RIGHT:
URL:
CITATION:
AUTHOR(S):
ISSUE DATE:
TITLE:
Relation between WaveCharacteristics of Cnoidal WaveTheory Derived by Laitone and byChappelear
YAMAGUCHI, Masataka; TSUCHIYA, Yoshito
YAMAGUCHI, Masataka ...[et al]. Relation between Wave Characteristics of Cnoidal WaveTheory Derived by Laitone and by Chappelear. Bulletin of the Disaster PreventionResearch Institute 1974, 24(3): 217-231
1974-09
http://hdl.handle.net/2433/124843
Bull. Disas. Prey. Res. Inst., Kyoto Univ., Vol. 24, Part 3, No. 225, September, 1974 217
Relation between Wave Characteristics of Cnoidal Wave
Theory Derived by Laitone and by Chappelear
By Masataka YAMAGUCHI and Yoshito TSUCHIYA
(Manuscript received October 5, 1974)
Abstract
This paper presents the relation between wave characteristics of the second order approxi-mate solution of the cnoidal wave theory derived by Laitone and by Chappelear.
If the expansion parameters Lo and L3 in the Chappelear theory are expanded in a series of the ratio of wave height to water depth and the expressions for wave characteristics of the second order approximate solution of the cnoidal wave theory by Chappelear are rewritten in a series form to the second order of the ratio, the expressions for wave characteristics of the cnoidal wave theory derived by Chappelear agree exactly with the ones by Laitone, which are converted from the depth below the wave trough to the mean water depth. The limiting area between these theories for practical application is proposed, based on numerical
comparison. In addition, some wave characteristics such as wave energy, energy flux in the cnoidal
waves and so on are calculated.
1. Introduction
In recent years, the various higher order solutions of finite amplitude waves based
on the perturbation method have been extended with the progress of wave theories. For example, systematic deviations of the cnoidal wave theory, which is a nonlinear shallow water wave theory, have been made by Kellern, Laitone2), and Chappelears> respectively. All are based on a perturbation expansion method, which suitably stretches the vertical dimension, as developed by Friedrichs4). Keller confirmed the results of Korteveg de Vrieso which is a primary intuitive theory. The second approximation of the cnoidal wave theory was derived primarily by Laitone and in succession the third one by Chappelear. However, the mutual relation between
them is not made clear, since expressions for these solutions are different from each other.
In this paper, the analytical relation between wave characteristics of the second approximate solution of the cnoidal wave theory by Laitone and by Chappelear is
presented and the limiting area between the theories for practical use is proposed from the comparison of the numerical results on the expansion parameters. In addition, some wave characteristics such as wave energy, energy flux in the conidal waves and so on are calculated.
As already pointed out by Stokeso, the physical definition is necessary to determine the wave celerity in the extension to the higher order approximate solution of finite amplitude wave theory. The Stokes second definition of wave celerity is used by Laitone and the first definition by Chappelear, although recently
218 M. Y AMAGUCHI and Y. TSUCHIY A
Tsuchiya, one of the authors and Yasuda" calculated a new cnoidal wave theory,
making use of the Gardner-Morikawa transform, in which the deficiency is overcome.
As the cnoidal wave theory by the same definition must be used in the comparison,
the cnoidal wave theory recalculated by the authorsg), using the second definition,
is utilized in place of the one by Chappelear.
2. Second Order Approximate Solution of Cnoidal Wave Theory
If the coordinate system x, z as shown in Fig. 1 is taken, the wave characteristics
by the second approximate solution of the conidal wave theory derived by Laitoneo,
which is converted from the depth below the wave trough It, to the mean water
depth h and from the coordinate system taken at the depth of the wave trough to
the one at the mean water depth are given by the following equations respectively ;
for the wave celerity c and the wave length L, they are written as
1_1.7 H ht h
1////)///1111/MMIIIIMMIIMIIIM1/l Fig. 1 Definition sketch of coordinate system
=1-L(M1—2N1)(Hh2)+(M+M2IN Ir 12N2X h)2 Vgh(1)
L 4KK 13.±( 70-2 3 NV H\ 2 2 (2) h 1/.3(H/h)8c2 2Vh)1
in which H is the wave height, g the acceleration of gravity, IC the modulus of the Jacobian elliptic function and K the complete elliptic integral of the first kind and MI, M2, N1 and Nz are given as
