The wave height variation for regular waves in shoaling water

Coastal Engineering, 1(1977) 261--284 261 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

THE WAVE HEIGHT VARIATION FOR REGULAR WAVES IN SHOALING WATER

I.A. SVENDSEN and J. BUHR HANSEN

Institute of Hydrodynamics and Hydraulic Engineering, Technical University of DenmarlL 2800 Lyngby (Denmark)

(Received March 2, 1977; accepted August 8, 1977)

ABSTRACT Svendsen, I.A. and Buhr Hansen, J., 1977. The wave height variation for regular waves in shoaling water. Coastal Eng., 1: 261--284. The theory of cnoidal wave shoaling has previously been connected to data in deeper water (described by sinusoidal wave theory) by assuming continuity in energy flux Ef at the matching point between the two theories (Svendsen and Brink-Kjeer, 1972) resulting in a small shift in wave height H at the matching point. In this paper we assume directly continuity in wave height and tables are given from which the wave height variation can be determined when the wave is specified at any water depth (including deep water). It is also shown that a deep water limit for cnoidal waves exists. The nature of this limit and the behaviour of the waves close to it are analysed. Experimental data for waves with no free second harmonic components are compared with a shoaling model based on linear theory for h/L o ~ 0.10 and cnoidal theory for h/L o < 0.10. The agreement is very good for deep water wave steepnasses HolLo up to 3--4%. For larger steepnesses the linear theory fails to predict the variation for h/Lo ~ O.10.

INTRODUCTION T h e l i t e r a t u r e s h o w s a c o n s i d e r a b l e n u m b e r o f e x p e r i m e n t a l investigations o f t h e s l o w t r a n s f o r m a t i o n o f waves o n a sloping b o t t o m , w h i c h is d e n o t e d as shoaling. I n p a r t i c u l a r , d a t a f o r t h e v a r i a t i o n o f t h e w a v e h e i g h t h a v e b e e n reported. T h u s r e f e r e n c e c a n b e m a d e t o Wiegel ( 1 9 5 0 ) , I v e r s e n ( 1 9 5 2 ) , I p p e n a n d Eagleson (1955), Eagleson (1956), Vera-Cruz (1965), and Iwagaki (1968) w h o all r e c o r d e d t h e w a v e h e i g h t v a r i a t i o n in regular, p e r i o d i c waves. I p p e n a n d Kulin ( 1 9 5 4 ) a n d C a m f i e l d a n d S t r e e t ( 1 9 6 9 ) i n v e s t i g a t e d t h e t r a n s f o r m a t i o n o f s o l i t a r y waves. T h e b o t t o m slopes r a n g e f r o m 1 : 1 0 . 8 (Wiegel) t o 1 : 1 0 0 ( C a m f i e l d a n d Street). M o s t o f t h e s e results, h o w e v e r , d o n o t c o n f i r m e a c h o t h e r , a n d w i t h a f e w e x c e p t i o n s t h e s c a t t e r i n g o f t h e results is c o n s i d e r a b l e . T h u s n o definitive c o n c l u s i o n has b e e n o b t a i n e d so far n e i t h e r a b o u t t h e real v a r i a t i o n o f t h e

262 wave height nor as to which theory will predict the variation sufficiently accurately. Most of the results were compared with linear (i.e., sinusoidal) theory, which appears not to fit very well (except in Wiegel's experiments). The solitary wave experiments, on the contrary, seem to follow the h -1/4 rule (h being the water depth) for linear waves quite well (which they should not; they should grow as h-l). Iwagaki compared his measurements with the theory for hyperbolic waves, and since this theory cannot be expected to predict shoaling very well, except for very large values of the Ursell parameter U = H L 2 / h 3 (Svendsen, 1974), the best fit was obtained for waves with small deep water steepness HolLo. (Here H stands for wave height, L for wave length, and index " 0 " means values at deep water.) One possible reason for some of the discrepancies is that all the experiments for pure solitary waves, and Wiegel's experiment with periodic waves may have been performed on slopes which are actually t o o steep to allow the shoaling assumption to be valid. Another important factor is the friction losses, which have a considerable effect on the shoaling process, in particular in a small scale experiment in a relatively narrow laboratory wave flume. Part of the surprisingly large scattering which appears in many experimental results for wave quantities is most likely due to the free second harmonic waves generated by the sinusoidal motion of a piston-wave-generator. Also the effect of variations in the deep water wave steepness appears as a scatter relative to linear theory predictions. The aim of the present investigation has been to try to clear up some of the uncertainties, using the facilities for generation of waves of extremely regular and permanent form, described by Buhr Hansen and Svendsen (1974). Also the results are compared with the theory of cnoidal wave shoaling in the region where this can be applied. After an introductory discussion of the shoaling assumption (next section) this theory is briefly outlined and discussed on pp. 263--270, including a table for the variation of the wave height. In the next section is sbown that for cnoidal waves a deep water limit exists in the sense that for a given value of the deep water steepness Ho/Lo there is a maximum value of h/Lo, above which no cnoidal solution exists. This value is determined and the behaviour of the waves in the neighbourhood of the limit is analysed. The experiments are described in the section on p. 273, and in the section on p. 275 is presented a comparison with theory in which the linear theory is applied for depths outside the deep-water limit, cnoidal waves inside, the theories being matched as described on pp. 263--270. In this comparison the friction losses in the wave flume are included in the theoretical evaluations. In general the conclusion is that with a few quite obvious exceptions the two theories considered work remarkably well. Notice that in the Appendix a brief account is given (including tables) of the numerical evaluation of cnoidal waves specified by various combinations of parameters.

263

THE SHOALING ASSUMPTION

The notion of wave shoaling or wave transformation on a beach was introduced on an intuitive basis by Rayleigh (1911). In his approach there are three more or less independent assumptions involved. (a) The wave will -- to the first approximation in b o t t o m slope -- continuously adjust its form so that surface profile, phase and particle velocities, pressure variation, etc. can be determined from the horizontal b o t t o m theory, applying the local values of water depth and wave height. (b) The wave energy flux through a vertical section is constant, which implies that the reflection is negligible. (c) The number of waves remains constant during the shoaling process so that the wave period T is conserved. Essentially each of these assumptions requires a "sufficiently gently varying water depth", b u t h o w gently will actually depend on the wave theory considered. This question can be analysed theoretically by rigorous perturbation expansions including the effect of the b o t t o m slope h x . Rayleigh, of course, presented the ideas in terms of the linear wave theory, and for that case it may be shown that the shoaling assumptions will be satisfied provided the relative change in water depth over a wave length is of the same order of magnitude as the wave steepness (or smaller), i.e.: S = hxL/h

= O(H/L)

(1)

For higher order Stokes waves only smaller values of h x are allowed (depending on the order considered), and for (first order) cnoidal waves, Svendsen {1974) showed that a consistent shoaling theory requires S = o ( h / L ) ~ In conclusion we notice that in all cases the parameter S occurs and that shoaling conditions imply that S is t o o small to be of importance. Since the wave period is assumed to be constant, one of the principal problems in wave shoaling is to determine the wave height H as a function of the water depth h. In the following this is analysed both theoretically and experimentally. SHOALING OF CNOIDAL WAVES

The transformation of wave heights in shoaling water is a well established topic in the linear (sinusoidal) theory, and the theory of cnoidal°wave shoaling has been described recently by several authors. The version used here for the transformation of wave heights is almost identical with that developed by Svendsen and Brink-Kjaer (1972). In the following it is briefly outlined in a more general form with particular emphasis on the principal aspects and with an important improvement in the relation to sinusoidal theory. According to the shoaling assumptions the wave will locally be described by the cnoidal theory which means that the surface elevation ~ is described by the cnoidal wave profile, and that (using the notation in Fig.1):

264

c2

H

--

=

i

+

gh

--

A

16

HL 2 U

(2)

h -

h3

=

-

-

m K:

3

(3)

where c is the phase velocity, g the acceleration of gravity, h and H water depth and wave height respectively, and: A = A(m)

= 2/m -1-

(4)

3E/(mg)

Further K = K ( m ) and E = E ( m ) are the complete elliptic integrals of the first and second kind, respectively, m being the parameter, and U is the Ursell parameter. The energy flux Ef becomes:

(5)

E f = pgI-I:cB

1

rnl

m~---

+ (2-4ml)

E

with ml = 1 - m, and on the gently sloping b o t t o m assumed Ef is conserved from one depth to another.

