BREAKING WAVE CRITERION ON A SLOPING BEACH Richard Markley Smith "•"iJSSSmtBWi NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS BREAKING WAVE CRITERION ON A SLOPING BEACH by Richard Markley Smith September 1976 Thesis Advisor: E. B. Thornton Approved for public release; distribution unlimited. T176100 SECURITY CLASSIFICATION OF THIS RACE (Wt\un Dote 1 Sntered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO. I. RECIPIENT'S CATALOG NUMBER 4. T|T1_E ,«r»<* Subttilai Breaking Wave Criterion on a Sloping Beach S. TYPE OF REPORT ft PERIOO COVERED Master's Thesis September 1976 ft. PERFORMING ORG. REPORT NUMBER 7. AUTHOR^*; Richard Markley Smith ft. CONTRACT OR GRANT NbMIEROI 9. PERFORMING ORGANIZATION NAME ANO AOORESS Naval Postgraduate School Monterey, California 93940 10. PROGRAM ELEMENT. PROJECT, TASK AREA ft WORK UNIT NUMBERS 11. CONTROLLING OFFICE NAME ANO AOORESS Naval Postgraduate School Monterey, California 93940 12. REPORT DATE September 1976 13. NUMBER OF PAGES 97 14. MONITORING AGENCY NAME ft AOORESSff/ different from Controlling Office) Naval Postgraduate School Monterey, California 93940 IS. SECURITY CLASS, (ol thle report) Unclassified lt«. DECLASSIFI CATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION STATEMENT (of thle Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the omotrmct entered In Bloek 20, If different from Report) tft. SUPPLEMENTARY NOTES 1*. KEY WOROS (Continue on NfWM aide If neceemmry end Identity oy mlock nummer) Breaking waves, wave theory, water waves 20. ABSTRACT (Continue on rereree tide II neeeeemry end Identify ey eteek numket) The various wave theories, theoretical breaking criteria and derived breaking criteria are reviewed for shallow water waves. To account for the non-linear hydrodynamics present in a shallow water wave breaking on a beach with a sloping bottom, the pertur- bation technique of Iwagaki and Sakai is used to derive a second order expression for the horizontal water particle velocity for do ,: (Page 1) Fan"M7, 1473 EDITION OP I NOV «• IS OBSOLETE S/N 0102-014-460 1 | SECURITY CLASSIFICATION OF THIS PAOE (When Ddtm Entered) JuCUWITY CLASSIFICATION OF THIS P»CEf*>i.n r>»f« Enimrad 20. Abstract (Cont'd) long waves. The kinematic breaking criterion is applied to the derived c(2) and uA^J values to establish breaking. The results indicate that the ratios of 7^-^/l,0 and h^/HQ provide reliable breaking criteria. Each of the parameters is dependent only upon beach slope and H0/LQ. Theoretically derived values for h^/HQ compare favorably with field measurements and offer significant improvement over previous theory. Predicted breaking depths are less than those present in experimental data, suggesting extension to higher orders may be warranted. DD Form 1473 1 Jan 73 b/N 0102-014-6601 SECURITY CLASSIFICATION OF THIS PAGEr*h»n Dmtm Enfrmd) Breaking Wave Criterion on a Sloping Beach by Richard Markley Smith Lieutenant, United States Navy B.S., United States Naval Academy, 1971 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL September 1976 D'JOLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA 93940 ABSTRACT The various wave theories, theoretical breaking criteria and derived breaking criteria are reviewed for shallow water waves. To account for the non-linear hydrodynamics present in a shallow water wave breaking on a beach with a sloping bottom, the perturbation technique of Iwagaki and Sakai is used to derive a second order expression for the horizontal water particle velocity for long waves. The kinematic breaking criterion is applied to the derived c'2' and u'2' values to establish breaking. The results indicate that the ratios of /^ /LQ and hi /HQ provide reliable breaking criteria. Each of the parameters is dependent only upon beach slope and H0/L0. Theoretically derived values for h^/H compare favorably with field measurements and offer significant improvement over previous theory. Predicted breaking depths are less than those present in experimental data, suggesting extension to higher orders may be warranted, TABLE OF CONTENTS I. INTRODUCTION -- - 12 II. THEORETICAL BREAKING CRITERIA 17 A. KINEMATIC BREAKING CRITERION 18 B. DYNAMIC BREAKING CRITERION 18 C. GEOMETRIC BREAKING CRITERION --- 19 III. DERIVED BREAKING CRITERIA 20 A. BREAKING WAVES OF PERMANENT FORM (SHALLOW WATER, CONSTANT DEPTH) 20 1. Kinematic Breaking Criterion 20 a. Crest Angle 20 b. Wave Steepness 22 c. Wave Height to Depth Ratio 23 2. Dynamic Breaking Criterion 24 a. Water Particle Acceleration 24 b. Vertical Particle Acceleration 24 c. Vertical Pressure Gradient : 25 B. WAVES WHICH DEFORM AS THEY SHOAL (SLOPING BOTTOM) 25 1. Kinematic Breaking Criterion 25 2. Geometric Breaking Criterion 26 IV. REVIEW OF LONG WAVE SOLUTIONS - 29 A. METHOD OF CHARACTERISTICS --- - 29 B. IWAGAKI AND SAKAI PERTURBATION TECHNIQUE — 37 C. PHASE SPEED RELATION -- 39 V. DERIVATION OF WAVE-INDUCED VELOCITY AND WAVE PHASE SPEED USING IWAGAKI