheet (x, t) is divided into bound vortices
(x,t) and free vortices (x,t), as in the paper by Schwarz [11], so
that
u t) i *(x,t) + E*Cx (1 (3. 6)
we may substitute this expression in the second term of (3. 5) to get
SX + ( \ _i aP (3.7)
T6 X
Now, according to the KuttaJoukowski law
(3. 8)
Using this in (3. 7), the equation becomes
E"
= O (3. 9)
giving a relation between the total vortex sheet and the free vortices.
Integrating (3. 9) for
E (x,t),
x
n (dt) i n  r l 3t) w
and inserting the result in (3. 6), we obtain
and inserting the result in (3. 6), we obtain
(x,t) + 1
v
; 1 ;x < I J
aLDt) f
In Chapter II we assumed the vortex distribution Y(x, t) to be
given by the Glauert trigonometric series
C, cot e
2
L t
QKs;v\ v\e
o7
4 E
^%z I
where x = cos e Placing this expression in (3. 10), there results
~e(t)= Ke,t) + c) E
ero
L+ t eif + I
&
+21w I
(3. 10)
Y(x, t) =
1Ye,t) = zu
= (e, t)
(3. 11)
e
L / Cot^ Sc v (p dq>
aK Si1 n s;I( d
Carrying out the indicated integration for I, and 12, we obtain
*x
I,= ^o( i ne
Z LQ s~iv\(j1 a sN1(v\j j1)e 1
SYl ,
hr
giving for t (e ,t)
Y*(9t) =
S(et) + 2 iL e
+ (3. 12)
Finally, after a little manipulation, the pressure distribution Ap(e, t)
is obtained from (3. 8) as
ZCV P Liot f co
ap(et) = zl e ao cot
+
ct,
00
 n \ wzn
Cn2'
 a_ + 2(2az+0tc)(1
hI
It is pertinent to note that this expression reduces to that de
rived by Siekmann [8] for the case D0 (See Appendix C.)
3. 2 Lift and Moment
With the expression for Ap(e t) given by (3. 13) at hand, it is
easy to compute the lift and moment acting on the plate. In computing
the lift, the pressure should be allowed to act in a normal direction to
the surface of the plate (which is constantly changing), but as small
ALt (I + YG
) J Y1 10\ (3. 13)
deformations have been assumed, it is within keeping with the linear
theory to assume that the pressure acts normal to the xaxis. Hence,
the lift is given by
L p(X, t) dc. (3. 14)
J1
and the moment by
M = A p(x,, ) x (3. 15)
J.I
where the moment is considered positive in a clockwise sense and is
calculated with respect to the origin, x = 0, y = 0. Inserting Ap from
(3. 13) and changing the variable according to x = cos E we may
write
7T
L = o v Let) s;y e de (3.16)
o
and
M = f (0,tg) s;viecose S (3.17)
0
x
It remains now only to use the expression derived for X (G t) from
Section 3. 1 and carry out the indicated integration with respect to e.
The resulting expressions for the lift and moment are respectively
L 2r e ( + and (s (3. 18)
and
M, TUe )I (3. 19)
In calculating the moment, the pressure difference Ap has again been
assumed to act normal to the xaxis, and the leading edge suction
force [8] has been neglected.
3. 3 Drag and Thrust
The drag or thrust experienced by the undulating plate will re
sult due to the xcomponent of the hydrodynamic pressure difference
A p. It can readily be seen that the horizontal hydrodynamic force on
a plate element of projection dx on the xaxis is given by
Re (ap] Re dx. (3. 20)
The real parts, denoted by ResU, are taken because nonlinear terms
enter the product which involve mixed expressions in a. Thus, over
the whole plate the hydrodynamic force in the positive xdirection is
given by
1
D = Op+ Ds = e [Ap }Re }dX (3. 21)
1
where Ds is the leading edge suction force.
The plate displacement Y(x,t) given by equation (1. 8) is an even
function of e where x = cos due to the fact that the plate is very
thin, thus allowing a point on the bottom side of the plate to be repre
sented by the same coordinates as the adjacent point on the top side of
the plate. Hence, Y(x,t) can be expanded in a Fourier cosine series
of 9 as
y(it) =
Lot
where
= H () Cos 'e de
+flu cosrhe) ] e L ,
TT
77
= H G) cosne de .
"
The slope, .Y can also be expressed as a Fourier cosine series
since the plate is infinitesimally thin. First, note that from
y(e,t)
= (e) e
there results
(e,9t)
X
and as x = cos e we have
where
de
de
Now we can write
00
77"
f H (e)
o4
Cos e*A
S~v\Q0
CLt
(3. 26)
(3. 22)
(3. 23)
(3. 24)
 I _
s;nG ~G
H e
0
(3. 25)
0(e,t'
where
C.h
de
(3. 27)
D
O
2
SO [ +
There is obviously a relation between the C's and the B's. This
follows from (3. 22) with
= 2 (3. 28)
Upon comparing (3. 28) and (3. 26), we get
C C = 2 (3. Z9)
or
00
C 2 .(Zni) + J 01, 2... (3.30)
2yt Z ZwA+ I
L1Z.