~

MWL

l,

X

h(x)

~

f .....

Fig. 1. Definition sketch.

The problem of cnoidal wave shoaling is then: given the wave motion at water depth h r (e.g. by the wave height Hr at that water depth in addition to the wave period T), find the wave height and other wave parameters at water depth h. In essence this means solving the equations described above with E f a known quantity determined from the data at depth hr, and L and c being related by L = c T. It is obvious, however, that among the five unknowns, H, L, c, m, and U, only m appears in transcendental form in the equations, namely as the independent variable in the elliptic functions E and K. It turns out to be convenient to eliminate H, L, and c from eqs. 2, 3, 5, and L -- c T. This can, e.g.,

265 be executed by the following operations. In eq. 2 we write c = L / T and for (L/h) 2 substitute U h/H from eq. 3 which yields: h h H U--= I+A-H gT 2 h From this equation H]h is eliminated using eq. 5 again with c = L I T and L/h from eq. 3. The result is: - E ~ h~ 2 V -1/3 B -2/~ A + E~ 2 h~ U 413 S 213 = 1

E1 = (Ef T/pg)l/a/(gT 2)

(7)

hi = h / ( g T 2) = h / ( 2 ~ L o )

where L0 is the deep water wave length as determined by linear theory, and A and B in eqs. 4 and 6, respectively, are regarded as functions of U. Eq. 7 is called the shoaling equation (for which there is no parallel in the linear theory). This is an expression containing both m (in transcendental form) and U. Although U could be substituted by 16/3 m K 2 from eq. 3 (giving an equation for m alone), it is more convenient to consider U as the u n k n o w n of the shoaling equation and m = m (U). Hence eq. 7 must be solved for U in combination with U = 16/3 m K ~, and eqs. 4 and 6. Notice that if we follow a particular wave train with specified wave period and energy flux, E~ is a constant. Thus the transformation of the wave is in eq. 7 represented by the changing value of hi alone. When U has been determined by solving the shoaling equation, B and L can be determined from eqs. 6, 2, and 4, and hence the wave height can be determined from eq. 5. Actually in the form of eq. 7 the shoaling equation clearly demonstrates that the problem has t w o independent parameters, represented by E~ and h~. The form of these parameters also suggests a simple transition between linear (sinusoidal) and cnoidal theory b y assuming that the energy flux is the same for the wave described b y either of the t w o theories. If this assumption is introduced, Ef can be evaluated b y linear theory in deep water, as: 1 gT Ef = El,0 = - ~ p g g ~ 2~

(8)

(index " 0 " denoting deep water). E1 can then be written: 3

Ez = ( 2 n ~ - ~

-1 (HolLo) 2/3 ~ 0.06316 (HolLo) 2/3

(9)

where HolLo is the deep water wave steepness. It turns o u t that cnoidal theory cannot be applied at water depths larger than a certain deep water limit (about h/Lo = 0.10, see the following section), so the transformation of the wave seawards of that point must be evaluated from e.g. the linear theory.

266

By using eq. 9, however, we can describe the variation of a wave in the cnoidal region, the energy flux of which is specified by its deepwater steepness HolLo and its wave period T. Svendsen and Brink-Kjaer (1972) did that, and their method has later been evaluated in direct form by Skovgaard et al. (1974) as a table for H/Ho in terms of the two parameters Ho/Lo and h/Lo (= 2 u h l ) . In these tables the sinusoidal and the cnoidal theories are matched at h/Lo = 0.10 by the assumption that the energy flux determined by the two theories is the same at that point. This procedure results in a small discontinuity in the wave height at the matching point, the magnitude of which varies between 7% for small deep water wave steepnesses and 12--15% for steepnesses of 10% or more. It may well be argued, however, that since neither of the t w o theories yields the exact energy flux for a given wave height, it is equally correct to match the wave heights, and to accept the resulting discontinuity at the matching point in the theoretically determined energy flux. In other words, for h/Lo > 0.10 the wave heights (determined by linear theory) are based on one value of the energy flux, and for h/Lo < 0.10 the heights of the wave (calculated by means of the cnoidal theory) correspond to a slightly different ("equivalent") energy flux, which is chosen so that H is continuous through the point h/Lo = 0.10. This complicates the computations somewhat because eq. 9 is no longer valid, i.e., the parameter E1 in the shoaling equation is n o t immediately known from the actual deep water steepness Ho/Lo. In analogy with the above-mentioned shoaling tables, however, we choose the actual deep water steepness Ho/Lo as input. Thus the shift in energy flux at h/Lo = 0.10 corresponds to a formal shift in the deep water wave steepness, so that all the cnoidal computations are carried out with an equivalent (slightly larger) deep-water wave steepness He/Lo from which the equivalent parameter E , e can be found using eq. 8 in the form: E f = Efe

at h ~ ~

(10)

so that eq. 9 becomes: Ele

=

3 ( 2 7 r v / - ~ -1 (HelLo) 2/3

(11)

It turns out that no explicit expression can be obtained for the parameter

He/Ho = f(Ho/Lo). In Table 1 and Fig. 2 numerical results for He/Ho are shown. When U is determined from the shoaling equation, the wave height may be found from eq. 5 which by means of eq. 3 can be written in the form:

H Hr

hr fH(V) h fH(Ur)

fH(U) = U-1/3 B-2/3

(12)

267

TABLE 1

HolLo

HelHo

HolLo

or

or

HelLo

Hell o

0.0001 0.001 0.010 0.020 0.025 0.030 0.035 0.040 0.045

1.075 1.075 1.076 1.080 1.082 1.086 1.090 1.094 1.098

0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.090 0.100

HelHo

1.102 1.107 1.111 1.115 1.120 1.123 1.127 1.134 1.137

]

1.15 HolLo

1.10

1.07 1.05 0.0001 0.0002

HolLo 0.0005 0.001

0.002

0.005

0.01

0.02

0.05

0.1

Fig. 2. Ratio He/H o between equivalent and actual deep water wave heights.

in which Ur is the UrseU-parameter at the reference water depth h r where the wave height is supposed to be specified as Hr. Using deep water as a reference (i.e., hr -~ o~) we notice that Br = (1 + 2krhr/ sinh 2krhr)/16 -- 1/16. The value of Hr will n o w correspond to He, so the r a t i o hr/fH (U r) can be written (using the definition of U): hr fH(Ur)

_

~,/3 r2/3.16-2/3 ~e

~

0

Thus eq. 12 becomes:

Hell_ 16_2/3(~o)113(L~)-IfH or

(13)

268

IAI{I.I: 2 IIo/l'o

II/110 VI:HSUN h / I . ° a n d l [ o / L ° W I I ' I I CON3 INlJi'l Y IN II AT h / L ° .0001

.0[}05

_

It 0. I 0 f o r U > 1 a n d ~ - <

1.85

.0010 .0020 .0030 .0040 .0050 .0060 .0070 .0080 ,0090 , 0 1 0 0 .0120 .0140 .0160 , 0 1 8 0 .0200

"~/':o ..................................................................................................... • 100 ,095 .090 .085 ,080 .075 .070 ,065 .060 .058 .056 .054 .052 .050 .048 .046 .044 .042 .040 .039 .038 .037 .036 .035 .034 .033 .032 .031 .030 .029 .028 .027 .026 .025 .024 .023 .022 ,021 .020 .019 .018 .017 .0~6 .015 .(]14 .013 .012 • 011 .o10 .009 .008 .007 .006 .005 .004 .003 .002

U<

1

•213 .224 .336 • 249 • 262 • 276 • 291 .307 .324 .342 .3[;2 .:HI3

.I(/7 .432 .461 •4!)3 .529 .570 •620 •680 .760 .876 2.071 2.457 3.344

].039 1.0,18 1.057 1.066 1.077 1,988 1.100 1.106 1.]12 1.119 1.126 1.133 1.14i 1.146 1.156 1,165 1.174 1,183 1.193 1.203 1.214 1.225 1.237 1.250 1,263 1,277 1.2!13 !.309 1.327 1.3.16 1.3fi8 1.3!)1 1.418 1.,I.13 1.484 1.528 1.562 1.852 1,748 1.882 2.077 2.370 2.833 3.634 5.268