M2 K(1 2tel M—1{140(0+14/09) + E(E +-43-t2—1)) K K
11 N1=(22--1 + K ).A72— 4z4 {2(1 —,e2) — (2_K2) N1(3) in which E is the complete elliptic integral of the second kind.
The water surface displacement )2 and the horizontal and vertical water particle
velocities u, w are expressed respectively as
— (cn2—N1)( hI4) +13cn2+34cn4 — N2)(LI)2(4)
Relation between Wave Characteristics of Cnoidal Wave Theory 219
Ni (i • vgh—(-1+201+Mi+cn2)( )+ hL2\2K2 cn2) + M+ MINI 22
21K4— 6K2— 97K2—2 +3/ _1(2z t+(z)211cri2 (5) 4004K22 \K2)h
±r-±f2z±(zlen,1012zH L44 th+(h )2}4 \ K2)1.h \12).A h 13(H 50+
ghh 1VK2Vhsn cn dn[1. +2N, 8,22 (6)
(1 _ 21e ) {2hz+(zh)2}21012+1 + (z)2} cn2( )1 2hhh
in which sn = sn (2K (x — ct) / L), OA= cn (2K (x — ct) / L) and dn = dn (2K (x — ct) / L) are
the Jacobian elliptic functions with a real period.
Also, tne wave pressure is expressed as
pgh= (cn2 Nh+433cn2+cn4 N2)( H )2
4
(7) 3 {2+(z)2}P.K2 + 2 (2x2 1)cn2 3K2H )2z 4K2hhh h
in which p is the density of fluid.
As already pointed out by Le Mehalitelt3), N2 in the Laitone theory falls into an
error in addition to the expression for the vertical water particle velocity expressed
by the mean water depth.
On the other hand, the wave characteristics in the Chappelear cnoidal wave theory
of the second approximation by the Stokes second definition of wave celerity are
given as follows ; for the wave celerity and the wave length, they are given respectively as
=1+ L3+0 KE { L0 (2±K2 K )22±5L0L3} (8) gh
L 4K (9) h V3L0
in which Lo and L3 are the small expansion parameters. The small parameters Lo and L3 can be calculated from the following equations as function of K and H / h.
—H = K21,11 + —4°(10+70) +6.L31 (10) 2L3+Lo(K2+—K)+ Lo2——5(1 — 6K2— 9n4) +2 (1 +K2) E
(11) +6L0L3(tc2+ K)+ L32-0
Next, the water surface displacement and the horizontal and vertical water particle
velocities are written respectively as
1711/, 3 30° (2+0(1 ±C2) — L0K25212} ±[L32+L2(12 +23K2+ 12K4) 20
, (12)
+6Lo.L.3(1+K2)— {2-L02‘2 (1+x2)+.6L0L3r2} sn2+3Lo2K4sn4
,
220 M. YAMAGUCHI and Y. TSUCHIY A
K 7,t °esa2}°42(10+ 2)_L,02(K2+3)_E.
±1.02( )2+5L0L(1KE5L°20(1 +x2)sn2— 5L0L.,,,asn2 32
5+(zh)24°(13) )( +—4Lo2r4sn4 — {23L2K2—2°3L1/42 (1-I-K2) sn2
—9—L2‘44}-1 4sn
— (1+ 2 ) 2f2 -Srl co dnr3 L2 ± 1-3 (1 ±L2-)2L3(1 +t2 -3x2sn2) Vgh h 1/31.0 2°4h° (14)
3 15 LD3+r2) +--Lo2L3+3r2Lossn2}
2
Also, the wave pressure is given as
pgh2 (2L3+Lo(l+K2)Loesn21 +1/324- 30L02(120± 23/C2+12)
5
+6LoL3(1 +r2) —6 [6LoL3r2+1.L02,20 +0} sna (15) 3 3
+ 7-1-Lo2c4sn4 — 2i +GI} { 4 L.0262 7L02 (1 + r2)sn2 9 , —z
h
3. Relation between Wave Characteristics by Both the Theories
It is assumed that the small expansion parameters Lo and L3 are respectively
expanded into series forms of H/h as
/\H Lo—alcji)+a2c h) + (16)
La= b,(1)+1),(li)2 + (17) in which a, and bi are the coefficients to be determined. Then, substituting Eqs.