00
CZ 2 1(2n 4Z) z+ f, Z ,... (3.31)
In Section 2. 1 it was assumed that the portion of the downwash given by
[ H C')O + LK HCx eWut (3. 32)
was expressible as a Fourier series like
L0 + A os "n e (3. 33)
This assumption, together with equations (3. 22) and (3. 26), allows the
conclusion that the following relation must exist between the A's, B's,
and C's:
h = C + L k (3.34)
As only the real parts of Ap and are used in computing the
horizontal force D, the notation is best altered to denote this fact.
Following Siekmann [8], we define
38
*t i ~L t 
An e (AV + A,) e = A, L A (3. 35a)
Liot ,, t , _< 1
 II
8 e. = ( B,+ S A e. = 6 + L y, (3. 35b)
aL.3', iot 
where
A = AL Cos Lt s A (3.36a)
II II
A = A" Cos wt + A sv (3. 36b)
and so forth. Then, for the real part of ./ we have
i0
Re c + Cos n (3. 37)
?7 V I
nsi
The notation will be simplified somewhat if the pressure dis
tribution Ap( ,t) is written from (3. 13) as
ap(,t) = 2 z a.ocot + d1 Sie + d, s. ne (3.38)
Li
where
d, = I K La30 a, + _3 (3.39)
da= + Q. + (zC Q.1)lvi .(3. 40)
Analogous to equations (3. 35) and (3. 36), we define
dae O ( 4 dt) e L = d 4 ( (3. 41a)
d, e = (d' + c ^)e ^ d, +td, (3.41b)
where
Oa Cos Wt,
+ S;i Lot .
Then the real part of Ap(e ,t) is given by
Re p7l =
Z'r '
2I o, cot a + d. s;ye
Z
The portion of the horizontal force D denoted by Dp follows
from integrating
77
2 e
0
0o cot +
Z
d, sI; +
d 5 8
+ d:
}'z
sin V, e *
C0 cs
'',z^0 00
siVeI de. (3.44)
Multiplying sin e with the first braced quantity and employing the
trigonometric identity
we may write (3.44) as
we may write (3. 44) as
D= 2e
07
o
1 [cos (n) cos(+))oe ,
C co siv+e + 4 S;n e +
+ dL [cos (.vie cos('it1)r6
31: Z
O 
+ C coM C me de
m 1
SI
0.1h
CII II
c, = aCz CoS (t
(3.42a)
(3. 42b)
00
+
sI19 ne
(3.43)
{ I+
2
(3.45)
Using the idea of the Cauchy product in the integrand of (3. 45) and
interchanging the order of integration and summation, Dp may be
written as
C Co d cos (.1)e cos (nIcI)e d e +
I 0
+D(p.0 Cvn Co4 i2i on o ',,e o " J s.n
0
To. T0
So I ( cc3.46)
t d C e V Yos a +
t I Z [, COI ( Ni 1) 9  COS C ni 1 e2 c  O Csa
Ir = (3.46)
The definite integrals in equation (3. 46) are evaluated as
T"
o
cos ( &i)O de a= o 
0
(3.47a)
(3.47b)
(3.47c)
.icotI s5ie cos tye de .=
7
cot Si e CoS y e d =
0
of Si 9 Cos rriede
77
cos (ni1) cos ede
C5S(I+1) e cos 'vO e de
7rT
)i = h7
n? = 'i+ 7
Using these definite integral values in equation (3. 46), we write
Dp as
P =r \U i I5
Dp = 70~(U Co(ao
00
+ tEj i,
dl
2
C
yt*1jC,
(3.48)
Recalling equation (3. 29), the last term in (3. 48) may be
written as
oo
 n .
)1Z
(3. 47d)
(3.47e)
"m = "1
.W.= ?
v>n=2
>v2
o
Il
0
O
(3.47f)
(3. 47g)
(3. 47h)
(3.49)
From equations (3. 39) and (3. 40), keeping in mind the notation indi
cated by equations (3. 41) and (3. 42), we may write
ii
d = , K( o + 0 T ) (3. 50)
and
+ n 1 2 J (3.51)
Using (3. 50) and (3. 51) in equation (3. 48), the horizontal force Dp is
written as
+ +
~ 1  , \
The suction force due to the singularity at the leading edge of
the plate is discussed by Siekmann [8]. The contribution of this singu
larity to the horizontal force acts in the negative xdirection, hence
1 Yna B + X k( a C^ + C// > +
0 "
the terminology "suction force. Its magnitude is
Os = 2T ( 1 C .o (3. 53)
Therefore, the total horizontal force acting in the positive xdirection,
or the drag, is given by
0 = D D = Dp 2r ( .
(3.54)
The thrust is clearly the negative of this quantity and is written from
equations (3. 52) and (3. 53) as
T o a.Co + )+
2 O
vI
Y, 5_' G
, ( a, 1 I1 ") I/
yi2.
I (za.+ a, ) Z :*' (3. 55)
?2._
Instead of this expression for the thrust, it is more meaningful
to calculate its time average value over a period Z. The time aver
age value of a function f(t) is defined as
S (3.56)
0
Because of the

*
a,
relations (3. 36), need for the following will arise:
o Cos Ot dt = I
2
I siV Ot cos t d t = O (3.57)
o
00
h>iZ
Keeping equations (3. 36) and (3. 57) in mind, the
various products in the time average value of the thrust,
T = T(t) dt
are
are
= (' Ca
 i
2 K;
II
 Co ')
S'
:3 )
Bra .
0f
S'(aIt ) B'
2.
nzz
2 M2
B, dt
00
 .i (1) (o~ a 8 3 ).(3.59e)
2 .