0.980 0.994 .009 1.01o 1.023 1.031 I.Ii39 1.0.t3 1.057 1.067 1.077 1,088 1.100 1.106 1.112 1.119 1.126 1.133 ] ,141 1.149 1.157 1,165 1.174 1.184 1.194 1.204 1.215 1.227 1,239 1,252 1.266 1,282 1,298 1,316 1 .{37 1 .:t59 1.389 I.'114 1.I.19 I.,lYl 1,542 I ,(107 1.891 1,800 1.945 2.141 2.413 2.808 3.416 4.455

0.942 0.949 0.958 0.968 0.980 0.994 1.009 1.016 1.039 1.031 ] .039 1.048 ] .057 ].057 ].077 1.088 1,100 1.106 1.113 1,120 1,127 1.134 1.142 1,150 1.159 1.168 1,177 1.187 1.197 1.209 1.221 1.234 1.248 ].263 ].280 ].298 1.319 1.343 1.370 1.402 1..I:t9 1 ,-t84 1.537 1.fi02 1.6152 1,781 1.905 2.061 2.262 2.527 2.889 3,406

0.933 0.937 0.942 0,949 0.958 0.968 0.980 0.!]94 1.009 1.016 1.023 1.031 1.039 1.046 1.057 1.067 1.077 1.089 1.101 1,107 1.114 1.121 1,128 1.136 1.144 1,152 1.161 1,171 1.181 1.192 1.203 1.218 1.230 1.244 ].261 1.279 1.299 1.923 1.34!5 1.379 1.415 1.456 1.505 1.562 1.(;31 1.713 1.813 1,934 2.083 2,270 2.508 2.820

0.933 0.937 0.942 0,949 0.958 0.968 0.980 0.!]94 1.009 1.016 1.023 1.031 1,039 1.048 ] ,057 1.067 1.078 1.089 1.102 1.108 1,115 1,133 1,130 1.199 1.147 1.156 1.165 1.176 1.187 1,199 1.212 1.226 1.241 1.256 1.278 1.299 1.324 1.352 1.384 1.420 1.463 1.513 1.571 1.639 1.71!) 1.514 1.928 2.066 2,235 2.446 2,713

0.933 0.937 0.942 0.949 0.958 0.968 0.980 0.!)94 1.009 1.016 1.023 1.031 ].039 1.0,t6 1,058 1.068 1,079 1,090 1.103 1.110 1.11"7 1,125 1.133 1.]41 1.150 1,160 1.170 1.182 1.194 1.207 1.222 1.238 ].255 1.275 1.297 1.322 1.351 1.383 1.420 1.463 1,512 1.568 1.634 1,711 1.800 1.905 2,032 2.184 2.370 2.600

0,933 0.937 0,942 0.!549 0.958 0.968 0.980 0.!194 1.00!t 1.016 1.024 1.031 1.040 1.049 1.058 1.068 1.079 1.091 1.105 1,112 1,119 1.127 1.136 1.145 1.155 1.165 1.176 1,189 1.202 1.217 1.233 1.251 1.271 1.293 1.319 1.347 1.379 1.416 1.457 1.505 1.559 1,622 ].G94 1,778 1,675 1.990 2.127 2.291

0.933 0.937 0.942 0.949 0.958 0.968 0.980 0.994 .009 ,016 ,024 ,031 ,040 .049 .039 .069 .080 .093 .t07 .114 .122 .131 .140 .149 .160 .171 .183 1.197 1.212 1.226 ].248 1.266 1.288 1.313 1.341 1.373 1.408 1.448 ].494 ].546 1,GO5 1.[;73 1.751 1.941 1.945 2,088 2,214

0.933 0,937 0,942 0,949 0.958 0.968 0.980 0.994 1.009 1.016 1.024 1,032 1,040 1,049 1.059 1.070 1.082 1.094 1.109 1.117 1.125 1.134 1.144 1,154 1.165 1,178 1,]91 1.206 1.222 1.240 1,259 1.281 1,306 1,333 1.364 1.396 1.437 1.461 1.530 1.586 1.649 1.721 ].804 1.900 2.011 2.141

0,933 0.933 0.937 0.937 0.942 0.942 0.949 0.949 0.958 0.958 0.998 0.968 0.980 0.980 0.994 0.994 1.009 1.009 1.016 1.017 1.024 1.024 1.032 1.032 1.041 1.041 1.050 1.050 1 , 0 6 0 1,061 1 . 0 7 ] 1.072 1.083 1,085 1.096 1.099 1.112 1,114 1.120 1.123 1.129 1.133 1.138 1,143 1.149 1.154 1.160 1.t66 1.172 1.179 1.1851.193 1 . 2 0 0 1.209 1,216 1.226 1.233 1.245 1.252 1.266 1.274 1.269 1.298 1.315 1.324 1.343 1.354 1.375 1.387 1.410 1.424 1.450 1.460 1.494 1.512 1.543 1.565 1.598 1.624 1.661 1,691 1.732 1.768 1.612 1.955 1.904 1.950 2 . 0 0 9 2,073

0,933 0.937 0.942 0.949 0.958 0.968 0.980 0.994 1.010 1.017 1.025 1.033 1.0,t2 1.052 1.063 1.075 1.088 1.104 1,121 1.131 1.I42 1,153 1.166 1,179 1.194 1,211 1.229 1.248 1.270 1.294 1.320 1.349 1.381 1.416 1.456 1.499 1.548 1.602 1.663 1.731 1.808 1.696

-H- > h

0.933 0.937 0.942 0.949 0.958 0.966 0.980 0.994 1.010 1.017 1,025 1.034 1.043 1.054 1.065 1.078 1,093 l.llO 1.130 1,141 1.152 1.165 1.179 1,195 1.211 1.230 1,250 1.272 1.296 1.322 1.351 1.383 1.418 1.457 1.500 1.547 1.600 1.659 1.724 1.798 1,880

0.933 0.937 0.942 0.949 0.958 0.968 0.980 0.994 1.OlO 1.018 1.026 ].035 1.045 1.056 1.069 1.083 1.099 1.117 1.139 1.151 1,164 1.179 1.194 1,211 1.230 1.250 1.272 1.296 1.322 1.351 1.383 1.417 1.455 1.497 1.543 1.594 1.650 1.712 1.782

0.933 0.937 0.942 0.949 0.958 0.968 0.980 0.994 1.011 1.019 1,028 1,037 1.048 1.059 1.073 1.088 1.106 1.126 1,150 1.163 1.177 1.193 1.210 1,229 1,249 1,271 1.295 1.321 1.349 1.380 1.414 1.451 1.491 1.636 1.584 1.638

0.933 0.937 0.942 0.949 0,958 0,968 0.980 0.994 1.012 1.020 1.029 1,039 1.050 1.063 1.077 1.094 1.113 1.135

11161 1.175 1,191 1.208 1.222 1,247 1,268 1.292 1.317 1,345 1.375 1.408 1.444 1.483 1.526

1.573 1.625

1.681

1.699

1,33

and since He/Lo = f(Ho/Lo), H/Ho may be determined from eq. 14 as a function of h/Lo and the actual deep-water wave steepness HolLo. The results are given in Table 2. From Table 2 the height of a wave at a point in the cnoidal region can be determined directly when the wave is specified in deep water. If the wave is specified at another point, h = hr say, in the cnoidal region, we have to determine H0 before Table 2 can be utilized. This is analogous to

269

1l'~' 11 (}'>'1(1 ()'~li0

U)II0

03(J0

0 . 9 3 3 0.933 0 . 9 3 3 0.937 0.997 0.997 0.942 0 . 9 4 2 0.942 0,949 0.949 0.949 0.958 0.958 0.958 0.968 0.968 0.968 0 . 9 8 0 0.981 0.981 0.(i95 0.996 0.997 .014 .016 1.018 .02:] .02(; 1.028 .099 .036 1.099 .045 .0.18 1.052 .058 .0(~2 1,0(17 .072 .078 1.083 ,089 .095 1.103 .108 . l l G 1.125 .130 . 1 4 0 1.150 1.156 .167 1.179 1.186 .199 1.213 1.203 .217 1.232 1.221 .236 1.252 1.240 1.257 1.273 1.261 1.279 1.297 1.284 1.303 1,322 1.308 1.329 1.349 1.335 1.356 1.378 1.363 1.386 1.409 ].394 1.419 1.443 1.402 1.428 ] . 4 5 4 1.480

0.933 0.937 0.942 0.949 0.958 0.959 0.982 0.999 1.020 1.031 1,043 1.057 1.072 1.090 1.110 1.134 1,161 1.191 1.227 1.247 1.268 1.290 1.315 1.341 1.369 1.399 1.432 1.467

