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CHAPTER 2
ELEMENTS OF WAVE THEORY
R. L. WIegel and J. W Johnson Respectively, Institute of Engineermg Research, and DiviSlOn of MechanIcal Engmeering UmversIty of Callforma Berkeley, Callforma
INTRODUCTION
The first known mathematical solution for finite height, periodic waves of stable form was developed by Gerstner (1802). From equations that were developed, Gerstner (1802) arrived at the conclusion that the surface curve was trochoidal in form. Froude (1862) and Rankine (1863) developed the theory but in the opposite manner, i.e., they started with the assumption of a trochoidal form and then de- veloped their equations from this curve. The theory was developed for waves in water of infinite depth with the orbits of the water particles being circular, de- creasing in geometrical progression as the distance below the water surface in- creased in arithmetical progression. Recent experiments (Wiegel, 1950) have shown that the surface profile, represented by the trochoidal equations (as well as the first few terms of Stokes' theory), closely approximates the actual profiles for waves traveling over a horizontal bottom. However the theory necessitates molecu- lar rotation of the particles, while the manner in which waves are formed by con- servative forces necessitates irrotational motion.
The first satisfactory treatment of two dimensional wave motion in water of arbitrary depth was given by La Place (1776) for waves of small amplitude. Airy (1845) developed an irrotational theory for waves traveling over a horizontal bottom In any depth of water. This theory was developed for waves of very small height. Airy (1845) showed that the velocity of propagation of the wave form was dependent upon the wave length as well as upon the water depth.
stokes (1847) presented an approximate solution for waves of finite height which satisfied the boundary conditions of waves In water of uniform depth and, in addition, required irrotational motIon. The series was to the thIrd approximation for finite depths, or to the fifth approximation for infinite depths, but there was no proof of their convergence. The most interesting features of the solution, apart from the irrotational motion, were, first, the dependency of the wave ve- locity upon wave height as well as upon wave length and water depth and, second, the fact that orbital motion of the particles was open rather than closed, indI- cating a mass transport in the direction of wave travel. Experiments (Mitchim,
- have shown both of these findings to be correct.
Levi-Civita (1925) proved that Stokes' series was convergent for "deep-water" waves and Struik (1926) proved that it was convergent for "shallow-water" waves.
Reynolds (1877) and Rayleigh (1877) worked on the problem of the difference between the energy transmission velocity of a wave group and the velocity of the wave form. They concluded that the energy of the group of waves was propagated with a velocity less than that of the individual waves. In deep water, the "group" velocity was found to be one-half the wave velocity.
The problem of the maximum steepness (the ratio of the wave height to its length) that a wave could attain without breaking was worked on by Stokes (1847), Michell (1893), and Havelock (1918). Their conclusions were in close agreement. A crest angle of 120 degrees, or a steepness of H/L = 0.142,* was found to be the theoretical limit.
Recently, many field and laboratory studies, as well as analytical studies, have been made. These observations, together with the mathematical studIes, lead *See list of symbols.
COASTAL ENGINEERING
to the conclusion that Stokes' irrotational theory represents the natural phenomena more closely than the other theories.
Waves in nature vary considerably in height and period over a relatively short length of time at any pOint of observation. In a generating area, the wave charac- teristics show the maximum variability; however, even after the waves have passed into a region of relative calm, considerable variations in wave characteristics exist. In theoretical problems, such variability cannot be treated mathematlcally and certain idealized conditions must be assumed. Accordingly, the flrst step in the analysls of oscillatory waves is to study the behavior of single wave trains of unlform period and amplitude as they progress in water of constant depth. Present-day wave theory deals 'nth periodic waves of stable form in which all ele- ments of the wave profile advance with the same velocity relative to the undis- turbed water. Complete development of the various analyses of Lamb (1932), Stokes (1847), Gerstner (1802), and others are not presented herein as they are readily available in the original references.