,I __
C'
(3.58)
Z"
0
Jo
r
0
czlu
(3. 59a)
(3. 59b)
(3.59c)
B ld
1 )1
0 200
o k=2
Si0
0
9, Q 07)
o >=1
(3. 59d)
where = 2 7T/ .
 .,) d t =
Inserting these representative values in (3. 55), the time aver
age value of the thrust is given by
S7 a. C + C,
l tru c CT is efne a
00
o C
2. sZ
aC ( (3. 61)
77 9 a
Substituting (3. 60) in (3. 61) gives the thrust coefficient in terms of the
Q's (the solution of the infinite set of algebraic equations), the B's (the
Fourier coefficients of the deflection function's expansion), the C's
(the Fourier coefficients of the slope function's expansion), and the
flow parameters U, and k.
CHAPTER IV
NUMERICAL EXAMPLE
In accordance with the idea of representing the motion of a thin
flat fish with a twodimensional thin waving plate, we consider an ap
propriate displacement function and calculate numerical results for the
thrust coefficient in this chapter. It is desired to compare these re
sults with those of Siekmann [8] for the case of swimming at infinite
depth so as to effectively isolate the influence of the free surface on
the thrust.
4. 1 Quadratically Varying Amplitude Function
Photographic analyses of swimming fish indicate that the mo
tion of the fish is perhaps best represented by a quadratically varying
amplitude function like
Y = h (X,1t) = (C e C + CXz) Cos (oxcWt), (4.1)
where the c's are constants. The notation here is in conformity with
equation (1. 3). The phase angle S has been taken as zero.
In the last chapter the thrust coefficient was shown to be a func
tion of the Fourier coefficients, Bn and Cn, and the 0. which are the
solution to the truncated set of equations represented by equation (2. 18).
Also, the inhomogeneous terms of this set of equations are related to
the Bn and Cn coefficients through equations (2. 16) and (3. 34). Thus,
we first calculate the Bn and Cn coefficients associated with an ampli
tude function like that given by equation (4. 1). Corresponding to the
notation
Y(x,t) = H () e
H(x) is given by
H( x = (co + c1x + xC ) e
With x = cos e we have
H e) = C c cose + ccos e
Therefore, from equation (3. 23) we have
(4. 2)
(4.3)
_IT
Lox cos .
S = 2 (C C cos + c, cos e) a cos vY de. (4.4)
Tr
0
Using the integral formula [25]
.ey 6 cosa5
o<) e cos e de ,
7r
o
there results
i c, J J (.^1
where the identity
cose cos Me =  [cos (i+ e 1 cos (nr) e
(4. 5)
(4.6)
has been used. A similar calculation starting with equation (3. 27)
yields the following for the Cn:
C L r[ [ (O)7 , (T 
^.Z7r[ c, S] 3o( + zc,[r1o 
Iz VI [ U, +1
3IT(.] CW [ T^ T.( ] ) (4.7)
To solve the truncated set of algebraic equations for the Q,,
we will separate the set of N complex equations into two sets of N real
equations each. This operation is necessary because the coefficients
and the inhomogeneous terms are themselves complex. We have al
ready indicated the complex nature of the Bn and Cn coefficients
through the primed notation of the last chapter. Adhering to this no
tation, the real part of the Bn is
= 1 Zc(CoT () + c,[T.( T C) + T.a }] (4. 8)
0 O, 2, o....
S.Ef, c,[() o (a) T (4.9)
The imaginary part of the Bn is
a K C J I (4. 10)
1n+
c, T c[_)]} + (4.11)
1 J 'i ., 5Z...
For purposes of calculation it is desirable to eliminate the possibility
of a negative order for any of the Bessel functions. Using the following
recursion formula [26]
(o) + (X = JT(oc)
as many times as necessary, there results
S ) + 2 (c c.) T + (4. 12)
So 9~ 0 ,...
8 = (i) 2c C ) T. ) (4. 13)
" 1 3, ,5, ...
for the real part of Bn, and for the imaginary part
Ii J = o, z,v,..
B; (1C 2 ( ,,, n(T1 C1
Similarly, the real part of Cn is
u+z
S = (I) a c ('C +1C ( (n + +) )T
Sn ~l 0 Z
cif; afrwc c,, + C K,)] J^ Oc)
and the imaginary part reduces to
C= (1)z 2 iTr C+ C ) T) +
C = C 2 c +1) )
Next we discuss the calculation of the An coefficients.
equation (3. 34) there results
A' C x <
V% /
S(4. 15)
S1, 3, 5, ..
(4. 16)
(4. 17)
(4. 18)
o, ZIi, ...
(4. 19)
From
(4. 20)
and
AnI/
n, C
+ k B
(4. 21)
4. 2 Solution of the Algebraic Equations
The set of N complex algebraic equations given by
N
a J cM M n + r V, 0, 2 N
M=o
is separated into two sets, using the following notation:
Cr = Cr + iCC
m +
(4. 22)
(4. 23)
We obtain
01
N
= C in + C ) + ; .1,v,..,n/ (4. 25)
I ii
To solve these equations for the (, and Qa, we use an itera
tive method, whereby we assume some arbitrary values for the un
knowns in the first equation of the first set and compute an initial value
for the first unknown to the left of the equation. This value will be
used in the second equation and so on until initial values of the first
2N unknowns are found. Then the equations are solved again for a
second set of values for the 2N unknowns, making use of the first set
for starting values. This procedure is continued until the difference
between values for the unknowns obtained from two successive itera
tions is less than some preassigned value. The computer program
used for this operation is designed to provide this check for accuracy
in solving the equations. This iteration method was found to be effec
tive for computing the 0a and aM for all values of the reduced fre
quency greater than the critical frequency. For frequencies in the
range 0 k < critical, the method fails. This failure is noted for
depths that are small compared to the halfchord only. The method
is satisfactory for all frequencies at large depths. Note that critical
is the value of k for which no solution exists.