0.939 0.997 0.942 0.949 0.958 0.9(;8 0.980 0.995 1.019 1.0'2'2 1.091 1.042 1.054 1.067 1.083 1.101 1.121 1.145 1.1'73 1,189 1.206 1,224 1.244 1.205 1.288 1.313 1.340 1.370

0350

0.10 ) .(1'150 .0500 ,(j550

0.933 0,983 0.937 0.937 0 . 9 4 2 0.942 0,949 0.950 0.958 0.960 0.970 0.072 0.984 0.988 1.003 1.009 1,028 1.037 1.040 1.051 1.054 1.067 1.070 1.085 1.OSB 1,105 1.108 1,129 1.131 1.155 1,158 1,185 1.189 1.218 1.223 1.256 1.263 1.300 1.285 1.324 1.308 1.349 1.33,3 1.360 1.389 1.419

0.933 0,997 0.948 0.!151 0.961 0.975 0.992 1.016 1,0,18 1.064 1.082 1.102 1.125 1.151 1.180 1.212 1.249 1.290

0.933 0,!137 0.944 0.952 0.963 0.978 0.998 1,025 1.061 1.079 1.0!19 1.121 1.146 1.174 1.205 1,240

0.933 0.938 0.944 0.954 0.966 0.983 1.005 1.035 1,075 1.095 1.118 1.141 1.168 1.198 1.232

06 )0 0.933 0.938 0.946 0.956 0.{)70 0.!t88 ].013 1.047 1.090 1.112 1.135 1.161 1.190

) 5 )

I'

I

0.933 0.933 0.939 0.939 0.!!47 0.9.19 0.II58 0,961 0.974 0 . 9 7 9 0 . 9 9 5 1.002 1.022 1,093 1.05!t 1.072 1.1[1~; 1,122 1.12!) 1.147 1.154 1.182

,0750 .(IIIIIG ,( !H I . 000 0.933 0.!140 0.951 0.!165 0.984 1.009 1.042 1.085

0.!)93 0,941 0.959 0.968 0.989 1.017 1.052 1.098

6.939 0.949 0.957 0.970 1.001 1.039 1.073

0.933 0.945 0.962 0.!185 1.013 1.048

1.437 1.465 .1.492 1.519 1.474 1.504 1 . 5 3 8 1.510 1,54"t 1.561 1.594 1.010

the linear theory where transformation from one depth to another is established via the (fictive) deep-water data. From eq. 10 we get:

He

=

4Hr(BrLr/Lo) '/2

(15)

from which He can be determined. Table 1 then yields Ho when He/Lo is used as entrance.* Fig. 3 shows a comparison between the present theory (continuity in wave height at h/Lo = 0.10) and that of Svendsen and Brink-Kjeer (1972) (continuity in energy flux at h/Lo = 0.10). Finally it is mentioned that the theory described above is based on c = x/-g-h~l+ A H/h. A slightly different result will appear if x/l+ A H/h is substituted for by 1 + 1/2 A H/h, and other differences of similar nature are also *It should be noticed that actually Table 1 ought to be two independent tables with H o / L o and HelLo as entries respectively. The difference, however, would be so small, that by letting H e / H o be the m e a n values of the two tables the m a x i m u m error in the combined Table I is 0.25% on l"le/Ho.

270

'HIH o ~--Linear --____ C~:iidd:I :iitt~ HE ' ::~tiinnuU2:

HolLo: 0.00] 4.0

\\~\\\\\\\\~oos

3.0

2.0 - ' ~ \ ~

u.u~

\

1.0 h/L o 0.001

,

0.002

i

0.005

,

,

i

J

0.0]

i

0102

i

i

i

0.05

,

,

'~.~

Fig. 3. H/Ho d e t e r m i n e d f r o m linear theory, and f r o m cnoidal t h e o r y with c o n t i n u i t y in energy flux and wave height, respectively, at h/Lo = 0.10.

possible. Of the same category, though of a somewhat more principal nature, are the differences that occur if c is approximated simply by v z ~ either in eq. 5 or both in eqs. 2 and 5 (as it may be shown that a higher order perturbation expansion in the b o t t o m slope requires). Formally all these versions (as e.g., Shuto, 1974, and the sketch by Ostrovskiy and Pelinovskiy, 1970) are equal in the sense that they only differ in the way the small terms are handled. For practical applications, however, where H/h is n o t really as small as envisaged in the theory, they result in considerable differences in the numerical results for e.g. the wave-height variation, in particular as we approach the breaking point. We have tried some of the other versions and found that the one described above seems to yield the best fit to the measurements. D E E P W A T E R L I M I T F O R C N O I D A L WAVES

It is generally accepted that cnoidal wave theory cannot be applied for depths which are t o o large. Although this is a consequence of the long wave assumption a specific analysis has never, to the authors' knowledge, been given either of the behaviour of the cnoidal waves as the depth increases or of the largest possible water depth for which a cnoidal theory can be applied. The first explicit analysis of the problem seems to be due to Svendsen

271 (1972). It turns out that there is a well defined deep water limit: b e y o n d which no real solution exists to the problem of finding the wave-length for a cnoidal wave with a given period T and wave-height H (which is the problem d e n o t e d case (b) in the Appendix). This result can be obtained by solving eq. 28 (see Appendix) with respect to (L/h) 2 after substituting H/h = U(h/L) 2 in the square root. However, it appears to be more convenient for the numerical evaluation to find the result by using the shoaling equation which can also be written as: E~

---AU+~[=I =

hi

hi E~

(16)

U 4Is B21S

which has the solution:

E:~

= 2-h~ (l(_) x / l + 4 h ~ A U )

(17)

From this we see that real solutions for L/h and hence for U can only exist if:

I+4h~AU>O

or:

h - - ~<

Lo

2A U

For large values of the water depth, A < 0 so t h a t this represents a relevant deep water limit (denoted DWL) for cnoidal waves, and this m a x i m u m value of h/Lo depends on A U. Since U and hence m decrease when h/Lo is increasing, it is natural first to consider the value of A U for m -* 0, which can actually be obtained only if H/h ~ O. For m -* 0 we have K -~ ~/2, so the relation eq. 3 between m and U yields: 3

m = 4~ 2 U + O ( m 2)

(18)

and hence by the definition of eq. 4 of A: lim A U = - 4 ~2 = _ 13.15 m-*0 3

(19)

Thus, for infinitely small waves we have the deep-water limit: h -

-

Lo

<

3 81r

- 0.1193

for H -~ 0

(20)

In the general case of finite values of H/h at the deep water limit, the upper limit for h/Lo depends on U at t h a t limit, so t h a t h/Lo is determined by:

272 h -- = L0

at the DWL

2A U

(21)

and U satisfies eq. 17 which by substitution can be rearranged to yield:

Ir3/4 B_,,2 He(

h 1-9/4 L0 \Lo I

U = T

(22)

valid along the deep water limit. Hence this limit is found b y solving eqs. 21 and 22 using the definitions of A = A(U) and B = B(U). Obviously Ho/Lo is the parameter of these equations and the result for h/Lo is shown in Fig. 4 together with the corresponding values of A, U, A U, h/L, and H/h. We notice that H/h b e c o m e s v e r y large when Ho/Lo increases towards the largest possible value of 0.143. Hence these waves break in the neighbourhood of the deep-water limit. Fig. 4 also shows the variation of h/Lo along the DWL. It can also be shown that at the deep water limit we have for all values of HolLo :

H

1

h

2A

......

or

1 c: = ~ g h

(23)

which is far below the result c:/gh = tanh kh/kh ~ 0.80 found by linear theory, which suggests that cnoidal theory may n o t yield the best prediction close to the deep water limit. This is further confirmed if we consider the variation of U and H for a particularwave train close to the DWL. From eq. 17 we find by substituting for the relevant parameters: u

I +

:

+

Lo

(24)

Realizing that A U is nearly constant near the DWL, this implies (usingeq. 14) that:

HDWL where:

near the D W L

_ hDw L -h

(25)

<< 1

hDWL This shows that the wave height as a function of water depth has a vertical tangent at the DWL, which is illustrated in Fig. 5. A similar behaviour can be shown to exist for the wave length.

