February, 1970 WW 1 Observations In thetwo-layered Frlerfjord indicated large seasonal vari-ations in Se/Sd. Flux estimates based on simpler methods using either the mechanism of entrainment or diffusion, while neglecting the other mechanism, must be applied with caution.
ACKNOWLEDGMENTS
This investigation was sponsored by The Royal Norwegian Council for Sci-entific and Industrial Research under a contract with The River and Harbour Laboratory at the Technical University of Norway, Trondheim. Most of the field measurements were part of a feasibility study by NORSK HYDROA/S for a nuclear powerplant.
APPENDIX. REFERENCES
I. Batchelor, 0. K., "Diffusion in Free Turbulent Shear Flows." JournalofFluidMechanics.Vol. 2. 1937. pp. 67-80.
Ellison, T. H., and Turner, J. S. "Turbulcot Entrainment in Stratified Flows," Journal of Fluid
Mechanic,,Vol.6, 1959, pp. 423-448.
lppcn, A. T., L.rluary and Coastline hydrodynamics. McGraw Hill Book Co., New York, Chapt. 11-18, 1966.
Macagfa, E. 0., and Rouse, Hunter, "lntcrfacial Mixinj in Stratified Flow. Transactions,
ASCE, Vol. 127, 1962, pp. 102-27.
Pickard, 0. L.. "Oceanographic Features of Inlets in the British Columbia Mainland Coast," Journal ofihe FishResearchBoardofCanada. Vol. 18, No.6, 1961, pp. 907-99.
Phillips, 0. M., The Dynamics ofthe Upper Ocean. Cambridge University Press. Cambridge. Mass., 1966.
Rouse, Hunter and Dodu, .1., "Diffusion Turbulentc a Travers une Discontinuité de Densite," La flouil!e Blanche, Vol. tO, 1955. pp. 522-9.
Turner, J. S., "The Motion of Buoyant Elements in Turbulent Surroundings," Journal ofFluid Mechanics. Vol. 16, No. 1963, pp. I-IS.
7092 February, 1970 WW I
Journal of the
WATERWAYS AND 1-IARBORS DIVISION
Proceedings of the American Society of Civil Engineers
TECHNISCHE UNIVER$[IEJT
Laboratórfu voor
Archfef
Mekolweg 2,2628
CD Delft
TeL 015- 782373 - Fax 015- 781838RELATIVE VALIDITIES OF WATER WAVE THEORIESa
By Robert G. Dean,' A. M. ASCE
(Reviewed by the Technical Council on Ocean Engineering)
INTRODUCTION
The major IniLial developments In water wave theory were conducted in the mid-1800's by Airy and Stokes. In the last two decades, increased offshore and coastal construction activities have placed new demands on our capabili-ties to calculate realistic representations of water wave motion. An increase In the amount of environmental data for waves,wlnds and currents, and an proved understanding of wave generation relationships, have required
im-proved theories in order to develop designs warranting higher levels of
confidence.
At present, there are at least twelve available wave theories which can be selected for design wave representation. The engineer confronted with this choice, however, has only general guidelines pertaining to the relative depth conditions for which a particular theory was developed, and also an intuitive belief that the additional effort required to use higher order wave theories should be rewarded by more accurate results. Because most of the theories are quite difficult to apply, the differences in particular design parameters
Note.Dlscussion open until July 1, 1970. Separate discussions should be submitted for the Individual papers in this symposium. To extend the closing date one month, a
written request must be filed with the Executive Secretary. ASCE. This paper is part of the copyrIghted Journal of the Waterways and Harbors Division, Proceedings of the American Society of Civil Engineers, Vol. 96, No. WWI, February, 1970. Manuscript
was submitted for review for possible publication on April 23, 1968.
a Presented at the September 6-8, 1967, ASCE Conference on Ocean Engineering,
held at San Francisco, Calif.