WAVES OF SMALL AMPLITUDE
If waves are of small amplitude compared to their length and to the depth of the water, the wave profile closely approximates a sine curve. The equation for motion (Lamb, 1932), considering both gravity and surface tension, is:
(gL/2 rr + 2 7ro/pL) tanh 27r"d/L (1)
For water deeper than one-half the wave length, tanh 2 rr^ d/L is almost equal to 1 and the equatlon reduces to:
0 z (^00) '"If) a: '"II. ~ '"'"...
I- <> 0 ...J '"> '"> ~ 0: '"~ ~ II. '"'" 0
24
20
16
12
08
04
o
o
1'-" l 'II I' , f f
~PE CEN^ DIFFERENPE / c!. 9 Lo^ .~^ ,/
2iT (Lo I~ /'
~ V
\ V
\
V/
l! ~ ~~ (^) 2rI9Lo^ I'--I---
" (^) k::~ =~
(^02 04) 0.6 08 DEEP WATER WAVE LENGTH, FEET
20
I L, :jfl"
I
I
4 , -'
I
] r 2 " 1=
Io - 1=
~
'~+ ~~'"
- , 8 ~ .:J ,o (^) - '" ..J 6 ~
2
o
'"<>z '"a: ...^ '"... is I- z '"<> a: '"II. 10
Fig. 1. Effect of surface tension on deep water wave velocity in fresh water at 70 oF.
The relatlve effects on ve- locity of the gravity and the sur- face tension components for deep- water waves are presented in Fig. 1. Experimental data by Chinn (1949) and Kaplan (1950) verifies the equation. It can be seen that for any wave over a foot in length, the effect of surface tension may be neglected. In practice, these small waves are usually called ripples as distinguished from the longer waves •
Neglecting the effect of sur- face tenSion, the equation for ve- loclty of propagation of gravity waves (Airy, 1845; Lamb, 1932)
C^2 = (gL/27r) tanh 21fd/L (^) (3)
and for "deep-water:"
Co 2
= gLo/21f (^ 4a)
or, in Engllsh units:
2 Co = 5·12 Lo (^ 4b)
Since the relationship between length, period and velocity of all periodic wave phenomena is defined by: L = CT (5)
COASTAL ENGINEERING
The horizontal and vertical displacements from its mean posltion, a distance z (measured negatively downward) below the still-water surface, are: cosh cos 2IT(x/L - tiT) sinh
~ = iH sinh 2~(d + z)/L sin 2rr(x/L - tiT) (9b) sinh 27Td!L
From these equations it can be seen that the semi-orbital amplitudes of the sub- surface particle's motions are:
a' = cosh^ 2~(d^ +^ z)/L sinh 27Td/L
b' = iH sinh 2;(d + z)/L sinh 27"d/L
with the ratio of the orbital amplitudes (Fig. 5) being:
~: = tanh 2JT(d + z)/L
(lOa)
(lOb)
(ll)
Recent experiments (Morison, 1948) have verlfied these equations (Figs. 6
and 7) except that, in addition, there is some mass transport. The full amplitude of the orbital motion at the surface (2a~ and 2b~) may be expressed as:
2a s ' = H coth 2/Td/L
2b S ' = H
(12a)
(12b)
When the equations are converted into their exponential form,it is found that as the water depth approaches infinity:
a'~ tHe~z/L
b'~ iHe 27TZ^ / L
(13a)
(13b)
However, the horizontal and vertical semi-amplitudes approach these limltlng values at different rates with respect to z (Fig 5). So, although the orbital motion near the surfac·e becomes nearly circular In shape very rapidly as the depth of water increases, the orbital paths become flatter and flatter with increasing dis- tance below the surface until, at the bottom, the vertical motion is zero and so the particle moves back and forth with a purely horizontal motion. Only when the water depth becomes "lnfinite" are all the particle paths circular. Fig. 8a shows the vertical amplitude of oscillation for various depths and wave lengths, and Fig. 8b shows the horizontal amplitude of oscillation for various depths and wave lengths.