273

t 0,

°

15

0.15

HIHo 10

D.W.L. for HolLo= 0.03

0,10

1

1.00

0,95

5

0.05 0.90

h/Lo 0.05

0.10

0.08

0.10

0.12

0.1~,

Fig. 4. Variation of various parameters along the deep water limit (DWL). Fig. 5. The variation of the wave height near the deep water limit.

In the practical applications we therefore choose h / L o = 0 . 1 0 as the largest depth for which the cnoidal theory should be applied (Skovgaard et al., 1974) and as the matching point between the t w o theories. (In principle any point shoreward of h / L o ~ 0.10 could be chosen as the matching point.) DESCRIPTION OF EXPERIMENTAL FACILITIES AND PROCEDURE

The waves are generated by a flap-type wave generator in a flume 33 m long, 60 cm wide, with a plane beach sloping 1 : 35 (see Fig. 6), The m o t i o n of the wave generator is controlled by a PDP 8 mini.computer, Which generates a c o m m a n d signal of the form: = el sincot + e2 s i n ( 2 ~ t +/32)

....

(26)

(Buhr Hansen et al., 1975). Fig. 7 shows a comparison between a resulting wave profile measured in front of the slope (with almost no free second harmonic c o m p o n e n t ) and a second order Stokes wave. The parameter U=

274

PISTON TYPE WAVE GENERATOR

~

I

'-'-

. . [-+---

[

~,~

~,e" ~ x

. . . .

....

~ ~-

ii

~-----------4 I ~ - - I ~

•

13S~"nl I"I I

!4.5m __ . . . .

SIGNAL GENERATOR POP 8/M MINICOMI~

~ ~

WAVE HEIGHT METERS ON CARRIAGE

/

----~..~

--'"7 "~

32m

DATA ANALYSER PDP 8/E MINICOMR

'

~ - -

!2,6 L _ _ _ _

-

1

3> DATA IN DIGITAL FORM: H.T.1l, c. SURFACE PROFILES

Fig. 6. Experimental facilities.

60

"~ " m m

,-, °k

/

/

/3

\

\

-'

\

Fig. 7. Comparison between wave profile measured in constant depth part of flume and Stokes second order approximation.

HL2/h 3 is only about 2. The wave steepness H/L, however, is as much as 8.5%. It also appears that even for U as large as 40--50 (which is far up in the cnoidal region), the waves generated by eq. 26 remain of clean and constant form. This is important because the waves in the constant depth part of the wave flume represent the initial conditions for the shoaling process. (It may be noticed tllat it has no meaning to consider whether the wave produced by eq. 26 is a " S t o k e s " or a " c n o i d a l " wave, as long as it is of constant form.) The water-surface variation is recorded by a resistance wave-height transducer (twp silver wires, 0.17 mm diameter, 5 mm apart), the signal of which is scanned by the computer 400 times per second. The transducer is m o u n t e d on a carriage which moves slowly (2--4 cm/s) along the flume. Vertical irregularities of the rails for the carriage are eliminated by storing in the computer a zero-level Correction, which is obtained from the wave transducer during a carriage-run without waves. In the e~xperiments, the computer determines on line the height H of each wave, the mean water level ~, and by means of an additional wave height

275

transducer, the phase velocity c. At the same time selected wave Profiles are stored. After each experiment the results may be plotted out on an ordinary pen-recorder. In this paper we concentrate on the wave-height measurements. In all experiments reported, the still water depth was 36.0 c m and the plane slope was 1 : 35. The wave frequency varied between 0.3 Hz and 1.2 Hz, the wave height in the constant depth part of the flume between 3.5 cm and 10 cm. The reproducibility of the experiments is illustrated in Fig. 8 showing t w o records of the same experiment. The curves in the figure are a moving average over a number of waves. The figure also shows that even most of the small and apparently inexplicable irregularities in the records are obviously repeated the H- trim 80 70 60 5O 2.0 J -c-m/s

15

1.0

0.5 I

0

,

I

0.04

i

24

,

I

22

1:35

,

I

O.06

I

20

h/Lo

.:,

0.0829

,

,

18

16

I

14

x~m i

I

12

~

I

,

I

10

8

Fig. 8. Illustration of reproducibility of experiments.

same way from test to test. S o m e may be due to irregularities in the wave flume (though the accuracy of alignment of sides and b o t t o m is well below 1 m m ) but m o s t of them seemed to be generated by either capillary waves, secondary waves generated by the breaking process, or perhaps the rest o f the free second harmonic. As the development of the system af~d the measurements was spread over a long period, this reproducibility was used as one of the checks against errors in the rather complicated system. t

COMPARISON BETWEEN THEORIES AND EXPERIMENTS

In Figs. 9 and 1 0 the experimental results are shown for t h e variation of H

276

ill-ram 50 &0 3O cn

sin

I H/h = 0.104

2O 10 0

L

L

I

,

i

i

,

0.1

J

,

,

,

,

J

0.2

0.3

, hlLo

"-0.332

H- mm~-. "". 60 50 40

I

30

HolL o = 0 0099; T~/g/h = 8 7 0 Holh = 0.106

20 I0

h/L I'

0

,

[

0.02 H-ram

,

0.04

I

,

o

i,=,

0.06

' 0.0829

\ C

60 70

i.~..

SINUSOIDAL

60

~;LX.~" '%~

CNOIDAL CNOIOAL i CNOIDAL ii

50 40

I

H o l L o = 0.0039;

30

Tg ' , / ~ = 13.05

Holh = 0.111

20 10 I

o

o'.ol

'~'

o!o2

' 0.03

J 0.04

, I,,

h/L°

0.05

0.0576

16

14

1:35 x

24

22

20

18

i

1

12

10

~

i

8

Fig. 9. Variation of the wave height for three relatively small deep water wave steepnesses.

277

H~ mm -

110

-

-

-

SINUSOIDAL CNOIDAL

-

------

T g'./-~"= 5.22 I-~ :,-

100

c 90 80 70 60

J HolL o = 0.0458 / H I h = 0.185

50 40

I Ho/Lo = 0.0270

30 c-n

H / h = 0.109

I si~

20 10 0

b

I

0.05

,

,

,

,

I

,

0.10

,

h

J

I

,

,

0.15

,

,

[

0.20

,

, ~

hlLo

0.230

1:35

Fig. 10. Variation o f t h e wave height, rather steep waves.

for some values of the deep-water wave steepness Ho/Lo ranging from 0.0039 to 0.064. Fig. 9 represents the smaller values, Fig. 10 the larger valUes of Ho/Lo. The experiments were carried o u t with different wave periods, so the h/Lo scale is n o t the same. In the figures are also shown the values of other dimensionless parameters such as T~/g/h and H/h in the constant depth part of the flume. They represent the actually measured data there (H, h, and T), from which h/Lo and Ho/Lo have been determined theoretically as described in the section on pp. 263--270. Though the experimental curves seem to show a continuous variation of the wave heights, they are actually step-curves. This is because the carriage with the wave transducer moves 2--4 cm (depending on the wave period) along the wave flume during one wave period, i.e., b e t w e e n each new result for the wave height.

Energy loss due to friction This effect was taken into account in the theoretical curves b y reducing the energy flux at each station by the energy lost since the previous station in the calculation, due to friction along the b o t t o m and along the side walls. Thus the local energy flux El(x) can be written as:

278

Ef(x) = Ef,h + f

3Ef ax

d x ~ Ef,h + ~ a E f Ax Ox

where Ef, h corresponds to the wave height measured at the foot of the slope. In the calculations laminar boundary layers were assumed in all cases although the longest and highest waves according to Jonsson (1966) might have turbulent boundary layers, at least close to the breaking point. The effect, however, of introducing the turbulent value of the wave energy loss on the last part of the slope appears to be insignificant. The wave height attenuation thus calculated showed agreement within 25--50% with measurements in the constant depth part of the flume. This comparison also seems to confirm that a surface film, like the one described by Van Dorn (1966) did not exist. In the calculation of friction losses, wave particle velocities determined by the linear theory were used. This also applies to the region where the wave height variation was calculated from the cnoidal theory. In fact, this is a reasonable simplification since the friction only "eats" a minor part of the energy flux anyway. Preliminary calculations also indicate that the real energy losses will n o t be significantly different from those determined by linear theory. As an example it can be mentioned that omission of the friction losses in the flume would increase the theoretical wave height at the point of breaking by 6.1% in the case ofHo/Lo = 0.0039 and T x ] - ~ = 13,05 at h = 0 . 3 6 m. Linear wave shoaling