AND SAKAI PERTURBATION TECHNIQUE 44 A. DERIVATION OF u^ - 44 B. SECOND ORDER EQUATIONS FOR n AND u 52 C. DETERMINATION OF THE PERTURBATION PARAMETER, « 54 D. SECOND ORDER EXPRESSION FOR c 58 E. DETERMINATION OF a 60 VI. BREAKING CRITERION DERIVED 63 A. APPLICATION OF KINEMATIC BREAKING CRITERION - - 63 B. RESULTS 64 VII. CONCLUSIONS - — 74 APPENDIX A - DERIVATION OF n^ - 75 APPENDIX B - COMPUTER PROGRAMS - 80 LIST OF REFERENCES 92 INITIAL DISTRIBUTION LIST --- 96 LIST OF TABLES I. Numerical Results of Derived Shallow Water Breaking Criteria 65-66 LIST OF DRAWINGS 1. Waves on a Sloping Beach 11 2. Enclosed Crest Angle for Kinematically Limited Wave 21 3. Integration by Finite Differences 32 4. Propagation of Disturbances 42 5. Applicable Range of Solution 56 6. Plot of hb/HQ vs. H0/LQ 68 7. Breaking Wave Profile, Period =10.6 Seconds 70 8. Breaking Wave Profile, Period = 15.0 Seconds 71 9. Program 1 10. Program 2 11. Program 3 Approximation of a 86 Refinement of a 87 First Approximation of x and t at Breaking 88 12. Program 4: Determination of x and t at Breaking 89 13. Program 5: Calculation of n^/h^ and h^/L 90 14. Program 6: Plot of n at time of Breaking 91 8 TABLE OF SYMBOLS AND ABBREVIATIONS a Constant Related to Wave Height «< Perturbation Parameter 0 Angle of Bottom to Horizontal c Wave Phase Speed g Gravitational Constant H Wave Height h Depth of Water Referenced to Still Water Level = i*x i Beach Slope ■ tan 3 JQ Zero-Order Bessel Function J-, First-Order Bessel Function L Wave Length (feet) N Zero-Order Neumann (Weber) Function o N, First-Order Neumann (Weber) Function tj Elevation of Free Surface Referenced to Still Water Level p Pressure p(x) 2 a (x/gi)% p(x) la (x/gi)% - -£- p Density of Water ■ I U < 03 a z a. O _i < z d (A > 0) fa -11 I. INTRODUCTION Attempts at deriving breaking criteria have been made since Stokes (1847) presented his classical development. Derivations of the many available water wave theories all involve the solution of Euler's equations of motion coupled with the continuity equation for incompressible, inviscid, irrotational flow subject to certain boundary conditions. Breaking, or near breaking waves have very steep profiles in which the wave height is large compared to the relevant length scale implying the hydrodynamics are highly non- linear. At the onset of breaking strong vorticity is introduced at the surface near the crest and the assumption that the motion is irrotational is no longer valid. The strong non-linearities and induced vorticity make the analysis of breaking waves mathematically very difficult. This thesis is concerned with finding an incipient breaking criterion for waves shoaling on a beach before vorticity is induced but including non-linear effects. The discussion will be limited to shallow water wave theory. Solution of the equations of motion has required the application of physical assumptions associated with various wave characteristics. Thus, each formulated theory is 12 limited in its range of applicability to regions where its underlying assumptions are valid. Shallow water wave theory may be classified according to the bottom being horizontal or sloping. This is an important restriction because field and laboratory measurements of breaking waves suggest that the bottom slope is an important parameter in the classification of breaking waves. The simplest form of solution to the wave problem is to linearize the equations of motion assuming a horizontal bottom. Peregrine (1972) has shown that the linearized equations apply to regions where the ratio H/L and H/h are both much less than one, where H is wave height, L is wave length, and h is water depth. Consequently, this solution is restricted to waves of infintesimal height and not applicable to steep breaking waves. Stokes (1847) was the first investigator to present a higher order solution applicable to finite amplitude waves, though limited to a horizontal bottom. In his development, he transforms the basic equations to an equivalent set by using a velocity potential, $ . The solution is obtained by expanding the velocity potential using a perturbation scheme which employs H/L as a perturbation parameter. To the lowest order, Stokes1 method results in linear theory. The accuracy of Stokes' solution at a particular order 13 decreases as the ratio h/L decreases. Dean and Eagleson (1966) attribute this inaccuracy to increasing bottom influences and a decrease in the importance of vertical particle acceleration. De (1955) concluded that Stokian theory should be discarded for values of h/L of 0.125 and less. Dean (1968) expanded a stream function using a numerical perturbation scheme and was able to raise a "Stokes" type wave to any desired order. In this manner he was able to solve for incipient breaking. When the relative depth is very small, as in very shal- low water, the vertical acceleration can be neglected and the fluid path curvature is