(16) and (17) into Eqs. (10) and (11) and truncating the series expansions to the second order of H/h, the expressions Lo and L3 can be given as
LoK2 (Rh.) 410005E212{)(1)2 (18) L3 2E2 (r2 KE )( h_E 40K4i(- 6r4 + 26,0+4)
(19) +5 K30+2 —K*LIhT
In the next, if Eqs. (18) and (19) are substituted into Eqs. (8) and (9) in order
to convert the wave celerity and the wave length in the Chappelear theory into
the explicit expressions of the ratio H/h, the calculation finally yields
Relation between Wave Characteristics of Cnoidal Wave Theory 221
11_ (2 r,11`‘±-16+16E2-6E4 1/gh2E2\K I\ h40E4 (20)
+5—K-(2- K2 -23K)) (h)2 L _ 4KK h 1/3(th){1+812fie5E,12KE )(11h )} (21)
Also, the water surface displacement, the horizontal and vertical water particle velocities and the the wave pressure can be expressed respectively through a similar calculation as
h-E2(1K )1\ h-E1(\+4e11f(E2-KE'2 -E2) , (22) -3E4cn2+3E4cn41(4)2
/2
,1E)+cn _2_ 4c 12(e2 1) (110±r-4 gh!C-
y22E VE 1 (52E)254tz K K 4E2K2Kcn+4-cn-32h- (23)
2)132
h2 f 4,o(11c2) + 2,2(-1 ±2e2) cn2 -4-cn2} w- -(1+))/3VI)31 sn cn driEl+(-18+11E2+16—K
^gh hK2hSet (24)
4‘20-12) 4(2 h±(h )2} (1 2‘2+3k2cn2)1(—)1 H 1
p;h 1(K)+cn2) ( h)+4K4 C2 (K2 — 1) + (2 -E2)-K.1;20 e2 — 324012 2- 3K4C114}4K2i2hh)2} 11 —r2+2(2E2-1)cn2 (25)
—3r2cm4J igh)2 z If brief calculations are performed by substituting Eq. (3) into Eqs. (1), (2), (4),
(5), (6) and (7), it is clear that the expressions for the wave characteristics in the
Chappelear theory agree exactly with the ones in the Laitone theory. The second
approximate solution of the cnoidal waves by Laitone corresponds to the one by
Chappelear in which the expansion parameters Lo and L3 are expanded into the
power series of H/h and the expressions for the wave characteristics are truncated
to the second order of H/h. Conversely speaking, this is due to the fact that in
the Laitone theory, the original expansion parameters are expanded into the power
series of H/h and truncated to the second order of H/h as well as the standing
long wave theory of finite amplitude derived by Shutom as the nonlinear interaction
problem of the cnoidal waves.
Apart from the practical applicability for the prediction of the wave character-
istics, it may therefore be concluded that the solution by Chappelear is more exact
than the one by Laitone, as far as mathematical formulation is concerned.
222 M. Y AMAGUCHI and Y. TSUCHIY A
4. Numerical Computation and Consideration
In this section, the computed results for some wave characteristics based on both
the original mathematical solutions are compared to each other and the limiting
condition for practical use is considered
10 101 I 1 I 13 =Nun Try/n-7.14 8 =Nun Tgr717 I =30 6 111111
ME^^ ^:111
4
Airy in Chappelear Chappelear • MIIffira. ^ ^NNE.
Laitone2.Eggicifrone^
0 I 1 c/117i 1 1.11.01.1 12
c/ SW (a) (b)
Fig. 2 Comparison between numerical results for wave celerity by both theories
Fig. 2 shows one of the comparisons for the wave celerity, in which T is the wave period. The solid line, the broken line and the one-dotted chain line in the figure indicate respectively the theoretical curve by the Chappelear theory, by the Airy wave theory and by the Laitone theory. The theoretical curves by both the
0.5 I — Chappefear
N N Airy 0 1 /
I /\Laitone
-0 .5
V 7-/gi=17.13 h/H= 1.6
P1
0II 0 05 1.0
Fig. 3 Comparison between numerical results for vertical distribution of horizontal water
particle velocity of wave crest phase based on both theories
Relation between Wave Characteristics of Cnoidal Wave Theory 223
4 2
MIK=0.7 IIMIIIIII^ 2illI toirAin^milmn ,i8or08EIMMIlln Sae4 4111111PIERSIIIIM ..m...^
4:10."M117-11E -ILMIIMMIIII _32.:,..... ....,Al -LMVO".o9.5-rilIM 20111--tillal_id%.ble,O=.1111 8WAVMSIMIN=IIMIN 099991ft0996VAIMW.MII^II 61 IMIMMil .110=1.11 4 0.9999 ̂inummi 8m^Es^w•^^•IMIII,Il 6^1I 2
IIIIINS`1%. :2
k
262 8 --...11--..k I041.M111111.1..11\
8
X 6 111111111=1111
2
2 4 6 8/02 4 6 8102 I22 4 6 9102 4 6 8/02
'
h/H h/H (a) (b)
Fig. 4 Comparison between numerical results for expansion parameters in Chappelear's theory.