'Chairman, Dept. of Coastal and Oceanographic Engrg., Univ. of Florida, Gaines-ville. Fla.
106
MEAN WATER LEVEL
February, 1970 WW 1
predicted by several theories are not determined readily.
The purposes of the present study are: (1) To examine the fits to the two free surface boundary conditions as indicators of wave theory validity; and (2) to determine which wave theories provide the best boundary condition fits for important ranges of wave conditions.
THEORETICAL WAVE FORMULATION
It is important to define clearly the problem under consideration. Because the wave problem is usually formulated as a boundary value problem in p0-tential flow, the most reasonable test of the validity of a wave theory would appear to be the fit to this formulation. If a theory could be developed which
fit the formulation exactly, then at least our mathematical representation
would perform its purpose precisely. This would not guarantee, however, that the resulting equations would provide an accurate representation of the natu-ral wave phenomenon. That is,we do not know that the formulation has includ-ed all essential features of the wave motion In nature such as any possible rotation, capillary effects, and so forth. In order to resolve the problem of correct formulation, it Is first necessary to establish theories that faithfully satisfy the specified conditions of the problem.
W
VELOCITY COMPONENTS
lFF (l.'.'FFFF,'l1lIlIF (I .1(Fi'J.'IfI'FFJ'lF #1
FIG. 1.DEFINITION SKETCH. STATIONARY WAVE SYSTEM
The boundary value problem will be formulated for a periodic progressive wave system propagating in an incompressible fluid of uniform mean depth. The motion will be regarded as irrotational and the wave system will be ren-dered stationary by selecting a coordinate system moving wit!i the wave form (see Fig. I).
The differential equation is the Laplace equation In terms of either the ye-locity potential, , or stream function, , I.e.
v2=v2ip=0
in which 2
82 82
-
+The bottom boundary condition requires that there be no velocity component normal to this boundary
Ww 1 WATER WAVE THEORIES
w = 0, z = -h
The kinematic free surface boundary condition is a statement requiring that
the motions of the water particles at the free surface be in accord withthe
motion of the free surface (or, equivalently that water particles on the free surface remain on the free surface),
w
Bx
u-C'
=71 (3)in which C is the celerity of wave form propagation. The dynamic free surface boundary condition, expressing the uniformity of pressure on the freesurface,
is written as 1
ii + -
[(ii - C)2 + 102] = Q,2 -
77In which Q is a constant, and g denotes the gravitational acceleration. This formulation neglects any surface tension effects.
Because most of the available wave theories satisfy the Laplace equation
and the bottom boundary condition exactly (both linear), It seems correct to select the fits to the two free surface boundary conditions (both nonlinear) as indicators of the relative validities of the theories. In conjunction with the de-velopment of a graphical wave force calculation procedure, Dean (2)2 has cal-culated the boundary condition fits b.y several wave theories for two sets of wave conditions. Other indicators have been used as criteriafor establishing the ranges of validities of competing wave theories. Laitone (5) has evaluated the ranges of relative depth for Stokes third order theory and the Cnoidal the-ories; this evaluation was based, in part, on a comparison of the wave celeri-ties for these theories. The results of the study, as interpreted by Laitone, were that for breaking waves, the second order Cnoldaltheory is satisfactory only for Li/i0 > 5 (in which L and h0 are the wave length and depth below the trough level respectively) and the Stokes third order theory is most suitable for Liii < 8. In an evaluation of this type, it is not clear which of the wave theories should be selected as a standard for comparison for all wave condi-tions of Interest. In fact, a wave theory may provide reasonable valuesof wave
celerity whereas other parameters may be represented with less accuracy. The advantage of employing the, boundary condition fits as indicators of wave theory validity is that an absolute basis for a perfect wave theory is automatically incorporated into the procedure,i.e.,perfect fits to the two free surface boundary conditions. The errors in quantities of engineering signifi-cance associated with errors in boundary condition fits, however, remain to be established.