small. Hence, the pressure is assumed to be hydrostatic as the vertical component of motion does not influence the pressure distribution. The resulting equations are referred to as the "long wave equations." A sloping bottom and finite amplitude are allowed by the long wave equations. Freidrichs (1948) was able to derive the long wave equations by a rigorous mathematical approach. Utilizing quantities h and L which represent typical depth and length scales, the Airy equations were non-dimensionalized. This procedure resulted in a large stretching of the horizontal coordinate relative to the depth coordinate. A perturbation analysis was then applied with the perturbation parameter 14 2 2 9 = h /L . He found that to the lowest order the pressure was indeed hydrostatic and that the long wave equations resulted. A third length scale for shallow water theory, which utilizes the wave amplitude a, is the Ursell parameter £L3 L h Ursell (1953) showed that long wave theory belongs to the regime a L 3 L h Boussinesq (1872) assumed that the pressure was no longer hydrostatic, which allows inclusion of vertical water particle velocities, but results in a limitation on the wave height. The Boussinesq equations apply when a L 3 , — — ~ j_ h h implying the waves are not as high and the water is relatively deeper than for long wave theory. Korteweg and de Vries (1895) simplified the Boussinesq equation by considering waves which travel only in one direction over a horizontal bottom. Extending Boussinesq' s equation in this manner they produced a wave theory they termed "cnoidal." The limiting case of cnoidal theory is the solitary wave. The cnoidal/solitary theory has received 15 considerable attention by researchers in recent years. Keller (1948) extended the perturbation analysis employed by Freidrichs to the first approximation. He found that to the first order his results were those of Korteweg and de Vries. Laitone (1960) continued the process further, obtaining second order approximations to cnoidal/solitary waves by solving Freidrichs' method to the fourth order. The assumption of waves traveling only in one direction precludes a reflected wave and imposes the important restriction of a horizontal or nearly horizontal bottom. The long wave equations are used in this study because it is felt that properly including the sloping bottom is the most important next step in seeking a breaking wave criterion. The possible importance of vertical accelera- tions in the wave breaking process are recognized, but are assumed negligible in order to obtain mathematical tractability. A second order solution of the long waves is sought and a breaking criterion derived based on a kinematic instability condition. 16 II. THEORETICAL BREAKING CRITERIA It is desired to formulate breaking criteria which can be expressed in terms easily observable and measurable. The several breaking criteria which have been developed may generally be broken into two categories. First, there are those which are derived from waves of steady form. The waves considered are assumed to be in shallow water of constant depth. Both Stokian and cnoidal/solitary wave theories have been employed in these investigations. The second group of derived criteria consists of those con- cerned with waves which deform as they shoal. Research in this category of waves has been confined to long wave theory. The derived criteria for horizontal and sloping bottoms will be investigated separately. In order to determine wave parameters which can be used to predict the breaking of waves, it is necessary to first formulate some type of limiting value. The physically significant breaking criteria are the kinematic, dynamic and geometric criteria. 17 A. KINEMATIC BREAKING CRITERION Originally formulated by Rankine (1864), the kinematic breaking criterion states that the limiting value of the water particle velocity at the wave crest, uc, is the wave phase velocity, c, uc , a prop- erty of the wave, is restricted to %<$— 1, produced physically impossible velocities. Hence this ratio was established as the limiting value. As the value of Q approaches the limit of 1.0 the cnoidal wave approaches a solitary wave form. Therefore, the limiting value for a solitary wave to the third order is H/h = JL = 0.7272. 11 Laitone carried his solution for the solitary wave to the next higher order and found a value of H/h = V3 - 1 = 0.7321. These two results compare favorably. 24 c. Vertical Pressure Gradient Laitone (1963) developed a different limiting value for H/h than that previously discussed when he applied the vertical pressure gradient criterion. In this case he found that for H/h = (2 £/3)%, %<£<1, the pressure gradient is zero for cnoidal waves. This expression was derived to the third order. If the ratio H/h increased from this value, the sign of the gradient would reverse, a condition that he concluded could not exist. In the limit- ing case of (£ = 1.0, which gives a solitary wave, the limiting value of H/h = 0.812 is obtained. This differs significantly from his previous limit of 0.7272. B. WAVES WHICH DEFORM AS THEY SHOAL (SLOPING BOTTOM) As a wave shoals over a sloping bottom, the wave height and profile are altered. The theory of long waves has been most generally applied to research in this region. The derived breaking criteria have been formulated through the use of the kinematic and geometric breaking criteria. 