I -- [ ---
swave breaking Wm 4mon i 50% (Lo) = \ /de adidarailli larilinfirearelaWara 2mryziorawgir
i e
C5RIlilaMilireindirtiraW4 •Phi!'AgaIlil
4mompreApat —9 , 2waa...4"rwrin. .i_....zzi eni•ill/111 maeratranalmi Id2I=500--I-OW ^ -•M
Ila 6^(1-$11_,, 40!301110111P3/4111 4bIliN.IIIIPIMEAME MR ISMOIBILII08
EartilMEIMIEIN
1 (5: au a ok^ n so nitimmaames- inaimma 4 ELT I: '•..1iqkalailliiIIMEN
2 4 6 8 io 2 h/H
Fig. 5 Comparison of degree of deviation between expansion parameters computed
exactly and those approximately in Chappelear's theory
224 M. YAMAGUCHI and Y. TSUCHIY A
cnoidal wave theories agree with each other in the case of larger value of h/H,
while the result by the Chappelear theory becomes slightly larger than the one by
the Laitone theory, as the value of h/H decreases. The vertical distribution of the horizontal water particle velocity at the wave
crest phase is compared in Fig. 3. As already pointed out by Iwagaki and Sakaim,
the Laitone theory gives the shape of the vertical distribution of the horizontal
water particle velocity at the wave crest phase which increases rapidly toward the
water surface from the bottom, in comparison with the Chappelear theory and the
previous theories in the case of smaller value of h/H. This is caused by formulating the wave characteristics explicitly on H/h in the Laitone theory.
Fig. 4 is the comparison between the expansion parameters computed directly
from Eqs. (10) and (11) in the Chappelear theory for the given values of K and
h/H and the ones from Eqs. (18) and (19) taking into account to the second order
of H/h in the case in which the Chappelear theory coincides with the Laitone
theory, as described already. The solid and broken lines in the figure correspond
to the former and latter cases respectively. The results computed from the latter equations deviate rapidly from the ones from the former equations with decrease
of h/H and moreover the tendency for L3 is more prominent than the one for Lo.
Fig. 5 gives the degree of the mutual difference between the parameters computed
exactly using Eqs. (10) and (11) and those computed approximately using Eqs. (18)
and (19). In the figure, 50 % (L0) means that the degree of deviation between
the parameter Lo computed by the two methods is 50 percent and the broken line
indicates the breaking criterion computed by the Chappelear theory under the
assumption that the horizontal water particle velocity of the wave crest phase at
the water surface becomes to be equal to the wave celerity at the wave breakig. The figure shows that the region in which the degree of deviation for both the
parameters Lo and L3 is insignificant is limited to the higher value of h/H. It is also found that the region for L0 is considerably wider than the region for L3 in the same percentage.
Similarly, the ratio of the difference between the computed results on the wave characteristics by both the theories is given in Fig. 6. In the figure, no is the
wave crest height above the still water level at the wave crest phase, u, uo and uo
are the horizontal water particle velocities of the wave crest phase at the water surface, the mean water level and the bottom, po and p, the wave pressures of the
wave crest phase at the mean water level and the bottom and win.. (-0.2) is the
maximum vertical water particle velocity at the location of z/h= —0.2 respectively. It is found that although the magnitude of the errors due to the approximation
mentioned above is different from each other in the computed results of the wave
characteristics, the effects of the approximation on u0/ ̂ u0/ gh p,/pgh and w„,„0 (-0.2)/ gh are more clearly prominent than those on the other wave char-
acteristics. Also, these effects on the ratio of hit and wmax ( — 0.2): gh are 1/2
times and 3/2 times respectively of the effect on the parameter L0 in the case of
the larger- value of h/H, as would be made clear from the comparison between each equation.