For the range 0 < k < critical and for D < 4, the Crout method
[27] was used to obtain a solution for the equations. The Crout method
will give satisfactory results for all frequencies, but is somewhat
slower in regard to computer time than the iterative procedure de
scribed above.
All calculations were done by utilizing a highspeed digital
computer. Values for the real and imaginary parts of the Bn and Cn
coefficients are given in Table 2. These coefficients are not functions
of the reduced frequency k as are the An coefficients. The An values
can be obtained through the simple relations (4. 20) and (4. 21) and are
not tabulated here.
TABLE 2
REAL AND IMAGINARY PARTS OF Bn AND Cn
/ # I //
n Bn Bn Cn Cn
0 0.04084412 0.02390768 0.10066456 0.08960791
1 0.03316639 0.02193889 0.12262104 0.08028761
2 0.04143261 0.00205140 0.03433178 0.13348571
3 0.01402839 0.02095180 0.04310939 0.07208199
4 0.00361431 0.01181532 0.04983858 0.00888488
5 0.00574890 0.00155564 0.01419494 0.02244056
6 0.00142237 0.00204627 0.00765042 0.00778154
7 0.00058169 0.00061490 0.00287355 0.00211474
8 0.00019193 0.00013840 0.00049327 0.00082705
9 0.00002839 0.00004819 0.00019737 0.00009970
10 0.00001023 0.00000513 0.00001779 0.00004045
11 0.00000083 0.00000189 0.00000728 0.00000284
Note: a = 7, co = 0.023, c, = 0.042,
c = 0. 034.
4.3 The Resulting Thrust
The thrust coefficient at infinite depth is independent of the flow
velocity U as shown in Appendix C. For finite depths (D < 4) the flow
velocity enters the calculations through the Froude number. As the
time required for the execution of the computer program is fairly
great, only one flow velocity was considered. All the results reported
here are for a flow velocity of U = 3 units per second.
With the Q., Bn, and Cn in hand, the thrust coefficient may be
computed from equations (3. 60) and (3. 61). The first attempt to calcu
late the thrust coefficient was made using a depth of D = 50. 0. It was
correctly presumed that this depth was large enough to simulate an
infinite running depth. The results obtained are shown in Table 3 and
Figure 5. Also listed in Table 3 are the results obtained by Siekmann
[8]. The infinite sums involved in the expression for the thrust were
found to converge fairly rapidly so that six of the a. gave results
comparable to those obtained by Siekmann at an infinite depth. The
first six a. were computed to within 1 per cent accuracy. Looking
at equation (3. 60) for the time average value of the thrust, it is seen
that the last two sums involve only the first two a., but an infinite
number of the Bn. As the Bn are fairly easy to calculate and require
little computer time, these two sums are evaluated for n = 11. For
higher n, the Bn are so small that their contribution is negligible.
Several attempts were made, using successively smaller depths
beginning with D = 2. 0. Some differences were noted in the thrust co
efficient for depths just below 1. 0, but the depth must be smaller than
this for a clearly discernible deviation from the thrust at infinite depth
to be observed. Indeed, to register the effect of the free surface waves
on the thrust for the full range of reduced frequencies of interest, it
was necessary to decrease the depth to as little as D = 1/4. The re
sults for this depth and for several values of the reduced frequency are
shown in Table 4 and Figure 5.
The asymptotic behavior of the thrust in the neighborhood of the
critical frequency requires the observation that in the interest of
000000000000
I I I I I I I I
H
u
rA~
0"
cI~
H
000
I I I
n Noo rM\
"o so o Moo n o o t I
000000
0000000
00000000000000
O 4 O0000000000
O0OO000NLL4O
0000000c000000
00o r 0 o00000000000
6 6 6 6 6 6 6 6 (\ C4 tn so 06
E 5
2c
Uc
O o
C1
H
c. = 0. 023
c, = 0. 042
cz = 0. 034
critical = 0. 8935
0.06
0.04
D = 50. 0
0. 02
D = 1/4
0. 00
0.02 I l
0 2 4 6 8
k
Figure 5. Thrust Coefficient Versus Reduced Frequency
10
forward motion with as little effort as possible, the "fish" had best not
swim at this frequency. This singular phenomenon also indicates that
a hydrofoil experiencing vibration with an infinite number of degrees of
freedom at just the right frequency might expect to encounter a high
drag coefficient for the proper flow velocity. Again, note that the
critical frequency is a function of the flow velocity alone. It is not a
function of the depth.
The interaction of complicated wave trains on the surface pro
vides an interesting oscillatory behavior for the thrust coefficient at
frequencies in the range of 0N k < critical. This effect is shown
separately in Figure 6 where the scale has been lengthened. Note also
the wave drag [28] for zero frequency. At infinite or large depth the
thrust is zero at zero frequency, as required by D'Alembert's paradox.