1. Kinematic Breaking Criterion The first attempt at applying the kinematic breaking criterion to a deforming wave was made by Ayyar (1970). His derivation made use of the concept of a wave front. Simply stated, a wave front is the position where a discontinuity 25 in the surface slope occurs. The slope of the surface will be zero in front of the wave front and negative behind it. Ayyar's approach was to obtain the slope at the wave front, integrate to find the free surface tj , and then apply the kinematic breaking criterion. Assuming the geometry of the plunging breaker, he then derived the value of y^/h^ = 2.0, where y^ = height of breaker crest above the bottom and h^ = depth at breaking point below the still water level. Several problems exist in Ayyar's derived criterion. First, the derivation is based on the geometry of a plunging breaker and thus excludes the other categories of breaking waves. Additionally, his formulation assumes that breaking will occur at the wave front. Gaughan, Komar and Nath (1973) point out that this may not be a valid assumption. 2. Geometric Breaking Criterion Use of the geometric criterion has been made by several researchers in developing long wave breaking criteria. Stoker (1957) showed that long waves could obtain a surface slope of infinity. He extended his work and used a numerical methods technique to arrive at a solution to the problem. The method, however, is not satisfying in that it requires a number of approximations to be made and requires recalculation as the initial 26 conditions are altered. Further discussion of this pro- cedure will be presented in a subsequent section. Burger (1967) and Greenspan (1958) used the wave front concept and the vertical slope criterion to predict the horizontal distance traveled from the wave front at time t=0 to the point of breaking. The result was (l - |S_ \ \ S-H4 / Xb _±L fl -2S_W3 where, M = slope of the beach, h(x) = h-^-Mx, h-^ = initial water depth, and S = initial surface slope at the wave front. Gaughan, Komar and Nath (1973) discuss several limitations to this result. As was the case in Ayyar's work, the breaking is assumed to occur at the wave front. The surface slope behind the front is not examined. Some other point, such as the wave crest, may become vertical prior to this condition occurring at the wave front. Also, the use of horizontal distance to breaking is not a useful criteria. It is not easily measurable, having a somewhat arbitrary origin. Prediction of the wave height at breaking is a much more useful parameter. Another approach involving the vertical surface slope was taken by Mei (1966). The basis for his work was originally proposed by Carrier and Greenspan (1959). The technique is to produce a set of characteristic equations 27 from the long wave equations and then to make an additional transformation through the use of a final pair of inde- pendent variables, ( =0 Mei solves this equation to the first order. He then follows a procedure used by Carrier and Greenspan whereby the Jacobian J = d(x,t)/ Mc, X ) is investigated. This Jacob ian will vanish at points for which the surface slope is infinite. Mei was able to obtain an expression for h^ /H which was dependent upon HQ, LQ and the bottom slope. The subscript 0 denotes deep water conditions. Unfortunately, Mei found that his theory compared poorly with experimental data. Predicted breaking depths were too large, dependence on the beach slope was too great and the wave profiles were too sinusoidal. Mei suggests that these discrepancies could possibly be eliminated by extending the solution to a higher order. 28 IV. REVIEW OF LONG WAVE SOLUTIONS Essentially two approaches exist which can be used to solve the nonlinear long wave equations. The first pro- cedure, initially formulated by Stoker (1947), makes use of a solution technique known as the method of character- istics. A final solution utilizing this method can be made either through numerical calculation or by an analyt- ical approach. Iwagaki and Sakai (1972) propose a second solution procedure which involves a perturbation expansion. An evaluation of each of these techniques follows. A. METHOD OF CHARACTERISTICS The application of the method of characteristics to the long wave equations was explained by Stoker (1958). Neces- sary to this development is the acceptance of the wave phase speed relation c = [g(7^+h)]2. The validity of this equation is discussed in a subsequent section. From this expression it is seen that and ct=37?t/^<^- <2> 29 The long wave equations are ut+uu-x=-37x <3) and Cu(^+h)3x= "7ft . (4) Substitution of (1) and (2) into these equations yields ut + uux+Eccx-ghx=0 (3a) and 2ct+ 2acx + cax = O (4a) Relations (5) and (6) result from the respective addition and subtraction of (3a) and (4a) , {a/a-t +(u+c)VDx]-(u+ac-Qhx-t)= O (5) {Vat +(u.