By differentiating the horizontal and vertical orbltal displacements with re- spect to time, the horizontal and vertical components of the water particle veloci- ties occupying an average position at a distance z below the center of the surface particle (this neglect of second order quantities appears to be allowable) are found to be:
-ll- 1T_H cosh 21T(d + z)/L sin 2JT(x/L
- H - T- sinh 27Tli!L tiT) (14a)
vz -~-^ -7TH^ sinh^ 27!(d^ +^ z)/L^ cos^ 27T(X/L^ _^ tiT)
with the average velocities over one-half their cycle being:
(uz)ave. (^) = +^ 2H^ cosh^ 27T(d^ +^ ZlLL T sinh 27Td!L
(15a)
(vzlave. +^ 2H^
sinh 27T( d + z)/L
T sinh 2 7Td/L
(15b)
B
ELEMENTS OF WAVE THEORY
0.9 (^) (SUR ACE) Zld 0.0 V
V V
O.S
~a
/ V^ 1/
~
1// V
/
V (^) V r-/ V (^) V /"
V (^) V
V
0
./" ~
V
V ~Id'-~ ..,,-
/'
V
Zld· 0.
V /
V
lid· -O.S
V
V"Z/d / V
~ I---^ Zld=^ -
(BOT lid
V
V
V
~ -0.
OM) -1.
~ l--
./
V
l---
I---
!--
V
l-
Fig. 6. Photograph of water particle orbits for a wave with 0 0.1 0.2 0.3 0.4 (^) 0.5 the following dimensions. ...d... Lo Fig. 5 o
0.2^ I
.z.. 0. d
f
Vao
a.
0/
rtf
/ ?
O.S
O.B
o 0.2 0. b ,O.S O.B 1. --.o Comparison of the ratios of measured orbit axes with theory for d/L (^) o =0. Fig. 7
~ o.o",,~Tc!.!!!''F!....!!;F~=:t::::~
t.IL 0.2 1---+-
:!!.J 0.4~-,.0'1--+
~ ~ 0.1 1-+---+~~~+7""~""
IIL
BOTTOM 0.2 0.4 0.1 0.1 1. Zb/H
o
Q
o
Y,
8
O.
d = 2.50 feet, H = 0.339 feet, L = 6.42 feet, T • 1 12 seconds and d/Lo = 0.
~
l.-- ~
/
V
v?
THEORY
MEASURED o-b: 0;/^ I^ J^ )I-a I. 0 .02^ .04^ .06^ .08^ .10^ .12^ .14^ .IS b' 0' (^) FEET b. Comparison of measured orbit semi-major and semi-minor axes with theory: d/Lo= 0.
~ til 0.0 If b' J' o. 1/ /
% 0.2 I-- ';'/..'a'?f-l~.~-f--lJr-+-++--+-~++-...u~--+---- ~ (^) 0.4 ~ (^) b ./1; .;:: I--?,- "'" ~ .... I-- ~1-1-'''''~''---+--- ;I. J l) ~ ~ ~ " ~ 0.6 J-+--+I+-+-+-+-,J---j--+-I--+--I-----j~___j
UN 'ST RA~r
~ ~ ~ 0.8 H--H+--+-+--+-JH----+-I z~· • coshTCd+Z) f '1 .i~~ ~I 1.0 '-L--'--1-'------'----'---.L.J...B"'OT'-'T""01l"-.J....l...--.J^1 L-1L-^11 ---.JL- ° 1.0 2.0 3.0 4.0 5. Za"H RATIO OF VERTICAL AMPLITUDE OF ORBIT TO WAVE HEIGHT RATIO OF HORIZONTAL AMPLITUDE OF ORBIT TO WAVE HEIGHT a. Vertical amplitude of oscillation for proportional depth related to fraction of wave height.
b. Horizontal amplitude of oscillation for proportional depth related to fraction of wave height.