CT
co = 0. 023
c, = 0. 042 critical = 0. 8935
c, = 0. 034
S= T U=3
0. 02
0. 01 
0.00 D = 50. O0
0.01 /4
D = 1/4
0. 02
0 0.5 1.0
k
Figure 6. Thrust Coefficient Versus Reduced Frequency
for Small Reduced Frequencies
CHAPTER V
SUMMARY AND CONCLUSIONS
In this study the influence of free surface waves on the swim
ming of a thin, flexible, twodimensional, oscillating plate was inves
tigated. The motion of the plate was assumed to approximate the
swimming action of a thin flat fish near the free surface of a body of
water. The fluid was assumed to be perfect and infinitely deep.
The propulsive effect is due to the motion of the plate, consist
ing of traveling waves emanating from the "fish" nose which move to
ward the tail with increasing amplitude. The waves are harmonic with
respect to time.
The "fish" is replaced with a mathematical model consisting of
a vortex sheet of harmonically varying strength. Such a model re
quires that vortices be shed from the tail of the plate and carried
downstream with the flow. These wake vortices are adequately taken
into account by the theory presented. The assumption that they always
lie in the plane of the oscillating plate simplifies the theory, but it is
still an open question as to whether this assumption is the correct one.
Actually, photographs of an oscillating model indicate that the wake
vortices deviate in their downstream path from the plane of the plate
by perhaps as much as a quarter of the chord at a distance of only a
chord length behind the plate. Taking into account the fact that the
vortices in the wake are discharged at the tail of the plate with a ve
locity vector which is not in alignment with the free stream velocity
means that the vortices will continue to diverge from the plane of the
plate, or y = 0.
In addition to the plate system of vortices and its wake, an
image system identical to the plate system was postulated to lie just
above the surface so as to make the effect of the whole model on the
surface null. This sort of model was postulated to allow the use of
an exact potential theory. The velocity potential of the plate system
and its image was written down in terms of their unknown vortex
strength distribution. Then a third potential function was introduced
which took into account the free surface waves. This function was in
vestigated through a partial differential equation resulting from the
application of the free surface boundary conditions. Finally, the
boundary condition concerning the impenetrability of the oscillating
plate was used to obtain an integral equation for the vortex strength
distribution. The integral equation was solved by the assumption that
the vortex strength distribution could be represented by a Glauert
trigonometric series. This introduced a system of infinite algebraic
equations. The pressure distribution and the thrust coefficient are
functions of the solution to this system of equations. The equations
were solved by the approximate scheme of truncating the set and solv
ing repeatedly while increasing the number of equations solved until
values for the first N unknowns become apparent. The thrust coeffi
cient was then evaluated for a flow velocity of U = 3 and several dif
ferent depths.
The results of the calculations for a quadratically varying am
plitude function with coefficients which are thought to approximate the
swimming of a fish indicate the following conclusions.
1. The theory produces accurate results for the thrust coeffi
cient versus reduced frequency variation for a depth (D = 50. 0) large
enough to simulate an infinite depth when compared to figures obtained
by a different method.
2. The influence of the free surface is not a marked factor for
relatively shallow depths. One must decrease the running depth to
about onefourth the halfchord in order to note a finite difference in
the thrust from that observed at an infinite depth.
3. The influence of the free surface waves causes a varying
difference in the thrust coefficient. At high reduced frequencies the
difference is greater than at low frequencies, provided k is greater
than the critical frequency. At frequencies between zero and the criti
cal frequency the thrust (or in this case, drag) coefficient varies in an
61
oscillatory manner. The point of zero thrust does not occur at a re
duced frequency equal to the wave number of the plate motion, but at
a slightly higher value for small depths.
4. As a natural consequence of the theory used for this study,
the wave drag for zero reduced frequency is obtained without difficulty.
This might be of interest to designers working in the field of hydrofoil
craft equipped with slightly bent wings.
APPENDIXES
APPENDIX A
THE FREE SURFACE POTENTIAL
To solve the partial differential equation
Sx
I,
4 1
P1z
Io
[
iF
,rFL
(A1)
e ) d+l ,
CK 0)1 + D12
we proceed as follows, using the operator notation
M *~
A particular integral of (Al) is written in symbolic form as
1
c3 Q .S'}
S()1 o)
"(X f)? + 'Dz
So K
lK' e (xl d()
1
Z 2 il & k2.
Fz
zP3
^&x
(A2)
where
(A3)
f2 ( t j ) =
(A4)
k~3
64
Now it can be established that
)4 (D
I x F + L(tZD)2
dS, yzo
Then it follows that
= Re
0
00
0^^D
S;ih S(XFl
oo
Se s(A 6
0
and that the particular
(see [29, p. 103])
P
cp3
1
g( ,.s')
KP
ZTr 
P
integral P3 may be written symbolically as
Se e(A7)
o0
We also observe that for an equation like
a.x+ b4
the particular solution consists of terms of the type
P 1 a.x+ b,
00
= L eS
o
dS}
I
(x W)Z+(20)
P
so that the particular integral q3 may be written as
IP
^~~2T n 6Cs > ^
K I
z TrF j
where
pG, ( x,; ) = & 
(ss,K(ss,5
00L
LS(,) )
e ds
(ss) (ss,)
and
&z QI
S
For the complementary solution we write (see [29, p. 102])
1 ,
L= 1Jz4 1 G)x, C Jd) e (A9)
ZTTF~ Z rrF'
1 1
where
Gc(x, ; ) =
D,
+ D. 3
SOz e +
T.bzo (.X.)] S
+ D, e
Lb 20 (Y)3S
tij2O.( 3 *K~)15
and the Sj are defined as in (A8). Upon adding (A8) and (A9), the gen
eral solution of the partial differential equation (Al) appears as it is
given by equation (1. 44) in Chapter I.