-c)a/ax]-(a-ac-qhx-t)= O. (6) The interpretation of (5) and (6) is essential to the development. (5) implies that the function (u+2c-ghxt) remains constant for a particle moving with a velocity of u+c. A similar evaluation of (6) can be made. In other words, two characteristic curves, Ci and Co, are defined such that 30 Cx: dx/d-t = u + c and Cz: <±x/l± = u-c where U.-t"2c-qhx"fc a K±= A constant Along C , and U-2C- Q ^xt= K^- A COWSTA^T ALONG G ^ The system of equations given by (7) and (8) is equivalent to that defined by (3a) and (4a), hence, a solution of either set provides a solution for the other. It now becomes necessary to determine the character- istics so that a solution may be obtained. Stoker's technique was to make use of a method of successive approxi- mations. He assumed that the initial values of u and 7j could be observed. Use of c = Cg(7^+h)J 2 could then pro- vide the phase speed at time t=0. These initial conditions are assigned values such that for t=0 u(x,o)= u(x) 5 > (9) c(x,o) = c(x) ) The task is to approximate u and c for small increments of time. Figure 3 clarifies the discussion. A series of points along the x-axis, which are separated by a small distance 8 , are considered. Since the values J x' of u and c are known for each of these points from (9) , the slope of the characteristics C-, and CL at each point can 31 INTEGRATION BY FINITE DIFFERENCES Figure 3 32 also be obtained through the use of (7). These slopes are used to construct straight line segments from the points along the x-axis. Location of points 5, 6 and 7 is deter- mined by the intersection of these line segments. A source of error is inherent in the use of straight line segments to approximate the characteristic curves. This error is restricted to a minimum by using sufficiently small values of 5X. For this case, the tangents to the curves provided by the slopes give good approximations to small segments of the curves. Equations (8) and (9) give the character- istics issuing from points on the x-axis as along (^ : u+2c-Qhx"t= u.+ Zc along C2 : U-Zc -q ^x "t = U.- 2- C The values of x and t can be obtained for the points 5, 6 and 7 (this could be accomplished graphically, for instance) Equation (10) can then be used to determine the values of u and c at these points. The procedure can be continued to obtain values for u and c at the subsequent points 8, 9 and 10. In this manner, a net of points at which values of u and c are approximated could be constructed which covered an entire field of concern. Values at intermediate points could be found by interpolation. Stoker states that as 33 5 -»0, the process will converge to an unique solution of (7) and (8). Numerous objections have been raised against this particular numerical approach. Ayyar (1970) categorizes these into three areas. First, the solution is not explicit and requires re-calculation for a change in initial condi- tions. This alone makes the procedure difficult to use effectively. A second criticism is that a spilling type breaker is always predicted; thus the elimination of the several other breaker types places a severe restriction on the method. Finally, LeMeliaute (1968) has given evidence that the technique produces incorrect predictions of breaker points. An alternative and more analytical solution to equations (7) and (8) has been offered by Carrier and Greenspan (1958) and was discussed in Section III.B.2. Two independent variables ( (u) to which the authors propose a solution, 34 where A = constant to be determined and <£ = a phase lag. At this point, Carrier and Greenspan made use of the verti- cal surface slope criterion and determined values of a Jacobian, J = d (x,t)/ d (cT , A) , which would exist for the specific case of non-breaking waves. Consequently, the remainder of their study is of little value to our discussion, Mei (1966) considered the solution technique of Carrier and Greenspan as it applied to the case of breaking waves. He selects + ^(^''^^(x/el'^tirZ+S] (13) where B = a constant to be determined, i = slope of the beach, J = zero-order Bessel function, and NQ = zero-order Neumann (Weber) function as his solution. Using this rela- tion, Mei derived first order equations for u and t\ . The coefficient B is determined by matching the solution to that for an outer region where a horizontal bottom Airy Theory was applied. Mei determined a breaking parameter by applying the geometric breaking criterion. Comparison with experimental data indicated that Mei's results were not accurate. He attributed these