SCOPE OF INVESTIGATION
This investigation comprises a systematic evaluation of (he boundary con-dition fits for a number of wave theories over wave concon-dition ranges of engi-neering significance. (4) 107 (2) (la) (1 b)
00 5 2 to N.0 S.. 5 2 to to, H, BREAKING HEIGHT WAVE tO 2 5 100 2 h/T2 (ft./,.c2I 5 tO'
FIG. 2.WAVE CHARACTERISTICS SELECTED FOR EVALUATION
0.01 fps' to 10 fps2, covering most engineering Conditions of interest. The wave height ranges from one-quarterof the breaking height up to the breaking height, HB, thereby also covering most conditions of engineering interest.
The theories Indicated in Table 1 were included in the present evaluation. Although the Stream function wave theory can be extended to any reasonable order, it was decided to test the filth order version in the present study.
The errors in the kinematic and dynamic free surface boundary conditions as a function of phase angle, 9, are denoted as e (0) and e2 (0) respectively, i.e.
2 Numerals In parentheses refer to corresponding items In the Appendix
I.-iteferences.
WW 1 WATER WAVE THEORIES
and i(U) + [(u (0) - C)2 + w2 (0)] - Q a e2(0) (6) in which Q is defined such that the average of E2 (0) Is zero. For each pair of lilT2 and 1I/T2, the boundary condition errors were calculated at equally spaced
phase angles from the crest (0 = 0) to the trough (0 =r). (It can easily
beshown that the boundary condition errors (0) and e2 (0) are symmetric about
the crest position.) The calculations were carried out for those
theories proving a reasonablefit to the boundary conditions for the particular hIT2 and JIlT2 under consideration, e.g., the boundary condition errors for the Cnoldal theories were limited to hIT21.0 because it was found that the errors
were relatively large for greater values of hiT2.TABLE 1.WATER WAVE THEORIES INCLUDED INEVALUATION
109
Reference
1
Theory Linear wave theory (Airy)
Third order stokes (Skjelbreia
and Hendrickson. as summarized
by LáMohautd and Webb)
Fifth order stokes (Skjelbrela and
Hendrickson)
First order cnoidal (Laitone)
Second order cnoldal (Laltone)
First order solitary (Boussinesq,
as summarized by Munk) Second order solitary (McCowan.
as summarized by Munk) Sbeam function (numerical) wave
theory (Dean) (Fifth order evaluated In present study)
Comments
Terms of order (IL) neglected lu \"
Terms of order neglected
lii\'
Terms of order neglected
Terms of order neglected
/jj\3
Terms of order neglected
hA2
Terms of order neglected
Terms of order
()3
neglectedSatIsfies kinematic free surface
boundary condition exactly Extendable to any reasonable
order
Satisfies kinematic free surface
boundary condition exactly
For intermediate and deep water waves, the variations in wave form were reasonably well distributed from the crest to the trough and only a relatively small number, 21, of equally spaced points were required to obtain a signif I-cant sample of the boundary conditionfits. For shallow water waves, however,
the variations In wave form are concentrated near the crest region and a
greater number, 41, of equally spaced (with phase angle) points were neces-sary to represent the boundary condition fits. (See FIg. 3 for examples of the
waveform variations for shallow and deep water waves.) Undoubtedly, abetter procedure in evaluating the boundary condition fits would beto space the points at phase angles corresponding to equal changes in water surface displacement
(i.e., at equal Aj) rather than at equal O.