(A8)
zK)
ZK)
+ KZ 7
+
G,(x,~;~~ dF ,
(ss,)(ss, =
(sS3)Cs S =
To fix the values of the constants Dj, we proceed as follows:
If a small dissipative force
/1 gra.d (A10)
is introduced into the analysis, the external force potential .1 in equa
tion (1. 16) becomes
a. = 9 +0 /4 (All)
This leaves the Laplace equation
aZ = 0 (A12)
unaltered, but makes the boundary condition given by (1.41) become
"Vu*e K^t' ^K ^'^K1 4 y= Z (A 13)
Substituting from equations (1.38), (1.39), and (1.40) into (A13),
we get
,3 + aK(p+) ?3 + +KZL3 1)+P_
b xz Fz b
tg) cf f K / e (X d (A14)
Zf CKPz+ D2 7,r 0<z + D2
which governs the behavior of the spatial portion %3(x, y) of the free
surface potential 3(x,y,t). In equation (A14) the Froude number F,
the reduced frequency k, and a new parameter related to the dissi
pative coefficient are as follows:
F ,K =
Ir Uv zaO
A particular integral of (A14) is obtained just as for the partial
differential equation (Al) already treated. However, the function
G(x, y; ) of equation (A8) evaluated on the free surface y = D becomes
(x, M s So J edS, (A 15)
J L(5o;)(soG) csoa (scr,)
where
(so~(Uso,) = s5[ + ZK(1ip s + K 1( /) .
As both the integrals in (A15) are singular at 0C, Cauchy prin
cipal values must be considered. The O' are given by
ad Z= Fl 1 zKFZ(3)_ (l K^9F/F) (A16)
and
03 7 2F r^) 1+ KF p KF (A 17)
When / = 0, the roots ^,z and (,/, become
SoZ =  7ZFZ (K F F) (A18)
and
S3 1 [ YZKFZ + 4 KF)i (A19)
Now it is the behavior of the roots 0' when 3*0 which is criti
cal to this analysis. Upon examining equations (A16) and (A17) as P*0,
the results in Table 5 are indicated.
68
TABLE 5
VARIATION OF CO j = 1,2,3,4, WITH kF2
o
kF 2 Real Imaginary
a; a U o3 al a, 0 o 09
0 1/F2 0 1/F2 0 0+ 0 0 0
0< kF2< 1/4 (+) (+) (+) (+) 0+ 0 0 0
1/4
l/2
Notes:
These results may be found in Tan's paper [16] except for an
error which has been corrected here. Tan indicated that lim m m a=
2 'eo
0+ for 0 < kF2< 1/4. This led him to the conjecture that no waves
could exist upstream of the disturbance, or in this case the plate.
The notation "0+" or "0" indicates that as r 0, the imaginary
part of the root indicated approaches the real axis from the positive or
negative side respectively.
Return now to the integral in (A15). By writing
(s aC,)S a~)
0 1 [
7
Sa;
(A20)
 S,
and doing the same for the term involving CO and O, we may write
the function G(x, D; ) as
69
G(,O;, ) 3 I  r;I > (A21)
where
00
i 1 dS = Z (A22)
CI LoS(x)) S
S = e ds q3 (A23)
S O
These integrals may be evaluated by contour integration in the
complex z = S + m plane and use of Cauchy's integral theorem. The
os
proper contours must be chosen with care. Due to the term e in
the integrals, no part of the negative Saxis may be used. The choice
of the remaining first or fourth quadrants depends on the sign of (x ).
This will be evident later. The contours to be used are those shown in
Figure 7.
Lyn
(a) For x > 0
,, Efor Il',I2
'for 13, 12
(b) For x < 0
Sfor 13,14
Figure 7. Contours of Integration for the IIntegrals, j 2, 3, 4
Figure 7. Contours of Integration for the Ii Integrals, j = 1, 2, 3, 4
When a pole corresponding to C0 lies inside the contour of in
tegration, the result obtained using Cauchy's integral theorem includes
the residue at this pole. As we will see later, by allowing x to ap
proach ; these residue terms are responsible for harmonic wave
trains which propagate to infinity.
Lnv
LR
SRC
\^^"R
Figure 8.
Contour of Integration for I
Contour of Integration for I1
As an example of the integration procedure, consider I1 for the
range x J > 0. According to the scheme in Figure 7, the contour
El is to be used in this case. Integrating around this contour, shown
in Figure 8, and using Cauchy's integral formula, we have
R
e o+xys L o t(xt)l
e dS + e d> 
S  a 
SCR.
R
rr~1
0
[d ( 27 1r 1
ef 7r L e i
(A24)
where it has been assumed that OI does lie within the contour E1
In order for this to be so, the real part and the imaginary part of C0
must be positive. Whether or not this is true may be ascertained from
Table 5. Now, allow R to approach oo in equation (A24). The second
integral can be shown to vanish: We observe that
S= e C= S < 1 ior >0
D
and if R is sufficiently large,
I Z 0 I
so that
[o+ L(x ]z.