For h/Lo > 0.10 cnoidal theory is not applicable. The theoretical curve shown in Figs. 9 and 10 in this region is the classical linear wave shoaling. For comparison, in Fig. 9b the linear curve through to breaking is shown. We notice that for wave steepnesses HolLo less than 3--4% the linear theory predicts the wave height variation quite well seaward of h/Lo = 0.10. In fact the agreement is better than in the interpretation of Iversen's measurements (Iversen, 1952) given by Brink-Kjaer and Jonsson (1973). They found that even for smaller wave steepnesses an appreciable discrepancy seemed to develop between the linear theory and the measurement as h/Lo decreased towards the value 0.10. Fig.ll shows an example of this where HolLo is 3.58%, and the linear theory yields results up to 8% higher than measurements (i.e., a minimum value of H/Ho = 0.913 against 0.85 measured neglecting the scat. tering of the measured values). It has turned out that the major reason for this discrepancy is that friction has been neglected in Brink-Kjaer and Jonsson's calculation of the theoretical curves. In particular in Iversen's case, with a wave flume only 30 cm wide and a horizontal b o t t o m depth of 77.8 cm in the case considered, the friction along the side walls has a considerable effect. Taking this into account brings the theoretical minimum value of H/Ho down to 0.869 in the case shown in

279 Fig. 11, and this must be considered in fair agreement with the measured value. The theoretical variation with friction included is shown in the figure. For the experiment shown in Fig.9a, HolLo is almost the same {3.57%) and here the measured minimum value of H/Ho is 0.875 against the calculated value (including friction) of 0.889. 1.3o HH"~ 0

,.2o,.,o

'

'

i

-R;

0.~

~''~

0.0C 0.02

~--~NUSOIDAL

,,~.a 0.05

o.10

,, '* • • 0.20

WITH FRICTION 0.50

1.0o

Fig. 11. The effect of friction losses on Iversen's data.

Where HolLo increases b e y o n d 3--4% as in Fig. 10b and c, the difference between (linear) theory and experiments increases. The reason is probably that for these values of HolLo the wave-height to water-depth ratio is appreciable already outside the cnoidal region. In the case of Ho/Lo= 614% (Fig. 10) the wave actually breaks at h/Lo ~ 0.10, so that the entire shoaling process has been determined by the linear theory. And quite obviously, linear theory cannot handle the large values of H/h. Since at the breaking point we here have U m a x ~" 45 (H/h ~ 0.71 and L/h ~ 8), it seems likely that the problem could be overcome by using a second or third order Stokes shoaling theory. If a comparison is made with experiments in a wave flume, care should be taken to modify the theory to the condition of zero net mass flux, a condition which the prevailing versions of higher order Stokes shoaling do n o t satisfy. In particular in the third or higher order theory the results may be misleading if this problem is neglected. Cnoidal wave shoaling

As appears from Figs. 9 and 10a the combined linear-cnoidal shoaling model predicts the wave height variation quite well in those cases where the H/h-ratio remains small for h/Lo > 0.10. Close to breaking, discrepancies of up to 6--8% develop in some cases. This cannot be considered surprising,

280 however, since the energy flux used for the calculations was based on the assumptions that H << h, that the horizontal velocity u is constant over the water-depth, and that the excess pressure p+ due to the wave is also constant, and proportional to the local value of the surface elevation. As is evident from, e.g., velocity measurements in waves near breaking (see e.g., Iwagaki and Sakai, 1976), the velocity is far from constant over the water depth, and the constancy of p÷ actually represents neglection of the vertical accelerations in this context. In Fig. 9b and c the wave height variation for two of the other models of cnoidal shoaling mentioned previously is also shown, namely: (1) c = v / ~ in eq. 5; and (2) c = ~ g h in both eqs. 2 and 5. The curves appear to represent a typical pattern, version (1) predicting values 20--30% above those measured at the breaking point, version (2) giving values only a b o u t 10% t o o large. As mentioned in the second section, a proper measure of the steepness of the sloping b o t t o m is the parameter S = h x L / h . Since L is approximately proportional to ~ - h , a plane slope will correspond to S ~ h x h -in so that for fixed hx the value o f S grows with decreasing water depth, indicating that the slope appears steeper and steeper to the waves as they propagate shoreward. Hence the shoaling condition (which requires a small S) may sooner or later be invalidated. With reference to the assumptions in the second section, this would cause appreciable reflection and disintegration of the wave form (T not constant). No such phenomena were observed in the experiments recorded here, which suggests that S in all cases has been small enough. CONCLUSION A theory for shoaling of cnoidal waves is presented in the section on p. 263. The cnoidal waves are connected to deep water data for the waves b y matching with linear (sinusoidal) theory at the point h/Lo = 0.10 under the constraint that the wave height should be continuous at that point. In this respect theory represents an extension of Svendsen and Brink-Kj~er (1972) who assumed continuity in the energy flux at the matching point. Accordingly Table 2 for the wave-height variation as a function of depth h/Lo and deep-water steepness Ho/Lo differs slightly from previous tables. It is also shown that there is a well-defined deep-water limit for cnoidal waves b e y o n d which no real solution exists. In terms of h/Lo the limit varies between 3/8~ = 0.1193 (for infinitely small waves) and a b o u t 0.14 for very steep waves. The limit corresponds to a local wave length L close to 5 h (Fig. 4). In the immediate neighbourhood of the deep-water limit, the wave height is shown t o vary in an unrealistic way which is the reason for selecting h/Lo = 0.10 as the practical deep water limit and matching point (Fig. 5). Both linear ("sinusoidal") and cnoidal wave theories are compared with experimental results obtained with waves without free second harmonic

281

disturbance on a plane slope hx = 1 : 35. It is shown that: (a) Linear theory can predict the shoaling as long as the wave-height to water-depth ratio H/h is small (Fig. 9). (b) Cnoidal theory predicts the variation of the wave height quite well even close to breaking for waves with small deep water wave steepness HolLo (Fig.9). (c) For waves with large deep water steepness Ho/Lo (> 3--4%) the value of H/h is not small for h/Lo > 0.10. Hence linear theory fails (Fig. 10). Second or higher order Stokes theory is recommended in this case for h/Lo > 0.10. ACKNOWLEDGMENTS

The authors wish to acknowledge help and guidance from J~rgen Christensen and Paul Prescott in the experimental work, and from Ivar G. Jonsson in the calculations of friction losses. APPENDIX

Numerical evaluation o f cnoidal waves F o r reference a brief a c c o u n t is given of the following problem. Usually the wave is specified by either of the following series o f data: (a) h, L, H; (b) h, T, H (h being water depth, L is length, H height, and T period o f the wave respectively). The p r o b l e m is to calculate o t h e r enoidal properties for the wave f r o m these data.

Case (a) F r o m H, L, and h (i.e., Hih and L / h ) we find U = HL2/h 3. The p r o b l e m is to solve eq. 3 to find m. With m k n o w n we can find, e.g., K, E, A (from eq. 4), and B, and the surface elevation ~min in the wave trough in the expression for the surface elevation: = ~ m i n + H c n 2 (O,m) 0 = 2K

t

x

(27)

T-L

where cn is a Jaeobian elliptic function. Since m = m(U) all the quantities K, E, A, B, and ~min can be tabulated as a f u n c t i o n of U. This is done in Table 3. In o t h e r words, A and B can be d e t e r m i n e d f r o m U, and hence c f r o m eq. 2, Ef f r o m eq. 5. The variation of cn 2 is shown in Fig.12 for different values of U. Notice t h a t once U is k n o w n , all the cnoidal properties m e n t i o n e d ( e x c e p t cn2(o,m)) can be expressed in terms of U.