108 February, 1970 Ww1
The parameters, H (wave height), T (wave period), and h (water depth) uniquely define the characteristics of a periodic wave system propagating in water of uniform depth. These three parameters may be reduced to two Indé-pendent dimensionless parameters, i.e., H/gT and h/gT2. It Is more conve-nient and common to omit the gravitational term, resulting in the use of the pair of dimensional parameters HIT2 and hIT2 to define the characteristics of a regular (periodic) wave system. In the Investigation described here, the forty pairs of h/T' and HIT2 shown In Fig. 2, were selected for boundary condition evaluation. As seen from Fig. 2, the parameter hIT2 ranges from
1LL iv(0) Sx - u(0) - C (5) 3 6 B 4 4 7 7
To provide a singledimensionless number as a measure of the fit to each of the boundary conditions, the Simpson's rule numerical
approximations to the root-mean_square (rmS) errors were defined as follows for the
kinematic 60 40 20 0 -20 -40 80 60 17(H.l 40 20 0 -20
FIG. 3.EXAMPLES OF BREAKINGWAVE FORMS FOR DEEP AND
SHALLOW WA-TER WAVES
and dynamic free surface boundarycondition errors
v'r
%/(M
Mi
11
-in which Al is the total number (odd) of po-ints at which the boundary condition errors were evaluated.
RESULTS OF DOUNDy CONDITION CALCULATIONS
The majority of the results of the boundary condition calculations will be presented as the rms dimensionless errors expressed by Eqs. 7 and 8;
how-'r
I
l
3 (M - 1) i: ((E),_
+ 4 () +
(7)+ (2) + (2)+1] (8)
ever, an example of the distributed errors[E1 (0), E2 (0)]and the wave profile are presented in Fig. 4 for a third order Stokes wave of one-half breaking
height.
To illustrate the effect of change in wave height, the dimensionless rms errors In kinematic and dynamic free surface boundary conditions for the Airy wave theory are shown in Figs. 5 and 6, respectively, for Cases A, B, C, and D. It is seen, as would be expected, that the errors in boundary conditions increase as the wave height approaches the breaking value.
_!&iLO H 20 7(ft) 0 -20 2 diasl TI ni 113L2 ft. 7-20.0 sac. h 80.ofI.-IT
FIG. 4.EXAMPLE OF DISTRIBUTED FREE SURFACE BOUNDARY CONDITION
ER-RORS AND WAVE PROFILE; STOKES THIRD ORDER THEORY
The rms errors in tile boundary conditions for all theories are shown in
Figs.7 and8 forCase A ('/"
= 0.25). Two of tile theories, the second order Solitary wave theory and the Stream function wave theory, satisfy the kine-matic free surface boundary condition exactly. Note that because the Solitary wave theories are strictly applicable for an Infinite period, only theirasymp-a) Shallow' Firsl H-78.0 T-i0o.0 h100.0 woisi Ord.r ft. CaaIdoi 8isakln Wa. 1kb,7 S.0 Ti. -11/2 U l..dI,.l b) Dsp ois, BVICISQ Third Ordsr Stok..
H 87.511. Tk.,7 Wa.. T10.0i.c. hi000.0 ft. 112 9l.dlan.l It 110 February, 1970 'NW 1
112 February, 1970 WW 1 totic values have been plotted. In the case of the dynamic free surface bound-ary condition, Fig. 8, it is seen that the fifth order Stream function provides
the best fit for hIT' > 0.08. Considering only the
analytical theories, theStokes fifth order provides the best fit for hIT' > 0.2. It is interesting to note that for h/T' < 0.2, the fit to the boundary condition for the linear (Airy) theory is better than that for either the third or filth order Stokes wave the-ories. The reason for this is not known; however, a reasonably logical expla-nation is as follows. For the first order wave theory, the errors in the dynamic free surface boundary condiLion are proportional tozi + w in which the sub-scripts denote the order of the velocity component contribution.