EO
R
where = x Now
77
a.R sy e
S ,Iz
(A25)
oC< 7T
z
so that
r/z
DoZ i(.X) ;Ir
R e
and as 60 for R o the integral
tain from (A24)
S TTL e
za. R.
Tr = 9.___ CL1
a. L &I
clearly vanishes. Thus, we ob
00
L D +i 10
0
Proceeding in the same manner, taking care to observe the proper
/ / /
contours of integration, we obtain similar results for IZ, 13, and 14
for the case x > 0:
00
I o+. C( )3a r cD4 <^
I, = 2T e e advn ; a=7,2 (A26)
' + CT.
0
DLCx^ ^ /00 LD LCxpiC 0
I^+ = 2(A27 e ) e
3 3
For x 4 < 0, or upstream from the plate, we obtain
ro+i O(pIO, f +DLCYxF)l(i v"
oo
_^ =1 2nT le o e d_rn ;j1/ (A28)
00 L
ZTr e C4 fa3C*.. (AZ9)
Since what is sought are the flow conditions at infinity, the next
step is to note that the definite integrals in equations (A26), (A27),
(A28), and (A29) decrease rapidly for large x Thus, they con
tribute only local disturbances around the plate. Further, note that an
infinite propagation of waves can occur only when
Po
and even then a wave results only when the OC. is contained in the con
tour of integration.
It is helpful to write a compilation of the results for G(x, D; ~)
as x,o. This is shown in Table 6. A reminder of the contour of
integration used for each integral making a contribution to G is also
included in the table.
It is a simple matter now to compile the values of G+ and G_
for the ranges of the flow parameter kF2, i. e., from Tables 5 and 6
we have
(a) For kF2 = 0
i10'(xF)30o E [o+;(xF)]0
) e v ec
G = o. (A30)
(b) For 0 < kF2 < 1/4
SC 0, tLx:lo z rr. p
G+ r_ r
* ~ L
a, e
1/4
=  o
3^
1/2
6_ =
(A31)
t. . Cx ).Jo;C;f
S.
< kF < o
O .. rT0CxP]0' L0L( <op. "
e e J
J
(A32)
(A33)
for G(x, D; ) we are to write
S4
(A34)
6
(c) For
G,
G =
(d) For
.
 (.
74
i *
I 0
0 1
fQ)
b
Q
*
4 o
x *
sO

"^ ^ 3
r ^ 1 ,
It would appear that for the range 1/4 < kF2 < 1/2 another wave
train should be included since both 0 and 0. lie within the accepted
contours of integration. However, lirm m 0 0 0, and the resulting
wave trains are damped out as x,;o. For example, suppose OC is
written as
C7 = O<, t i ^ oc, I, + 0.
The corresponding G contribution would be
3_ y, _2___
+ 'Ll Y,DYOt,(KL.Y
(Oi + L Po
and as x  +oo the contribution damps out. The same argument holds
for Cj in this range of kF2. Hence, the two ranges, 1/4 < kF2< 1/2
and 1/2 < kF2< oO could just as well be considered as one range, i. e.,
1/4
Now consider the function G(x, D; ) as it is written from equa
tion (1.44b). Again using partial fractions, this function may be writ
ten as
L1G (X, D) 4
Die e DZ e +
DD3xlSa tOL(.<)3 S
+ D, e
(A35)
where
SCD + OL)IS
e ds ; i.= 12 (A36)
00 5
0 e SdS S =3 (A37)
o S s
In evaluating the above Ij integrals, the same quadrant scheme
for contours is adhered to as was used in evaluating the Ij integrals of
G(x, D; ). One exception is made in the form of the contour. It has
already been shown that no wave train exists at infinity unless the
roots O or Sj are real and positive. Thus, we use contours like
those shown in Figure 9 when working with the real positive Sj singu
larities. A semicircular indention of radius E is taken about the pole
Sj. Again, use of one contour or the other is determined by whether
x is greater than or less than zero.
o S
o S) R
Figure 9. Contours of Integration for Ij Integrals, j = 1, 2, 3, 4
Yn
LR
Figure 10. Contour of Integration for 11
As an example of this type of calculation, consider Ii for
x > 0. Using the contour shown in Figure 10 and Cauchy's inte
gral formula, we write
j.~ CD+ L^ s r [So + L x
e s + e d +
S S, f a 5,
o I
I LD +(XL)s(
s e s,
o L n S1
0
+ e + ((.<
o.
= 0
The fourth integral vanishes as R*o The first and third integrals
combine to give I1 as Ro The second integral is calculated for
E 0 as follows. On the contour
S= e + S1,
so that
f o + (x r3 (S,+ Se )
7 i e
'T
0
/
LD4, (X) Fe
de.
Now we expand the integrand in a Taylor's series like
te
,.E e a a .E
e e + o. e ...
where 0. = D
+ i(x ). Then
e a de
Tr
and
6 dtS7
* S,
= 77r e
0Thu
Thus,
I =S ri e d+ .S
As x t + o the definite integral above diminishes rapidly, and we
are left with a result similar to the one obtained for I1. The other Ij
are then calculated for x > 0 and for x < 0. The resulting
values are combined according to (A35) and are listed in Table 7.