Relation between Wave Characteristics of Cnoidal Wave Theory 225
I8I 8 I I 8 I I 6 5 4 3 2 , 6In wave breaking=Ha
4p588---or-4 ^mist.mmorm-raW.
2 2MOM i I Immanf irr Amoral 104 02% 10 =FPW•01 AMMIMMIll I= In rryA9rfp 12rA z 6 L05 6
4=:117r4eAAMEMITA
4
ME wave breaking iaarACTAIMISIAM= MitalAlimerAn
2
I anaprearranrp cadri 10`Maillal/ZIE Id'
a aPaIrd=1111/Afl1= 6 4IrolMtrainn,=
4=allinalnira
4
Mil^MPral=t4M^INI 2 2ralanina =EnMillrilainii
Ie i o'an inn 6 l 8MLn11•1^^^111111=11111^ 6 6 ̂ laninn1.111111111== 2I 4 Mitt_05 =Mi^n
ilkMMninMn
ErnEMMIIIIMIE 2 2 MIIII1U=IMn^MllIl
-.---11 /0 I I eammenn
I 2 4 6 8 10 2 ) 2 4 6 8 0 2 h/H h/H
(a) (b)
1 1 I I I I6aI I I
6 wave breaking 6 wave breaking
44_ aa^catarII 2/IAWIWISII2 40nok-papar ;_MIINIIIISNIIIrailway
4 0_, 0 8./Tar 6I6201111 84LI3ill 4 En —7/0/h andPd/pgh 1/411018sl.
I
i 4 2 1 98
2 -
I I 2 2 I 0.5 % _i_ n-a I I II- UvArgrl 10-2 I ‘' ---- NH N
M 8 I- 8 6 1 6 4 4
2 2
i d103 8 a I • 6 e
4 , 111
2 2
io -4rTh
10-• I I I _ i 2 4 6 8 10 2 1 2 4 6 e to 2 h/H h/H
(c) (d) Fig. 6 Comparison of degree of deviation between computed results of
wave characteristics by both theories
226 M. YAMAGUCHI and Y. TSUCHIYA
0—In=laill^•=1^0 MillavebreakingII"MMMII wave breaking=MEI. 6=W i___20NMIlngIn= 4MN= I,/CrieelliallneralPr4 mmlmmmis NM //.%/I.%%MIIPAINPAPfall= 2_Apsamora 2MI SAlwariallIMM•
i6,MI,Stalllia -,=granritrain.= 8^•IAISAn^AMMI 10=ENmaaanin^ nrawawnt:=11Fralral=r11111MME
6
IMAMA1•1111•111111111WI= %,I_Maar'.1/44EM—.—riain ,,2..031.r1 2 InigAlia5% altimiIlalaan=
IM
IFITINAME ,,megriForam
tar :natio:. 4 liral^^ ..111111•11111111111111= 4annanna
imp.ir 2mainumumma ....,....... ,,BS.,no'-IIMINIMIII^MIIII^I= Q^I=1111^1•1•M=SIIIMMIIIII
nall !IS II :flIIIMEMM.MMIN inp11.1 .IIRMIMMI^MEMMEM
2:EMMEN.II^MINI= 2MIMMIII BammiamIni leIMIIMMINIM I^_ 101 I 2 46 822 4 6 82 10I
h/Hh/H
(e) (f)
i 8 I I I 8 I I I
6—wave breaking 6 =wave breakingin 4,Pe.-r 4Malrerlattra
=smalliMargrare2 /r.,, 2MM.. -.r-arligralaWrIM11,21,WAII -,IM-^111."-a2nre td' 4° 1ArMt 6 c),NosaimemorarAis 8—I--I =1011111=11•1^11Pan
6 LAL_ " 4MILUMEMPS'lltIr rsEN, ,Es1 , ,2 •, 2 115 "- 8 6Mrb/pgh-
EN,,k,a,(-02)//g7—2 -4 2 1%
I IC? gam,Iv ̂^KmMkW=I•1 4Isktram4
)
10
68 6
al'aSIIIIIIIIM
MIWIllatE 2 2minatezaltamm1
1 6, =1111111111RIIIMIng/ I
iiturt- ramin 6 1
4MlaItvara, 1 •It2 ' -H----- 2 anmanma .