It is welt
known that the second and other higher order effects are relatively mucho. 'a. '0. H I0 10'
FIQ. 5.DIMENSIONLESS ERROR, F.DIMENSIONLESS ERROR,
'4. IN KINEMATIC FREE SURFACE '7ii. IN DYNAMIC FREE
SUR-BOUNDARY CONDITION; LINEAR FACE BOUNDARY CONDITION;
WAVE THEORY LINEAR WAVE THEORY
greater in shallow water than in deep water. Foran NLh order wave theory, the boundary condition errors under considerationare proportional to
N N
(uju, + wjwk) (9)
ja k=N+ij
It is evident from Eq. 9 that the higher the order of the theory, the greater the number of contributions to the boundary condition error. If the higher or-der terms converge and are negligible forN > N', then in order to obtain a
valid representation with small boundary condition errors It will be necessary to employ an order N > N'.It may be possible, however, to employ a theory of order N < N' which will provide a reasonably accurate representationbut
a poor boundary condition fit. The conclusion to be drawnfrom this discussion Is that the boundary condition evaluation may be biased in favor of the lower order theories.
There appears to be some tendency of the higher order theories to diverge in shallow water. This tendency could be due, in part, to the fact that, for the
10' 40' 40' l0' I0' 40' 0' hi T.l Il.i'I 0' I0' '0. I0' 40' 0'hiT.lII.#flI 10'
F. 7.DIMENSION LESS ERROR, F.DlMENSIONLESS ERROR,
, IN KINEMATIC FREE SURFACE IN DYNAMIC FREE
SUR-BOUNDARY CONDITION,H/IiB = 0.25; FACE BOUNDARY CONDITION, ALL WAVE THEORIES H/He = 0.25; ALL WAVE
THE-ORIES
10'
FO. 9.DIMENSIONLESS ERROR, FiQjO.DIMENSIONLESS ERROR,
'. IN KINEMATIC FREE SURFACE '4F/, IN DYNAMIC FREE SURFACE
BOUNDARY CONDITION.JI/1l 1.0; BOUNDARY CONDITION, If/ii 1.0;
ALL WAVE THEORIES ALL WAVE ThEORIES
To obtain a solution to the Stokes fifth order wave theory, it is necessary to solve two nonlinear algebraic equations for the wave length and a parameter
related to the wave height. Referring again to Fig. 8, It is interesting to note that a solution to this pair of simultaneous equations could not be found for H/He = 0.25 (Case A) for h/T' < 0.1. To the writer's knowledge, this is the first time this has been reported in the literature. The existing tabulations
S0L4TUN I -OTON 0TOIC0 '0 414 IWICTOT ITOO II II
A
4-
I
&4RT SOLITANY IPL;
ChI _,-$04.IT*NY I A YMPTOT SOLITflRT I ASYMPTOTO ITN000 FUNCTION FUNCtION V ITONCI Vww1
WATER WAVE ThEORIES 113analytical theories, the boundary conditions are effectively satisfied (to Nih order) on the (N - 1)th order wave form (e.g., the boundary conditions for the Airy wave theory are satisfied on the zero order or mean water surface).
O/T'.ClI.Ii.o) 10'
I
114 February, 1970
'NW 1 (9) of this wave theory do not extend to values of h/T2 less than 0.2. An addi-tional Interesting feature noted In the Stokes fifth order solution was that the wave form variation from crest to trough ceased to be monotonic for /z/T2 values near those where the solution could not be obtained.
Note from Figs.? and 8, that the errors for thesecondorderCnojdal theory are not shown. It was found that the boundary conditionerrors for the second order Cnoidal theory were generally greater than those associated with the first order Cnoidal theory. For example, for Cases C and D and hIT2 c 0.5,
the ratio of second order to first order Cnoidal
theory boundary Conditionerrors ranged from 1.5 to about 10.0. The reason for this is not known; how-ever, the same explanation offered previously for the higherorder Stokes theories may also apply to the Cnoidal theories.
Many of the features noted for the fit to the dynamic free surface boundary condition (Fig. 8) also pertain to the fit to the kinematic free surface boundary condition (FIg. 7) for Case A.