Again a reminder of the integration contour is provided.
STr
nI
oo
rI
0
!
IJ
I
.3
4
I:
0
(0
i>
~tS
.J
+
0
I
uJ.
+
t
~
v,
rr
r
~4r
I
~tS
.3
I
Q
I
LI
8
HH
+I
a O
x"
0
t0
I '
6
Q) 
I
ri
I*A
t o
xP
I
.3
0
rI
0
l7
t
w
*^I
^0
r5l
Ii
r.
1
i,
IC
4
I
U)
4m
rU,
4X
4
Q
LJ
'I
Q4
IWO
N+
Table 8 indicates the behavior of the Sj roots as the flow pa
rameter kF2 varies.
TABLE 8
VARIATION OF Sj, j = 1, 2, 3,4, WITH kF2
Real Imaginary
S1 S2 S3 S4 Sl S2 S3 S4
0 1/F2 0 1/F2 0 0 0 0 0
0< kF2< 1/4 (+) (+) (+) (+) 0 0 0 0
1/4 < kF2 < 1/2 (+) (+) (+) (+) (+) () 0 0
1/2 < kF< oo () () (+) (+) () () 0 0
The following compilation indicates the behavior of the G(x, D; )
function in an analogous manner to equations (A30), (A31), (A32), (A33),
and (A34).
(a) For kF2 = 0
G_ = 0.
(b) For 0 < kF2 1/4
6rL =e
+ Sr e
G_ i e.
S,SZ
+ e
S 3
(A38)
+ s[e l e 'j
(A39)
(c) For 1/4
G+ o. e
S S
G = O.
(d) For 1/2
G = 0,
where G(x, D; ) has to be taken as
a [2+(XO]Si
(A40)
(A41)
41 [DLOnSi
G= G, + G_ + L D. D e (A4Z
Upon comparing equations (A30) through (A34) with equations
(A38) through (A42), the appropriate values of the constants Dj can be
assigned. They are as listed in Table 1 of Chapter I.
)
L( 4^
>L(4()2S
APPENDIX B
COEFFICIENTS FOR THE INFINITE SET OF EQUATIONS1
R e [0 e z D SfZ{[ + 1 K S
S f t I _ _
S (K+S)(ss)(ss) ( ss)Sj))sS 3 s),, s
_ ___ _c/ 1 i C p 20^^ ^JfS kcos SZ1
S Se sJ ( s) { ss, ) (s]s )] (s)+Kcos
dS 2 (s)c ssx ( +)(s, 5)) f 
F(Ss,5z) k +L S k +
SZDS ZOSS
1Terms containing e and e are omitted when 5S and SL
are complex.
Re1 T les(0siFS T 1(a^ (^ SSSAS k
ZL SJ,) Ss (S Ks)(ss,(S (~)(5 S3) 5,,) s'
0
J i[ 1
,r e2DS,_T_(S _KS Co + ZS. .J,( kct cos Stl
7T Z zos AS3coSiS3] z 04[q W KScos1
7 L (JT)/ (5f) 0 (" ('
F'(Ss9)3e Ls) (K z 3(S+) L Z(K + y)J
1 /f 
z co s[ ,F kS(s )(S ) k S)(SS)(SfS jds
A r"cos rk(7)3
2 fJ
k Tr S, e" osg( s;)in Sf 7 ezs 2 S)s inSt
eF'(sf,Ss) i St K S
ki 7 eDss 1C3)s3 S, ezot(sy) sS,
2FZ (S,3Sy) LS
+ e(1) [(" e s ) e S) (S T 4
S71 (.1 J( es ) (s)~1 cs ss
F CSr T'z 2) [e C SJ %)e seD3Ls,) eT ,)J s]
(SZ 5 t[e) S3 Jt Oe Z y)
for v, Z, 3, ...
Re f (i) [ Lt ejI s) + {[ (( s (ss)] (s
Kcrs[s + (s)L17 ss) ( Fdss ] A
1
4 [^
^ I JS + Zic(I)
F2 (I s)(ssO(s s( ) (s)(ss()Cs v, iJ) ((i77 T'
S(1[) [S, eS's + sz zos )(.S) c 
(s, 30 k S,3 e (,T.,s,) kca S Syet"tesz k, C~s S~\
_ K r {,[s, e T (+s,1) S t7) e (o4
,~,i~iY 2f! S)[ ~ s ~ ) Xsy )
for vn = Y ,
+(IS( r
e~"{T.~VS I F s,(s s , s )
kLS;I]
42s./
( Js,} eT Z /J(s < s (D c s ,) L its S l
{ (k e .(s){ Ls,(~I +)(] s J(s)
IF' S(s s3)(SSO,) ( RS) (KSLK (J )
+ t) 1r7] e"^sM ^[)(s ,)s) s,s s j ts +r,,
cos [)15 )
V77
ft1) I e o 3 s,(s, X +.s) a, k4 SS
 r ) ( cos ,) s e2 DSyJ (s)(J.(s,
+C(,s ++.(I) V1 l15,e" ( ) ')+ s e ((J(s( )
I ~k4kS1

(S SO)
s e. ZS3J(s,5. ( (sj+ 9Sa) OSieK(s
for yn 1, 2,3, ..
(. f tt ___ I),
F+ ((s)(sS3)(ss,) (<+s(ss 1(. s.)1ds + 2k
V