•- mom aninvan= , I 0 I 6 ' -2 1 2 4 6 8
10 2 / 4 6 8 10 2 h/H h/H
(g) (h) Fig. 6 Comparison of degree of deviation between computed results of
wave characteristics by both theories
Relation between Wave Characteristics of Cnoidal Wave Theory 227
Since the Laitone theory is regarded as an approximation of the Chappelear
theory on the expansion parameters Lc, and L3, as mentioned previously, the limiting
region of applicability of the cnoidal wave theories for the various wave characteris-
tics has to be determined only by bounding the limiting region for the parameters
L, and L3.
6 4 M=MPMEIMINIMM=PE
ddll ,mmeorzaus zlymveMAIM= ci breaking/ IN 6900) ;
6 MI =EllaigME=1. MIMMI^MIN
Sig= ia/Min 2K311101.6
E region ofapplicability 4NMfor hyperbolic waves
MIMI= 2rinn O31^^MINE•M1111111111•11111 6,
la cm
wimmr..... 10 2 4 6 8 2 0
h/H Fig. 7 Limiting region between both theories for practical use
Fig. 7 shows the limiting area of applicability for practical purposes between
both the theories, in which the degree of deviation between the computed results
on each wave characteristic by both the theories is roughly 20 % of the one at
the breaking point except for the wave celerity in the case in which the difference is not so significant for practical computation. The limiting area is composed of
the overlapped area in which the degree of difference for L3 is smaller than 30 96
and the degree for Lo 6 % in the case of the smaller value of e and 2 % in the
case of the larger value of e. In the figure, the region corresponding to K3 is
the area of applicability for the Iwagaki hyperbolic wave theory's), which is derived from the Laitone theory under the condition that r= 1 and E =1 but IC=7 >0. Within
the limiting area, the degree of daviation between the computed results on the wave characteristics by both the theories is less than 0.7% for zzo h, 6% for u,/ 1./ gh ,
2 % for u0/ ̂ gh . 8 % for zit/ ̂ gh , 0.2 % for c/ ‘'gh, 3 % for h; L, 0.7 % for po'
pgh, 8 % for pb/pgh and 9 96 for wt,, (-2.0)/ ^gh respectively. Accordingly, the cnoidal wave theory by Laitone is only applicable in the shaded area under the
above criteria and the theory by Chappelear must be used in the other region.
228 M. YAMAGUCHI and Y. TSUCHIY A
Also, the hyperbolic wave theory by Iwagaki must be used in the region of K more
than 0.98 (K73) of the shaded area.
According to the above considerations, the errors due to the approximation should
be previously estimated from Fig. 6 wIllen applying the cnoidal wave theory by
Laitone and the hyperbolic wave theory for the calculation of each wave charac-
teristic.
5. Some Calculations on Wave Characteristics
According to Whithami,,, the density and flux of mass, momentum and energy in water waves are defined respectively as
(a) Mass : Po=5' pdz (26) Q0= pudz (27) -ft
(b) Momentum : P1=5 pudz (28) -h(p + ple2)dz (29) (c) Energy :- _„ t±w+pgzidz 2)(30)
Q2=-f2-1p(zi2-Ew2)+p+pgziudz (31) ft
in which P, and Q, indicate the density and flux respectively.
The mean values of these quantities on a wave length are calculated using the
second approximate solution of the cnoidal wave theory by Chappelear and by
Laitone. If the expressions for the wave characteristics in the Chappelear theory are
substituted into Eqs. (26), (27), (28) and (29), the calculation for the density and flux of mass and momentum yields
Po—ph (32) C20= P, =0 (33)
1 pgh212L3+L(,(K2i-x)+3L,32 +8E0E3(0+ r) (34) 02{_ 7 + 17K2+23K4+ 10(20+4 —EK )} 10
which are correct to the second order of the expansion parameters. The flux of
mass and the density of momentum in waves calculated using the wave theory
derived by the Stokes second definition of wave celerity vanish, as would be
expected from the definition.