The boundary condition errors for Case D (breaking wave height) are' shown in FIgs. 9 and 10. As for Case A, the Airy wave theory provides a better fit to the boundary Conditions than the higher order Stokes theories for small value of h/T2. The Airy wave theory alsofits the dynamic free surface bound-ary condition better than the Stream function fifth order numerical theory for Jz/T2 < 0.08. As mentioned previously, the numerical theory can be extended to any reasonable order; the improvement In boundary condition fit provided by an eighth order numerical theory is Indicated in Fig. 10 for hIT2 0.1.
Additional figures representing the boundary condition errors for Cases B (/'B = 0.5) and C (I!/HB = 0.75) were develpped, although for the sake of brevity they are not presented herein.
WAVE THEORIES PROVIDING BEST FITS TOBOUNDARY CONDITIONS
It would be desirable to establish the best wave theory for any particular set of wave conditions; this, however, is complicated by two prOblems. First, we have not established that of the theories tested here, the one providing the best fit to the boundary conditions provides all of the wave parameters of
in-terest (e.g.,
profile, water particle velocity components, celerity, and soforth) with the greatest accuracy. For example, It may be possible that a the-ory with relatively large boundary condition
errors will result in accurate
calculated crest elevations, but poor values of water pat tide velocity com-ponents. Secondly, the proper weights to be associated with the fits to the kinematic and dynamic free surface boundary conditions in establishing the abest theory is not known.In view of these problems, it isnotpresenUy possible to establish a "best" theory, however itispossibletoorganize,jngraphjcal form,the wave theories providing a best fit to either of the two boundary conditions. As mentioned previously, the second order Solitary and the Stream function theories fit the kinematic free surface boundary conditions exactly. The Stream function
the-ory can be regarded as fitting the kinematic condition best for all wave con-ditions. It is therefore of most interest to establish those theories that best fit the dynamic condition forwave conditions included in this study.
Using Figs. 8 and 10, and othersfor Cases B and C not presented here, the analytical theories providing the best fit to the dynamic condition were
deter-o,,I'.IfI_,,I
'0.
10"
0" ID., ID'
.fl..II,,,,,,. I
FIG. 11.PERIODIC WAVE THEORIES FIG. 12.PERIODIC WAVE ThEORIES
PROVIDING BEST FIT TO DYNAMIC PROVIDING BEST FIT TO DYNAMIC FREE
FREE SURFACE BOUNDARY CONDI- SURFACE BOUNDARY CONDITION
(Ana-TION (Analytical Theories Only) lytical and Stream Function V Theories) Fig. 11 is based on the analytical theories only. Including the numerical theory (Stream function fifth order), It Is seen (Fig. 12) that this theory pro-vides a better fit to the boundary conditions for the entire range previously covered by Stokes fifth order wave theory and also portions of the ranges covered by Airy and Cnoidal first order wave theory. The regions for which the Airy and Cnoidal first order provided best fits in Fig. 11, but which are cov-ered by the Stream function, are delineated by dashed lines In Fig. 12.
RELATIONSHIP BETWEEN DRAG FORCES AND FITS
TO BOUNDARY CONDITIONS
The total drag force was selected to Illustrate an example of differences between the available theories and also to examine any possible relationship between boundary condition fit and a significant engineering quantity. Two sets of near-breaking wave conditions were selected including a wave In interme-diate depth water and a wave in shallow water. The results of these calculations are presented in Tables 2 and 3 as percentage differences between forces
ww1
WATER WAVE THEORIES 115mined and are presented in Fig. 11. It is noted that the Stokes fifth order theory provides the best fit for deep water whereas the Airy and Cnoldal first order theories provide best fits for shallower water waves. It should be em-phasized that the better agreement with the specified boundary conditions does not necessarily implythebestoveralltheory.Thlswould be the case, however, for the limit of zero boundary condition errors. For example, although the Airy theory provides a better fit than Stokes fifth order for some conditions, we know from observation that the qualitative features of some quantities (e.g., the profile) associated with the Stokes theory are more realistic than those associated with the Airy theory.