Fig. 8
0: ::> o '"
o 3
o Z
o I
2 I 0
~ 0 ~ 0 ~ 0 III 0 0: ~ G ti... ... 0
7
5 4
I 0 o
ELEMENTS OF WAVE THEORY
MAXIMUM VELOCITY OF WATER PARTICLES ON THE OCEAN BOTTOM FOR VARIOUS DEPTHS AND PERIODS THE ORDINATE IS THE VELOCITY DIVIDED BY THE WAVE HEIGHT THESE CURVES ARE FOR
-H-::'T1,nhT^ (u.l^ .....^ x^ 1r^ (2TTd)
,\
~ ~ \
""
-:::::::: (^) t---t-- 1~5 ............^ ~ -:--t----^ -IS^
T-2e (^) SEC (^100 200 400 500 600 700 800 ) DEPTH OF WATER I FEET
Fig. 9
/
V^1000 • $^ Lfl^^2 84 HOURS /
/ /
WHEN L IS IN FEET
./
800
.,^600 0: ::> (^0 ) '"; /
V
/
V
ZOO
OZ 04 06 OB 10 10 ZO 30 WAVE LENGTH, FEET WAVE LENGTH, FEET 00 (~
/
/
40
Relationship between modulus of decay due to viscous damping and wave length.
." "- '" Z 0- IL '"^0 ..J '"Z 0 i=II: IL^0 II:^0 IL
00
02
04
06
08
o
I
Fig. 10
I J--
r- ...--....- 095 ~~ (^) ~ -- --
--
..-- / (^090) /" -
I
080 /" ~
j (^070) / ~ - II (^) II (^) V 016t -- /'
./
....- ~
I /^ I
050 /' /
/ (^0 40) L
I II^ II^
V (^) / ;3<: / II I I / /
;
I I (^) I I
II I
01 02 03
I
I
04 d/L
o{
/
I 05
SURFACE
~ f..--- ~^ f.--^ -^ f..--- --
f--
V ~
--
~
f--
V-
V
l-- ,./'
V
./ V K= 'Qlh 211 (I+Zldlll. cosh 211d/L where I • DISTANCE BELOW SURFAa d • DEPTH OF WATER L • WAVE (^) LENGTH
BOTTbM (^06 07 )
Fig. 11. Pressure response factor.
50
COASTAL ENGINEERING
as engineering applications are concerned, for waves of appreciable height. It has
also been observed that the very long, low ocean swell from distant storms are ap-
proximately sinusoidal in deep-water. However, for waves of greater height, theory
indicates that certain corrections are necessary.
Two theories have been developed for waves of finite height. The first
theory, developed by Gerstner (1802) and later by Froude (1862) and Rankine (1863),
is known as the trochoidal theory. This theory has been used widely by naval
architects and engineers in their studies. The second theory, developed princi-
pally by Stokes (1847) and later by Struik (1926) and Levi-Civita (1925), is more
difficult to apply but it predicts certain results that have been experimentally
verified which are not predicted by the trochoidal theory.
Trochoidal Theory - Infinite Water Depth. Tne trochoidal theory (Gerstner, 1802),
the first theory to be developed for waves of finite height, is often used for
engineering calculations. One reason for its use is the ease with which the equa-
tions may be used. It appears to represent the actual wave profiles as well as
actually satisfying the pressure conditions at the surface and the continuity con-
ditions. However, it requires rotation of the particles and does not predlct any
mass transport in the direction of wave propagation, while observations (Mitchim,
1940; Beach Erosion Board, 1941) show that there is mass transport. This theory,
developed for waves in water of infinite depth, has been well presented by Gail-
lard (1935).