deficiencies to the fact that the solution was not carried to higher orders. Previous studies conducted by Benney (1966) had 35 suggested higher order derivations were required for shallow water breaking conditions. Solution to a higher order is required in both the inner and outer regions if the coeffi- cient B is to be accurately determined. The procedure of Tlapa, Mei and Eagleson (1966) provides a perturbation expansion for the outer region. Review of this method shows that the third order solution must be considered to uniquely determine the second order coefficients. Assuming that a successful solution to the outer region expansion could be found, or that B can be found independent of the off-shore regime, difficulties still remain in the near shore area. This is the location of breaking and thus of concern. The solution would be vastly simplified if a method could be found in which only the velocity potential used by Mei need be perturbated. Unfortunately, the single equation for the velocity potential prevents this approach. The quantities hi^=o (19) Equations (16) and (17) together give two equations in two unknowns which can be solved to obtain first order expression for 7j and u of the form - sin -^H£YH81 (22) Iwagaki and Sakai presented experimental results which confirmed that their solution technique produced valid results for the wave form. C. PHASE SPEED RELATION In several of the previous sections mention has been made of the phase speed relation, c = £g(?7 +h)J 2. Although this equation is widely used in the literature on wave theories and is generally accepted; few discussions have been presented which establish its validity. The question deserves some attention prior to application of the kinematic breaking criterion. The first inclination toward the use of the relation comes from the linearized long wave equations for water of constant depth. Stoker (1958) showed that for these conditions u satisfies the relation u.xx- ^r- u_- 0 . 3* tt ?£ can be shown to satisfy the same equation. The implica- tion of this expression is that the speed of the wave dis- turbance is given by c = (gh) 2. Another indication that the expression may indeed be valid comes from what Stoker terms the gas dynamics analogy. Stoker credits the development of this idea to Riabouchinsky (1932). Consideration is made of a mass per unit area 39 expressed by f = I 6? + h) (23) where o = density of water. Thus (r?7f (24) The force p per unit width is defined as p = ) pda . (25) By using the hydrostatic pressure relation, p may be reduced to P=3«/£ C7?^) = y£? p . (26) Multiplying both sides of equation (3) by o(^+h) gives ^(?p+HX"t+"-3^(7?+h)7x <«> which may be re-expressed, using (23) and (26) , as f'(u^+UUx")=-px+|^Kx. (28) Equations (23) and (24) can be used to re-write (4) as f^")x= -\^ . (29) Equations (26) , (28) and (29) , when combined, give results similar to those of gas dynamics for one dimensional flow, 40 the only difference being the presence of the term g"ohx in (28) . For the case of constant depth, this term vanishes. In gas dynamics, the sound speed is given by c =L Vdoj • Applying this with (23) and (26), Although these discussions provide an insight into using the phase speed relation, they can hardly be consid- ered as a definitive argument. A satisfactory derivation can be obtained by returning to the methods of character- istics. The review of this technique explained that the phase speed was assumed to be given by c = Cg(7? +h)] 2« The theory could have been derived equally as easily by simply defining a quantity c = £g( r^ +h)J *. No physical significance need be immediately applied to c. This being the case, an identical development can be made resulting in the same characteristic equations. The task then becomes to discover the physical meaning of c. Stoker (1958) presents the following argument. It is assumed that the initial values of u and 77 are given for a body of water which is in motion. The value of c for this instant is given through c = [_g(^+h)J 2. Figure 4 will aid in the explanation. Consider a disturbance created over the segment of the x-axis Q-^ Q£. How will this effect the solution? Each point Q on the x-axis has what is termed 41 PROPAGATION OF DISTURBANCES Figure 4 42 as a range of influence. This is the region of the x,t plane in which the values of u and c are influenced by the initial conditions at Q. This area is defined by the characteristics issuing from Q. Consequently, for the particular case under consideration, the segment Q-, Q« will have an influence on u and c for the shaded region in Figure 4. The two curves are given by C-^:dx/dt = u + c and C£:d /d. = u - c. u is defined as the horizontal velocity of the moving fluid. The speed of the disturbance as it moves through the flowing water must therefore be. given by c. Thus, the validity of the phase speed relation seems apparent. One final argument can be formulated by the use of the method of characteristics. Although he does not discuss this aspect of his study, Greenspan (1958) outlines this proof. Again, consider the method of characteristics to be formed using a term c defined as c= CgC7?