In the next, the kinetic energy can be given from the first term of Eq. (30) as
1 P2k-2pglzilL2 fC2 I + 2 (2 —0EK3(E )2} +4Lo2L,k2— 1 3 K
+2(2 K2)EK3(EK)21 + L3 117K4+9K2-26+ (-340+190(35)
1
+101)c-45(r2+2)(K—E)2 +15(AK)311 \ ,1
Relation between Wave Characteristics of Cnoidal Wave Theory 229
If the potential energy is measured from the mean water level, it can be written
as
P2p=-2pgh2141,32 + 41,43(K2+K)+32 (3v4 Fe 1 E 4 (v2+ I) -EK L2L +4L33+4L33+-°53196v4+444(.2- 24 +120 (v2+ 1) 26LoL32 (36)
x (v24-K ) +15L°3(540+55v-4-3e-22+ (760+1340+76)-E Ij Using Eq. (11), the kinetic energy can be proved to be identical with the potential
energy to the first approximation of the cnoidal wave theory. The energy partition
to the second approximation, however is not in the same ratio.
Accordingly, the density of wave energy is expressed as a sum of Eqs. (35) and
(36) by the following equation.
P2= -21 pgh214L 32 + LIL oLa(K2+ KE ) + 1302 [3/C4 + 2/C2— 2+2 (K2+ 4) KE +- 3(Lif- )2}4-4L33+26L0L32(e+EK--)• L0:L3 {960+64e-44
(37) +40(20+5) K-60(—)21 +L°53[540+ 72K4 ± 6/C2-48
•1
+ (4201 1530+177)45 (K2I-2) ( EK )2Jr15(E--)1'1 K1
The energy flux in the Chappelear theory is calculated from Eq. (31). The result is
02= pgh21/ gh3 r_kpz (K21 2 (2 - e2)-K -3( )21 + 33Lo2L3 (K2—1 -2 (22— 2) KE 3( EK )2) ± 153 [18K4 +11E2— 29+ ( -36v4+6,v2 I (38)
54-124)KE--(7s2+ 25)(E)2+ 30( .4. )3I1 Therefore, the transfer velocity of wave energy cg, which is one of the definitions for the group velocity of waves, is expressed as the ratio of the wave energy flux to the wave energy density, although an explicit expression is difficult.
On the contrary, the similar wave characteristics in the Laitone theory are calculated respectively as
PO-ph (39) 0,=P, = 0 (40)
01= 2pgh2+2-0N-2-1+2(2 - v2)KE-3(ft-)21(-il)2 1(41)
P2k=-12ph21 31,54 {2 — 12(2- K2)Ej{-— 3(Eic)2} ()2+ 301S K.4 (42)
+20-2+2(0-K-2=1)fc.+15 (K2 -2)(E4+ 30(KE--hH2)3-1 K. ,
230 M. YAMAGUCHI and Y. TSUCH1Y A
1 1 P2p — 2 pgh2[ 3E4 (E2 1+2(2 E2) E3(EJ(\2+1 K\K)1U2) 100 (43)
E\ 2 H X fr4 —3K2 + 2 — 2(er— 6/C2+6) 5 (K2— 2) ( K) h
1
P2—2pgha 3,c4 {K2 1+ 2 (2 K2) t3(EK)1(11h)2 +1 1E5
} (44) E +15(H \3-1 x(r4_30+2+ ( — 2E4+17E2— 17' )K\Klfhl
02= pg1121/ gh 3‘14 [K2 1 2(E22) KEV.1(
h
+14( —E4 +3K2 —2) + (8E4-53E2+53)+60(2z-2)(E)2 30E6 (45)
+75(EHV') \K-MhIj
These equations agree exactly with the ones by the Chappelear theory in the case in which the expansion parameters Lo and L3 are expanded into the power
series of H/h and the expressions for the wave characteristics are truncated to the second order of H/h.
Also, the energy transfer velocity can be given in an explicit form as
(H/h) c0=Vgh+ {0 +30 _2 2E2 fE2— 1 + 2 (2 —r2)(E/K) —3 (E/K)2; (46)
+2(+0_70+7)(_E K )+ 12 (E2 2) ( t )2+ 12(K )3)1 Since the preceding wave theories are derived using the perturbation method,
the expressions for the horizontal water particle velocity and the other wave
characteristics in the theories are expressed as the infinite power series of small
parameter e such as u= E eittt, in which Ili is the approximate solution correspond-
ing to each order of r. Accordingly, in the calculation of Eqs. (27), (28), (29),
(30) and (31) using the truncated series solution of the second order approximation, the contribution of the higher order terms beyond the third order e.g. ma to the
results exsists in the mathematical formulation. However, this contribution is neglected in the calculation.