116, February, 1970
calculated by the individual theories and those calculated by a high order Stream function theory. The order of the Stream function theory was chosen such that a further increase in order (by one) did not change the maximum velocity by more than 1%.
Results of the intermediate depth water calculations are presented for a wave of 86% breaking height in Table 2. If the drag force for the Stream
func-tion seventh order theory is used as a base, then it is seen that the Stream function and Stokes fifth order results each agree to within 1%. The differences increase for the lower order theoriesup to 30% for the Airy wave theory.
TABLE 2.PERCENTAGE DIFFERENCES IN TOTAL DRAG FORCES (Intermediate Depth Water Vave)a
Theoxy Stream function (7)b Airy Stokes UI Stokes V Stream function (5)
a H = 49.53 ft; h = 72.0 fI; T 12.0 eec; and H/II = 0.86. b Usedas a reference in calculating percentage differences,
H = 15.42 ft; h 20.0 ft; T = 20.0 eec; and H/JIB = 0.99.
b Usedas a reference in calculating percentage differences.
Difference, as a percentage
TABLE 3.PERCENTAGE DIFFERENCES IN TOTAL DRAG FORCES (Shallow Water
Wave)a
WWl
The results of the shallow water wave calculations are presented for a wave of 99% breaking height in Table 3.Again, the higher order (tenth) Stream func-tion results were used as a reference in calculating the percentage differences. There is a much greater range of differences for this case than for the case of intermediate depth waves. It is interesting to note that, of the analytical theories, the Cnoida.l first order and Airy theories provide the best fits to the dynamic boundary condition for the wave characteristics of this example, however the Cnoidal first order theory predictsa force approximately four
Theory Difference, as a percentage Stream function (l0)b Airy 49 Stokes III + 44 Solitary I + 30 Solitary II + 36 Cnoidal I + 105 Stream function (5) + 5
WWl WATER WAVE THEORIES 117
times as large as that predicted by the Airy wave theory.
The drag forcefor the fifth order Stream function theory agrees to within 5% of that determined from the tenth order theory. If the tenth order results are regarded as being correct,' then for shallow water waves there does not appear to be any clear relationship between drag force errors resulting from the various theories and the associated dynamic boundary condition errors alone; it appears that the accuracy of a theory may be closely related to the boundary condition errors only if these errors are very small.'
CONCLUSIONS AND PLANS FOR FIJRTHER RESEARCH
Conclusio,zs.This paper has examined the root-mean-square errors in kinematic and dynamic free surface boundary conditions associated with a number of analytic wave theories and one numerical theory. The principal conclusions derived from this study are the following.'
First, the general status of wave theories for hIT2 > 0.2, for instance, is much more satisfactory than for the smaller values of hiT2. In particular, for the larger relative depths, there is reasonable consistency between the fits to the dynamic free surface boundary condition and the maximum drag force as calculated by the various theories including a seventh order Stream function theory. In shallow water, however, two theories providing approximately the same fit to the boundary conditions predict total drag forces that differ by a factor of four. The force calculated from a Stream function tenth order theory
is considered most realistic and is in closest agreement with forces predicted by the Stream function fifth order theory and the first order Solitary theories,
although these latter two theories did not provide best fits to the dynamic free surface boundary conditions. In shallow water, it is not clear that the boundary condition fit is an appropriate measure of wave theory validity, unless the as. sociated errors are very small. In particular, the Airy wave theory provides a relatively good fit to the boundary conditions in shallow water; however this theory does not represent, many of the observed features of shallow water waves including the strong skewness of the wave profile about the mean water level.
Secondly, it is noted that the set of Stokes higher order wave theories con-verge to an accurate representation of wave motion In deep water; however, in intermediate and shallow water the boundary condition fits are relatively poor. Furthermore, no fifth order Stokes theory solution could be found for shallow water waves and the smaller values of the intermediate depth ranges. The limiting value of h/T2, for which a solution exists, depends on H/Ta and was in the range 0.1 < h/T2 < 0.5 for the conditions examined in this paper.