^)!] • A wave which is progressing into quiescent water is considered. Of concern dx is the forward moving characteristic curve, -r— = u + c, dt which contains the wave front. Since the water immediately preceding the wave front is quiescent, the value of u must be zero at the front. The characteristic, and consequently the wave front, must be progressing with speed c. Hence, the phase speed is given by c = £ g( 7f +h)J 2 . 43 V. DERIVATION OF WAVE -INDUCED VELOCITY AND WAVE PHASE SPEED USING IWAGAKI AND SAKAI PERTURBATION TECHNIQUE A. DERIVATION OF u^ The second order wave induced velocity u' ' is derived using the perturbation technique used by Iwagaki and Sakai to derive a second order surface profile. All terms used are defined in the list of symbols. The initial equations employed are (18) and (19), These equations result when u and 77 are given as power series expansions of a small quantity c< and these expres- sions substituted into the long wave equations. Eliminating 77^ ' from (18) and (19), substituting h = i*x and grouping terms of u^' yields Using the first order term for u given by Iwagaki and Sakai and differentiating with respect to x gives 44 + + cos^t^N^]-^) Nx[p6$)j (3D where p (x) = 2. W]--J-,N1[?M])] . (33) 45 and When u^ ' is differentiated with respect to t we obtain w~t (x/t J = a. Or) ^~* *" ( COSart'J^L p(x)J - S.K^t-N^p^]} . (34) Similarly, differentiating 77 '(x,t) yields a set of equations , -SlMTt- Nx[p(xTl 3 (35) - SfNTt-N^pOO 3] Equations (31) through (36) can be used to find expressions for the individual terms on the right hand side of (30) . When these expressions are combined, the right side of (30) is evaluated as af x-'f Ji JjpW] Nj?to] 46 +NfCp(x)]-^Ni[p(x)])} + The asymptotic expansions of the Bessel and Neumann (Weber) functions are Jr (w)~(^C) 'cos (w- r/£ -■%) and .. Nr(wV(%^)/2s.N(w-r^-y4)) r (38) the approximations being valid for values of w such that Jw| £ 1.0. Substitution of (38) into (37) gives as a final expres- sion for the right side of (30) , cos zfio ] W CD 47 2. 2- -3 ; ~ °- a >< (39) where <)slN^ \ _ + 5in2«rt [-4 (46) B(x)-#r(+)V,fec.sr2ffxV| and x r (48) C(x^ Substitution of (46), (47) and (48) into the left side of (30) yields 49 cos Similar to the development of Iwagaki and Sakai for Equation (51) can be simplified to %^l£(K)\-**io^Uz 4.0, the left gi side of (53) is evaluated as 1.044. Consequently, the use of this ratio as a limit on the range of applicability of the solution may seem appropriate. Further discussion of this parameter is presented in Section V.C. B. SECOND ORDER EQUATIONS FOR 7^ AND u Originally, it was assumed that r? and u could be expressed as and It is therefore necessary to combine the expressions for 7\ and 7T* ' and for u^ and u' ' to determine the final relations. It is noted that for 77(x,t), Iwagaki and Sakai neglected the second part of (22) which is inde- pendent of t. They felt justified in so doing since these terms effect only the Stillwater depth. Their study was concerned with wave heights and profiles and consequently the Stillwater depth was not required. This simplification is not valid for the present study since the phase speed, given by c = £g(7^+h)] 2, is effected by the Stillwater depth. Furthermore, the ratio of the coefficients of the first term independent of t to the first term on the right 52 side of (22) is ["JL_ (-&_)^ i] . Utilizing the upper limit for T (•&) 2 of 4Tr/i given by Iwagaki and Sakai, the importance of including these terms is apparent. (The upper limit of T (§) 2 is discussed in Section V.C.) n When the first order equations for u and 77 given by Iwagaki and Sakai, equations (20) and (21), are expressed using the asymptotic approximations for the Bessel and Weber functions the following results are obtained, * zr(-£f- - "A} £^ to (61), J-r./L = O. 015*7. (62) 57 Equation (62) will be adopted for the value of ©< throughout the majority of this presentation. Iwagaki and Sakai discuss the fact that use of (62) restricts the discussion to waves for which H/L £=, 0.006. o' o For greater H 0/L values, the theoretical energy flux curves predict that the deep water waves will break prior to arriving at the point where h-t/L = 0.0157. In order to extend the theory to situations where H /L exceeds 0.006, J o o ' it is necessary to increase the upper limit placed upon h/L. Several cases for which h/L is 1/15 are investigated in this study to determine the applicability of the theory to a domain of H /L values greater than 0.006. o o ° A final comment may now be made concerning the approxima- tion of (53). Use of (59) evaluates (60) as 1.16, not the assumed value 1.0. However, when these values are multiplied 2 by the quantity c* , as is