The equations of the surface profile (Fig. 12a) are,
x = Re - r sin e
y = R - r cos e
(22a)
(22b)
It can be seen that the wave length, La, is equal to 2ffR, while the wave height,
Ho, is equal to 2rs, where rs is the value of r for the surface orbit. In order to
plot the equation of wave shape in dimensionless form with the origin of the coor-
dinates at the crest and the vertical dimension measured negatively, downward,
these equations may be transformed to:
x'/L (^) O = 1 - [(rad e/2~) - (H (^) o/2L (^) o )sin6]
y'/H (^) o = 1/2(1 - cose)
(22c)
(22d)
where, x' and y' are measured from the wave crest. These have been plotted in
Fig. 13 with Ho/Lo as the parameter. It can be seen that as Ho/Lo approaches zero,
the curve approaches a sine wave and the surface is nearly that as developed in the
irrotational theory for waves of very small amplitude.
are,
The positions of the crest and trough relative to the undisturbed water level
Height of crest = Ho - [rs - (rs2/2R)] = 1/2Ho +rrHo2/4Lo
Depth of trough = 2"R[r (^) s - (rs2/2R)]/Lo = 1/2Ho - rrHo2/4Lo
(23a)
(23b)
Thus, the crest is more than half the wave height above the undisturbed water
level, while the trough is less than half the wave height below this level. Ex-
periments performed by the Beach Erosion Board (1941) verify these relationships
(Fig. 14). It should be noted that they verify the results of the theory of
Stokes (1847) as well.
The paths described by the water particles during one cycle are circles with
the radii decreasing exponentially with depth (Fig. 12b). This is expressed as,
a' - b' - r e2~z/Lo - 1/2H e 2"z/Lo
COASTAL ENGINEERING
The energy of the wave is equally divided between kinetic and potential, with
the total energy being,
Trochoidal Theory - Finite Depth. The trochoidal theory as extended to water of
finite depth has been presented by Gaillard (1935) and is widely used. There ap-
pears to be no published mathematical work which substantiates the conclusions pre-
sented by Gaillard (1935). Perhaps the facts that (a) the wave velocity, orbital
velocities and wave shapes as represented in the trochoidal theory were the same as
those in the theory of Airy (1845) for waves in deep-water, and (b) other equations
of the trochoidal theory reduced to those of Airy (1845) for small amplitudes led
Gaillard (1935) to examine the similarities b0tween equations from a reduced (el-
liptical) trochoidal theory and the Airy (1845) theory for waves in finite depth.
The equations of wave velocity, and orbital velocities and shapes as obtained from
the reduced trochoidal theory are the same as those of Airy (1845) for shallow-
water waves and for small amplitudes. Other reduced trochoidal equations are al-
most identical to those of Airy (1845). However, the reduced tr choid theory does
not satisfy either the conditions of continuity or dynamical equilibrium except at
the trough and crest (Gaillard, 1935) and hence, this theory, although widely used,
is not sound.
Gaillard (1935) states that a shallow-water wave differs from a wave in very
deep water in that the particle paths are elliptical rather than circular, with
the eccentricity of the ellipses depending upon the ratio of the wave length to
the depth of water. For a particular length of wave, the eccentricity increases
with decreasing water depth so that, in very shallow water, its particle paths are
nearly horizontal lines; while the orbits decr~ase in size with increasing dis-
tance below the undisturbed water level with the vertical axes decreaSing at a more
rapid rate than the horizontal axes until, at the bottom, the vertical motion is
zero and the particle moves in a horizontal line. The angular velocity is not con-
stant, but greatest in the vicinity of the trough and crest. It should be noted
that this theory predicts that the velocity at the crest of the orbit is the same
as the velocity at the bottom of the orbit. Recent experiments performed in the
wave channel at the UniverSity of California, Berkeley, show that this is not true.
The actual crest velocities are greater than the trough velocities.
The following equations, describing the reduced trochoidal surface, were de-
veloped and presented by Gaillard (1935) (Fig. 15),
x = Re - a'sine
y = b'cose
The velocity of propagation is,
C2 = gLb (^) s '/2rra (^) s ' = gL(tanh ~d/L)/2~
The equations for the semi-axes of the orbits are,
b' = 1/2H[cosh 2~(d + z)/L]/sinh 2ffd/L
a' = 1/2H[sinh ~(d + z)/L]/sinh 2~d/L
and the ratio of the semi-axes is,
b' a' =^
tanh 2~(d + z)/L
(26a)
(26b)
(lOa)
(lOb)
(ll)
The total energy of the wave, which is one-half kinetic and one-half potential, is
E (27)
where M, the energy coefficient, is
M = 7r 2/(2 tanh 2 2;>rd/L) (28)
ELEMENTS OF WAVE THEORY
y
~o~~____~____~~~____________-,X
a
Fig. 15. Shallow water wave, trochoidal theory.