6. Conclusion
The relation between wave characteristics of the cnoidal wave theory derived
by Laitone and by Chappelear was established. Expanding the parameters Lo and L3 in the Chappelear theory into the power series of H/h and rewritting the
expressions for wave characteristics in the Chappelear theory into power series of H/h to the second order, it was proved that the second order approximate
solution of the cnoidal wave theory by Laitone coincides with the one by Chappelear
to the second order of H/h. And based on the numerical comparison between the
expansion parameters computed exactly and the ones computed approximately to
Relation between Wave Characteristics of Cnoidal Wave Theory 231
the second order of H/h, the limiting region of wave characteristics in which the
approximate expansion of Lo and L3 in the Chappelear theory on the ratio of H/h
is allowable was proposed for practical use.
In addition, the density and flux of mass, momentum and energy were formulated using both the cnoidal wave theories of the second approximation as well as the
energy tranfer velocity.
Acknowledgements
Part of this investigation was accomplished with the support of the Science
Research Fund of the Ministry of Education, for which the authors express their
appreciation. Thanks are due to Mr. T. Shibano, Research Assistant, Disaster Prevention Research Institute, Kyoto University for his kind help in preparing this
paper.
References
1) Keller, J. B.: The Solitary Wave and Periodic Waves in Shallow Water, Commun. on Pure and Appl. Math., Vol. 1, 1948, pp. 323-339.
2) Laitone, E. V. : The Second Approximation to Cnoidal and Solitary Waves, Jour. Fluid Mech., Vol. 9, 1961, pp. 430-444.
3) Chappelear, J. E.: Shallow-Water Waves, Jour. Geophys. Res., Vol. 67, No. 12, 1962, pp. 4693-4704.
4) Friedrichs, K. 0. : On the Derivation of the Shallow Water Theory Appendix to the Formation of Breakers and Bores by J. J. Stoker, Commun. on Pure and Appl. Math.,
Vol. 1, 1948, pp. 81-85. 5) Korteveg de Vries : On the Change of Form of Long Waves Advancing in a Rectangular
Canal and on a New Type of Long Stationary Waves, Phil. Mag., Ser. 5, Vol. 39, 1895, pp. 422-443.
6) Stokes, 0. 0. : On the Theory of Oscillatory Waves, Trans. Camb. Phil. Soc., Vol. 8, 1847, pp. 441-455.
7) Tsuchiya, Y. and T. Yasuda : An Approach to the New Cnoidal Wave Theory, Proc. 21st Conf. on Coastal Engg. in Japan, 1974, pp. 65-71 (in Japanese).
8) Tsuchiya. Y. and M. Yamaguchi : Some Considerations on Water Particle Velocities of Finite Amplitude Wave Theories, Coastal Engg. in Japan, Vol. 15. 1972, pp. 43-57.
9) Laitone, E. V.: Series Solutions for Shallow Water Waves, Jour. Geophys. Res., Vol. 70, No. 4, 1965, pp. 995-998.
10) Le Mehaute, B.: Mass Transport in Cnoidal Waves, Jour. Geophys. Res., Vol. 73, No. 18, 1968, pp. 5973-5979.
11) Shuto, N.: Standing Long Waves of Finite Amplitude, Proc. 15th Conf. on Coastal Engg. in Japan, 1968, pp. 212-219 (in Japanese).
12) Iwagaki, Y. and T. Sakai : Studies on Cnoidal Waves (Ninth Report)—Applicability of Cnoidal Wave Theory to Near Breaking Waves—, Annuals, DPRI, Kyoto Univ., No.
14B, 1971, pp. 327-345 (in Japanese). 13) Iwagaki, Y. : Hyperbolic Waves and Their Shoaling, Proc. 11th Conf. on Coastal Engg.,
Vol. 1, 1968, pp. 124-144. 14) Whitham, G. B. : Mass, Momentum and Energy Flux in Water Waves, Jour. Fluid
Mech., Vol. 12, 1961, pp. 135-147.