Case (b) H, T, and h represent the dimensionless parameters H/h and T~/g/h. To find U we need

L/h which means to solve: L

= T x/g--~ ~

(28)

U

m

(-1)5.302

(-1)5.639 (-1)5.952 (-1)6.240 (-i)6.507

(-i)6.754 (-1)3.246

(-1)6.982 (-i)7.194 (-1)7.389 (-1)7.570

(-1)7.738

(-1)8.036 (-1)1.964 2.266 (-1)8.293 (-1)1.707 2.329 (-1)8.513 (-1)1.487 2.393 (-1)8.702 (-1)1.298 2.456

(-1)8.866

(-1)9.006 (-i)9.128 (-1)9.233 (-1)9.325

(-1)9.404

(-1)9.473 (-1)9.533 (-1)9.586 (-119.632

(-1)9.673

i0

ii 12 13 14

15

16 17 18 19

20

22 24 26 28

30

32 34 36 38

40

42 44 46 48

50

1.881

1.754 1.786 1.817 1.849

3.113

2.883 2.942 3.000 3.057

2.824

2.581 2.643 2.704 2.764

2.519

2.201

1.043

1.063 1.057 1.052 1.047

1.070

1.104 1.094 1.085 ].077

1.116

1.176 1.158 1.142 1.128

1.196

1.243 1.230 1.218 1.207

1.256

1.318 1.301 1.285 1.270

1.335

1.416 1.394 1.373 1.354

1.438

1.542 1.514 1.487 1.462

E A

-0.312

-0.334 -0.328 -0.323 -0.317

-0.339

-0.365 -0.358 -0.352 -0.345

-0.372

-0.402 -0.394 -0.386 -0.379

-0.410

-0.426 -0.422 -0.418 -0.414

-0.431

-0.449 -0.444 -0.440 -0.435

-0.453

-0.472 -0.467 -0.462 -0.458

-0.476

0.029

-0.056 -0.033 -0.011 0.009

-0.082

-0.204 -0.170 -0.138 -0.109

-0.243

-0.449 -0.387 -0.333 -0.285

-0.521

-0.712 -~.657 -0.607 -0.562

-0.773

-1.119 -1.012 -0.921 -0.842

-1.245

-2.151 -1.830 -1.588 -1.399

-2.596

-0.495 -13.152 -0.491 -6.565 -0.486 -4.365 -0.481 -3.261

nmin H B

0.1149

0.1173 0.1167 0.1161 0.1155

0.1179

0.1201 0.1195 0.1190 0.1184

0.1206

0.1225 0.1220 0.1216 0.1211

0.1229

0.1236 0.1234 0.1233 0.1231

0.1238

0.1243 0.1242 0.1241 0.1239

0.1244

0.1248 0.1247 0.1246 0.1245

0.1249

0.1250 0.1250 0.1249 0.1249

Integers in parentheses indicate powers of i0 by Iwhlch the following numb ...... to be multiplied I

(-213.274

(-2)5.268 (-2)4.665 (L2)4.139 (-2)3.678

(-2)5.959

(-2)9.937 (-2)8.720 (-2)7.666 (-2)6.753

(-1)i.134

(-1)2.262

(-1)3.018 2.073 (-1)2.806 2.105 (-1)2.611 2.137 (-1)2.430 2.169

2.041

(-1)4.361 1.912 (-1)4.048 1.944 (-1)3.760 1.976 (-1)3.493 2.008

[-i)4.698

(-1)6.345 (-1)5.884 (-1)5.457 (-1)5.063

1.723

(-1)3.655 (-1)4.116 (-1)4.543 (-1)4.937

6 7 8 9

(-1)6.843

(-1)3.157

5

1.601 1.631 1.662 1.692

X

(-1)1.410 (-1)2.038 (-1)2.619

(-1)9.268 (-1)8.590 (-1)7.962 (-1)7.381

mI

1 2 3 4

i (-2)7.317

[ I

Table 3 CNOIOAL WAVE FUNCTIONS (after Skovgaard et al., 1974)

50

(-i)9.673

1.000

500

1.000 1.000 1.000 1.000 i0000

30.619

1.000 (-26)4.066

5000 6000 7000 8000 9000

0.000

1.000 (-37)3.918 1.000

=

43.301

33.541 36.228 38.730 41.079

19.365 23.717 27.386

1.000 (-16)2.421 1.000 (-20)4.015 1.000 (-23)2.611

(-28)1.177 (-31)5.451 (-33)3.663 (-35)3.336

13.693

1.000 (-11)2.044

i000 2000 3000 4000

9.682

5.304 6.124 6.847 7.500 8.101 8.660 9.186

4.336

3.886 4.003 4.117 4.228

3.766

3.252 3.386 3.516 3.643

3.113

K

10.155 10.607 11.040 11.456 11.859 12.247 12.624 12.990 13.346

(-8)6.224

(-4)3.955 (-5)7.674 (-5)1.808 (-6)4.894 (-6)1.471 (-7)4.807 (-7)1.681

(-3)2.753

(-3)6.805 (-3)5.379 (-3)4.278 (-3)3.423

(-3)8.668

(-2)2.466 (-2)1.874 (-2)1.438 (-211.112

(-2)3.274

mI

(-8)2.419 (-9)9.803 (-9)4.122 (-9)1.791 (-10)8.015 (-10)3.682 (-10)1.733 (-11)8.333 (-ii)4.089

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

(-1)9.996 (-i)9.999 1.000 1.000 1.000 1.000 1.000

550 600 650 700 750 800 850 900 950

(-1)9.972

(-1)9.932 (-1)9.946 (-i)9.957 (-1)9.966

150 200 250 300 350 400 450

(-1)9.913

75

(-1)9.753 (-i)9.813 (-1)9.856 (-1)9.889

80 85 90 95

55 60 65 70

m

i00

U

1.000

1.000

1.000 1.000 1.000 1.000

1.000

1.000 1.000 1.000

1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.000

1.001 1.000 1.000 1.000 1.000 1.000 1.000

1.005

1.012 1.009 1.008 1.006

1.014

1.034 1.027 1.022 1.017

1.043

E

-0.000

-0.023

-0.030 -0.028 -0.026 -0.024

-0.033

-0.052 -0.042 -0.037

-0.073

-0.098 -0.094 -0.091 -0.087 -0.084 -0.082 -0.079 -0.077 -0.075

-0.103

-0.188 -0.163 -0.146 -0.133 -0.123 -0.115 -0.109

-0.230

-0.255 -0.248 -0.242 -0.235

-0.263

-0.301 -0.290 -0.280 -0.271

-0.312

~min 'H

1.000

0.931

0.911 0.917 0.923 0.927

0.902

0.845 0.874 0.890

0.781

0.705 0.717 0.728 0.738 0.747 0.755 0.762 0.769 0.7"25

0.690

0.434 0.510 0.562 0.600 0.630 0.654 0.673

0.308

0.227 0.250 0.271 0.290

0.203

0.072 0.111 0.145 0.175

0.029

A

0.0000

0.0149

0.0190 0.0176 0.0165 0.0156

0.0207

0.0318 0.0263 0.0230

0.0434

0.0560 0.0540 0.0522 0.0506 0.0491 0.0478 0.0465 0.0454 0.0443

0.0582

0.0902 0.0822 0.0760 0.0711 0.0671 0.0636 0.0607

0.1009

0.1061 0.1048 0.1034 0.1021

0.1075

0.1134 0.1119 0.1104 0.1090

0.1149

B

O0 tO

283

cn

2

10

05

0 0

oi

02

03

04

05

Fig. 12. Cnoidal wave profiles for different values of U ffi HL21h 3 (after Skovgaard et 81.,