The equations for the shape of the surface profile may be wrltten in a dimension-
less form,
x/L = [rad(arc cos 2y/H)/2ff]-[H sin(arc cos 2y/H)/2tanh 2ffd/L] (29)
This, together with the equation for the displacement of the crest and trough from
the undisturbed water level (Yswl)'
Yswl/H = 1/2 - (1TH/4L)tanh 21Td/L (30)
allows the plotting, in dimensionless form, of the wave profile, or, as x/L = tiT,
the variation of surface elevation with time. Experlments (Wiegel, 1950) have
shown that actual waves are very closely trochoidal in shape (Fig. 16). It should
be pointed out that these profiles (i.e., for these values of d/L) are v~ry nearly
the same as given to the third approximation by Stokes (1847). If the equations
for the trochoid are expanded into a series, it can be seen that to the third term
it is the same as Stokes I equation as well.
Trochoidal Theory - Rotation. Stokes (1847) has shown that the trochoidal theory
necessitates rotation and derives the following expression,
Vorticity = 2w.
YL • 049 'YL • 0 037 '!1. ·0.32 YL • 030. 'Yc.0026 ~. 022
T ·0885Eo. d. 189" T^ •^116 SEC^ d.^2 17 FT^ T • 114SEG d- 189FT T • I 43SEC d -217FT
- --- - EXPERIMENTAL - - - - EXPERIMENTAL - - -- EXPERIMENTAL - - - - EXPERIMENTAL ----- THEORY (^) ----- THEORY ----- THEORY (^) -- THEORY 06 I I
I
06 I I
_Ll I I
~~
OUtlOTION
r- Of -~r-
~ 'R~N - r-
04
I If" r-~-
DIRECTION ~~
Of
r-H.- PROPAGATION
_Ll
=~
_ DIRECTION
Of^ r-Ir-
-P~N r-f-r-
" I I I"
=
r--- DIRECTION
r- PROPAGATION Of^ r-tt^ -
--+-r-^ -
04
02
~ \ I
1
, (^) II ~O^ \ II.' Utc 1'"- \ /J
~ I
02 rl 1\
I I I), /'^ 0.2 02 0.2 -
i'-.t-
r .... 1-"
I 0.4 (^) ,.-'., 0.4 (^04) 02 04 06 08 10 YT
02 04 06 08 YT
- o^ 0.2^ 0.4^06 08
iT
o 02 04 06 08 YT
(0) (^) (b) (e) (^) (d)
Fig. 16. Comparison of experimental elevation-time curves with trocholdal theory.
ELEMENTS OF WAVE THEORY
The equation for the wave profile, to the third approximation, is:
y = a COS2~/L+(~a2/L)(cos4rrx/L)[(e2~d/~e-2rrd/L)(e4rrd/~e-4rrd/L+4)~ +
(e2rrd!L_e-21Od!L)
2 3/2 ( 6 / (e12rrd/L+e-12rrd/L)+14(eSrrd/L+e-S~d/L)+19(e4~d/L+e-4~d/L)+32]
rr a L) cos TrX L)[ 6-
(e2~d/L_e-2~d/L) (34)
The equations for the horizontal and vertical components of orbital velocities
are (according to verbal communication from R. A. Fuchs, Institute of Engineering
Research, University of California, Berkeley).