Finally, it is observed that the second order Cnoidal theory provided a worse fit to the boundary conditions thanthe first order Cnoldal theory for all examined wave conditions. It should be noted that there are other versions of Cnoidal theories; the boundary condition fits of these theories have not yet been evaluated.
Plans for Further Research.Future efforts will be directed toward: 1. Further definition and tabulation of the most realistic wave theories. Plans include utilizing higher order Stream function representations to reduce significantly the 'boundary condition errors and possibly to establish the limits
118 February, 1970
WW 1
ofhIT2and J1/T2 wherethesolitarywavetheories are most valid. In addition, useful engineering wave characteristics will be tabulated using the wave theory or theories In their respective regions of validities, This would eliminate the problem of selecting the best theory for a given set of wave conditions.
Experimental verification of the wave theories established as providing the best representation.
Examination of analytical or numerical techniques or bothfor represen-tation of a nonlinear wave system in which a spectrum of fundamental compo-nents (an aperiodic system) rather thanone fundamental component as in the present study. An attempt will be made to represent the directional features of the wave system.
ACKNOWLEDGMENTS
The investigation presented in this paper was supportedby the Coastal En-gineering Research Center under Contract No. #DACW72-67..C_0009.
APPENDIX 1.REFERENCES
I. Dean, R. G., "Stream Function Representation of Nonlinear Ocean Waves," Journal of Geo-physical Research. 70(18), Sept., 1965. pp. 4561-4572.
Dcan, R. G., "Stream Function Wave Theory; Validity and Application," Proceedings of the ASCESpecialty Conference on Coastal Engineering. 1965, Ch. 12. pp. 269-299.
Ippen, A. T. (Editor), Estuary and Coastline Hydrodynamics, McGraw-Hill Book Co., 1966, Ch. I,pp. 1-93.
Laitone, E. V., "The Suond Approximation to Cnoidal and Solitary Waves," Journal of Fluid Mechanics, Vol.9, Part 3, Nov., 1960, pp. 430-444,
Laitonc, E. V., "Limiting Conditions (or Cnoidal and StokesWaves," Journal of Geophysical Research, 67(4), April, 1962, pp. 1555-1564.
LcMéhaut, B., and Webb, 1. M., "Periodic Gravity WavesOver a Gentle Slope at a Third Order of Approximation," Proceedings of Ninth Conference on Coastal Engineering, ASCE, 1964, Ch. 2, pp. 23-30.
Munk, W. H., "The Solitary Wave and Its Applicationto Surf Problems," Annals of the New York Academy of Sciences v.51. 1949, pp. 376-424.
Skjelbreia, L., and Hendrickson, J. A., "Filth Order GravityWave Theory," Proceedings of Seventh Conference on Coastal Engineering, Council on Wave Research: The EngineerIng Foun. dation, l96l,Ch, lO,pp. 184-196.
Skjelbreia, L., and Hendrickson, J. A., Fifth Order Gravity Wave Theory With Tables of Func-lions. National Engineering Science Co., Pasadena, California, 1962.
APPENDIX 11.NOTATION
The following symbols are used In this paper: C = wave celerity;
WW1 WATER WAVE THEORIES 119
g = gravitational constant; II = wave height;
"B = conventional breaking wave height; It = water depth;
j =
index employed In summation; k = Index employed in summation; M upper limit of summation index;n = Index employed in summation; T = wave period;
Q = quantity in dynamic free surface boundary condition;
u = horizontal component of water particle motion in progressive wave system;
w = vertical component of water particle motion;
z = vertical coordinate, origin in mean free surface, oriented positively upwards;
= error in kinematic free surface boundary condition; = error in dynamic free surface boundary condition;
= displacement of water surfacefrom mean waterlevel,positive upwards; 0 = wave phase angle, zero at wave crest, positive on trailing portion of
wave;
= velocity potential; and = stream function.