1974). TABLE

L/h AS FUNCTION

OF H/h A~iO ~ g~/h I

0.010.020.030.04

6.7 8.0

6.7 8.0

6.7 8.0

6.7 8.0

9.2

9.2

9.2

9.2

0.05

6.8 8.0 9.2

0.100.150.20

6.8 8.1

6.8 8.1

6.8 8.2

9.2

9,3

9.3

0.25

6.9 8.2 9.4

0.300.350.400.45

7.0 8.3

7.1 8.4

7.1 8.5

7.2 8.6

9.5

9.7

9.8

9.9

0.50

7.4 8.8

0.60 0.70 0.80 0.90

7.6 9.1

7.9 9.4

8.2 8.5 9.710.1

1.00 1.10 1.20 I

8.8 10,4

9.1 9.51 10.711.1

i0.i

10.410.811.211.5

11,9

12.312.6

i1.4 12.7 13.9 15.2

11.812.212.613.0 13.113.514.014.4 14.414.915.415.8 15.716.216.717.3

13.4 14,9 16,3 17.8

13.814.2 15.315.7 16.817.21 18.318.6

10.310.310.310.3 11.411.411.411.4 12.412.412.412.4 13.513.513.513.5

10.3 11.4 12.5 13.5

10.310,410.5 11.411,511.6 12.512,612.7 13.613.713.9

10.6 11.8 12.9 14.0

10.710.911.011.2 11.912.112.312.5 13.113.313.513.7 14.214.514.714.9

14.514.514.514.5

14.6

14.614.815.0

15.2

15.415.715.916.2

16,4

17.017.618.118.7

19.2

19.720.3

15.615.615.615.6 16.616.616.616.6 17.617.617.617.6 18.618.618.718.7

15.6 16.6 17.7 18.7

15.715.916.1 16.816.917.2 17.818.018.3 18.919.119.4

16.3 17.4 18.6 19,7

16.616.817.117.4 17.718.018.318.6 18.919.219.519.8 20.020.420.721.1

17,7 18.9 20,2 21.4

18.318.919.520.1 19.620.220.821.5 20.921.522.222.9 22.122.923.624.2

20.6 22.1 23.5 24.9

21.221.8' 22.723.3 24.124,8 ! 25.626,3

19.719.719.719.7

19.7

19.920.220.5

20,8

21.221.521.922.3

22.7

23.424.224.925.6

26.4

27.127,8

20.720.720.720.7 21.721.721.721.7 22.722.722.722.8 23.723.723.823.8

20.8 21.8 22.8 23.8

21.021.321.6 22.022.322.7 23.123.423.8 24.124.524.9

21,9 23.1 24.2 25.3

22.322.723.123.5 23.523.924.324.7 24.625.125.525.9 25.826.226.727.1

23.9 25.1 26.4 27.6

24.725.526.327.0 26.026.827.628.4 27.228.129.029.8 28.529.430.331.2

27.8 29.2 30.6 32.1

28.529,31 30.030.7 31.432.2 32.933.7

24.724.824.824.8

24.9

25.225.626.0

26.4

26.927.427.928.4

28.8

29.830.731.732.6

33.5

34.435.2

25.725.825.825.8 26.826.826.826.9 27.827.827.827.9 28.828.828.828.9

25.9 26.9 27.9 29.0

26.226.627.1 27.327.728.2 28.328.829.3 29.429.930.4

27.6 28.7 29.8 30.9

28.128.629.129.6 29.229.730.230.8 30.430.931.432.0 31.532.132.633.2

30.1 31.3 32.5 33.8

31.132,033.034.0 32.333,434.435.3 33.634°735.736.7 34.936.037.138.1

34.9 36.3 37.7 39.2

35.836.7 37.338.2 38.739.7~ 40.241.2

29.829.829.929.9

30.0

30.430.931.5

32.1

32.633,233.834.4

35.0

36.137.338.439.5

40.6

41.642.7

30.830.830.930.9 31.831.831.932.0 32.832.832.933.0 33.838.933.934.0

31.0 32.0 33.1 34.1

31.532.032.6 32.533.133.7 33,634.234.8 34.635.235.9

33.2 34.3 35.4 36.5

33.834.435.035.6 34.935.636.236.8 36.136.737.438.0 37.237.938.639,2

36.2 37.4 38.7 39.9

37.438.639.740.9 38.739.941.142.3 40.041.242.443.6 41.242.543.845.0

42.0 43.4 44.8 46.2

43.144.1 44.545.6 46.047.1 47.448.6

34.834.934.935.0

35.1

35.736.337.0

37.7

38.439.139.740.4

41.1

42.543.845.146.4

47.7

48.950.1

35.835.935.936.0 36.836.937.037.1 37.837.938.038.1 38.938.939.039.1

36.1 37.2 38.2 39.2

36.737.438.1 37,838.539.2 38.839.540.3 39.940.641.4

38.8 39.9 41.0 42.1

39.540.240.941.6 40.641.442.142.9 41.842.543.344.1 42.943.744.545.3

42.4 43.6 44.8 46.0

43.845.146.547.8 45.046.447.849.2 46.347.749.250.6 47.649.150.551.9

49.1 50.5 51.9 53.3

50.351.6 51.853.1 53.254.6 54.756.0

39.939.940.040.1

40.2

40.941.742.5

43.3

44.144.945.746.5

47.3

48:850.451.953.354.7

56.257.5

derived by eliminating c from eq. 2 and from L = c T . Since A = A ( U ) = A ( L / h ) , eq. 28 has no explicit solution for L / h . The solution is given in Table 4. Once L / h is k n o w n , U can be determined, and w e are back in case (a). REFERENCES Brink-Kjaer, O. and Jonsson, I.G., 1973. Verification o f cnoidal shoaling: Putnam and Chinn's experiments. Inst. Hydrodyn. Hydraul. Eng., Tech. Univ. Denmark Progr. Rep., 28: 19--23.

284

Buhr Hansen, J. and Svendsen, I.A., 1974. Laboratory generation of waves of constant form. Proc. Coast. Eng. Conf., 14th, Copenhagen, Chapt. 17, pp. 321--339. (Identical with part I in Buhr Hansen et al., 1975.) Buhr Hansen, J., Schiolten, P. and Svendsen, I.A., 1975. Laboratory generation of waves of constant form. Inst. Hydrodyn. Hydraul. Eng., Tech. Univ. Denmark Ser. Pap., 9. Camfield, F.E. and Street, R.L., 1969. Shoaling of solitary waves on small slopes. Proc. ASCE, J. Waterways and Harbors Div., 95 (WW 1): 1--22. Eagleson, P.S., 1956. Properties of shoaling waves by theory and experiment. Trans. Am. Geophys. Union, 37 (5): 565--572. (Disc., 38 (5): 760--763.) Ippen, A.T. and Eagleson, P.S., 1955. A study of sediment sorting by waves shoaling on a plane beach. Beach Erosion Board, Tech. News, 63. Ippen, A.T. and Kulin, G., 1954. The shoaling and breaking of the solitary wave. Proc. Conf. Coast. Eng., 5th, Grenoble, Chapt. 4, pp. 27---47. Iversen, H.W., 1952. Laboratory study of breakers. Gravity waves. Nat. Bur. Stands. Circ., 521: 9--32. (U.S. Govt. Printing Off., Wash., D.C.) Iwagaki, Y., 1968. Hyperbolic waves and their shoaling. Proe. Conf. Coast, Eng., l l t h , London, Chapt. 9, pp. 124--144. Iwagaki, Y. and Sakai, T., 1976. Representation of particle velocity of breaking waves on beaches by Dean's stream function. Mem. Fae. Eng. Kyoto Univ., 38(1). Jonsson, I.G., 1966. Wave boundary layers and friction factors. Proc. Conf. Coast. Eng., 10th, Tokyo, Chapt. 10, pp. 127--148. Ostrovskiy, L.A. and Pelinovskiy, E.N., 1970. Wave transformation on the surface of a fluid of variable depth. Atmos. Oceanic Phys., 6(9): 552--555. Rayleigh, Lord, 1911. Hydrodynamical notes. London, Dublin and Edinburgh, Phil. Mag., Series 6, 21,122: 177--195. Also (1920): Scientific Papers, 1920, Cambridge University Press, Cambridge, 6: 6--21. Also (1964): Dover Publ., New York, N.Y. Shuto, N., 1974. Non-linear long waves in a channel of variable section. Coast. Eng. Japan, 17: 1--12. Skovgaard, O., Svendsen, I.A., Jonsson, I.G. and Brink-Kj~r, O., 1974. Sinusoidal and cnoida] gravity waves, formulae and tables. Inst. Hydrodyn. Hydraul. Eng., Tech. Univ. Denmark, 8 pp. Svendsen, I.A., 1972. Deep water limit for cnoidal waves. Inst. Hydrodyn. Hydraul. Eng., Tech. Univ. Denmark Progr. Rep., 27: 23--29. Svendsen, I.A., 1974. Cnoidal waves over a gently sloping bottom. Inst. Hydrodyn. Hydraul. Eng., Ser. Pap., 6. Svendsen, I.A. and Brink-Kjavr, O., 1972. Shoaling of cnoidal waves. Proc. Conf. Coast. Eng., Tech. Univ. Denmark 13th, Vancouver, Chapt. 18, pp. 365--384. Van Dorn, W.G., 1966. Boundary dissipation of oscillatory waves. J, Fluid Mech., 24: 769--779. (With corrigendum: J. Fluid Mech., 32: 828--829.) Vera-Cruz, D., 1965. Experimental correlation of gravity wave heights. Laboratorio Nacional de Engenharia Civil, Lisboa, Mere., 250. Wiegel, R.L., 1950. Experimental study of surface waves in shoaling water. Trans. Am. Geophys. Union, 31: 377--385.