d? _ ~.C [COSh2~(d + z + '1! )/L] cos[2lr(x +f _ Ct)/L] +
dt L sinh2~7L
3(TH~ 2 C [cosh4'1r(d + z -" )/L] cos[4 ..... (x + " 4 L' l (sinh2~d/L)4 > - Ct)/L] (35a)
which, upon expanding, substituting and neglecting terms of third order or higher,
becomes,
C
(sinh2"'d!L)
II
1 3 cosh L (d+z)
[-2 + 4 (sinh2'J1"d/L)2] cos (35b)
and,
E-'I! = 'JI"H ·C [sinh2lr{d+z+ 7J )/L] ( i 2 ( J-Ct)/L)-
dt L sinh2lrd!L J s n T x+
3 'lrH 2 sinh41r( d+z+ 'Ij )
1+ (1:) c [(Sinh2...-d/L)4 ] (sin4T(x+ f -Ct)/L) (35c)
which, upon expanding, substituting and neglecting terms of third order or higher,
becomes,
d 11 '/rH sinh2lr(d+z)/L 3 'JI"H 2 sinh4T(d+z)/L
dt =L· Csinh2'11tl/L (sin2"1T'(x-ct)/L)-4tL"J cf (sinh2'1rd/L)4Hsin41r(x-Ct)/L) (35d)
The equations for particle displacement about their undisturbed positions are,
f :!!. cosh2'!r(d+z)/L ( i 2'-' Ct)/L)
2 sinh2~7L s n "\x- -
'lrH2 [ 1 3 cosh4'1r(d+z)/L lrH 2C COSh4lr(d+Z)/L. t
4L(sinh2~d/L)2 -2+4 (sinh2'11tl/L)2 ](sin4'J1"(x-ct)/L)+(~)·2· (sinh2~d/L)2 (36a)
and,
Thus, the particle orbit lies a little above an ellipse at the crest and is a
little flatter than an ellipse at the trough while, at the same time, the particle
is moving forward (i.e., mass transport). This is shown in Fig. lS.
These equations show the most interesting result of the theory of Stokes
(1847). That is, by not neglecting the effect of height (the velocity of a par-
ticle depends not only upon its mean position, but also upon its displacement from
its mean position) it is shown that the particle velocity is greater in its forward
COASTAL ENGINEERING
CREST
-.
r-.... d'^2 OOOFT
/
~
T' 2 000 SEC H' 0.400 FT Z ' 0 (SURFACE PARTICLE)
~L'1440 FT
C' 7 198 FTISEC
J
I \
UNDISTURBED WATER SURFACE
I/X,y
j
""
~
~
v
" FEET Fig. 18 Theoretical orbit of surface particle - Stokes' irrotational theory, second order.
movement (with the crest) than in its backward movement (with the trough). Labo- ratory experiments performed at the University of California, Berkeley, confirm this conclusion. This results in the fact that the forward motions of the par- ticles are not altogether compensated by their backward motions. Hence, in addi- tion to their orbital motion, there is a progressive motion in the direction of propagation of the waves. The orbits are open, not closed (Figs. 6 and 18). This motion has become known as "mass transport" and is given to the second approxima- tion by
07a)
For deep-water, this becomes,
00 = (~Ho/Lo)2 Coe4~z/L (^) (37b)
which is identical with the equation expressing the horizontal velocity remaining (due to rotation) after wave motion has been destroyed ln the rotational trochoidal theory (Equation 31). In other words, in order for a wave of finite height to exist, it is necessary for this additional velocity to exist. In the trochoidal theory, it is in the form of molecular rotation (which is not substantiated by ob- servations) of particles moving ln a closed orbit, while, in the irrotational theory, it results from particles moving in an open orbit (which is substantiated by observations (Beach Erosion Board, 1941; Mitchim, 1940; Morison, 1948).
Maximum Theoretical Wave Steepness. Stokes (1847) came to the conclusion that for any wave whose crest angle was greater than 120 0 , the series would cease to be con- vergent and hence the wave form would become discontinuous. However, the possibil- ity of a wave existing with a crest angle equal to 120 0 was not shown until later. Michell (1893) found the theoretlcal limit was H/L - 0.14 and Havelock (1918) found it to be 0.1418.
COASTAL ENGINEERING
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