required by (56) and (57) , the difference becomes negligibly small. Hence the determina- tion by Iwagaki and Sakai for the limit 2 cT (-4-) 2 > 1.0 seems appropriate. D. SECOND ORDER EXPRESSION FOR c A second order expression for the phase speed c can be found by substituting (56) into the phase speed relation, c " [g( ^T+n)] > giving 58 (jr< J (63) The negative square root is utilized due to the fact that the wave is progressing in the negative x-direction. E. DETERMINATION OF a Prior to applying the kinematic breaking criterion to the preceding equations, the value of the term a in (56), (57) and (63) must be determined. This evaluation follows closely that which was outlined by Iwagaki and Sakai. As mentioned previously, these authors neglected those terms in n which were independent of t. Although these terms are included for the determination of c, they can be elim- inated for the purpose of establishing the value of a. The validity of this simplification stems from the fact that the value of the constant will be evaluated at the point 59 where h = h-, , the deepest depth in the applicable range of the solution. Examination of the terms independent of time in (56) shows that they decrease in absolute value for increasing x; the terms becoming negligibly small at the point where h = h-. . Consequently, for the determination of a, 77 will be assumed to be expressed by +* +ta*-[V0 ^-V^J (64) Substituting °< = h-j^/L into (64) yields where r{Ax - A coS Equations (67) , (69) , (71) and (72) can now be substituted into (70) to evaluate a. Simple computer techniques, as discussed in Appendix B, provide a determination of a for specific values of i, H /L and period T. r o o 62 VI. BREAKING CRITERION DERIVED A. APPLICATION OF KINEMATIC BREAKING CRITERION The derived horizontal wave induced velocity and wave phase speed are used to derive a breaking criterion. As stipulated by the kinematic breaking criterion, breaking will occur when the horizontal particle velocity equals the phase speed velocity. Theoretically, this condition may exist for several points in space and time. Of con- cern, however, is the specific case for which the horizontal distance from the beach at which u equals c is maximized. This will be the first position at which the approaching wave may break and hence all other cases are purely imaginary. The numerical solution technique employed is relatively simple. Subtracting the second order relation for u, (57), from that for the phase speed c, (63), equating the resulting expression to zero, and solving for x and t gives points in space and time at which the kinematic breaking criterion is satisfied. Examination of this solution set yields the maximum horizontal distance at which u equals c. The computer techniques employed are discussed in Appendix B. 63 B. RESULTS The numerical results obtained are summarized in Table 1 Several wave conditions are investigated in which the values for the beach slope, wave period, and deep water wave height to deep water wave length (H /L ) are varied. The specific selection of 0.119298 for the beach slope and 8.6 seconds for the period was made to conform with a future study (Hulstrand, 1976) in which experimental data will be used to verify these theoretical results. All other choices are strictly arbitrary. The first result of interest is that of the ratio 7[ ,/L , where *?, is the free surface elevation at the point of breaking. Table 1 shows that for each combination of beach slope and H /L in the second order solution, the ratio is essentially constant (some small variations occur in the second significant figure) . The value of the ratio is independent of the wave period. The consistency of the ratio suggests the use of this parameter as a breaking criteria for specific beach slopes and H /L conditions. A second parameter listed in Table 1, that of h, /H , where h, is the depth at breaking referenced to the still- water level, has often been utilized in the measurement of breaking waves. The theory under investigation yields consistent results for this ratio for specific beach slopes 64 •H 0) crj CU U PQ H CU -u o O O O o co as O o O o o o o O O CM vO vO O o O O CM as O m m o o o o O o m vO vO o O o O m m as m CM CM o o o o O o vO vO vO ffi o o O 00 r-l as r^» r^ t^. 00 00 00 CM CO CM r-» r^ r- *■*«* o o O as o as 00 00 00 o as O «vfr CO vO vO vO CM VO vO . oo 00 oo CM CM CM m m m OS o> o> r-l r-l r-l CM CM CM O CM CM > r^ K r^ CO CO co CO co co m in m co CO CO CO •H u cu Q M-l O 01 3 co CU Pi o o o o o o o o o CM CM CM o o o o o o o> OS as as CM CM CM CM CM CM CM CM CM ,£ cu as as as as o> 0> as as o> o a i— i rH rH rH rH tH r-i i—i rH m m m m m m m m m CTj o rH r-l rH rH rH i— 1 r-l rH rH o o o o o o o O o CU rH pa CO o O O O O o O O o O o o o o o o o o cd o •H u cu e 3 sll O O O O O O O O O o o o O O O o o o t-4 CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM c o U -H CU 4J T3 T3 T3 Tj Tj "O Tj Tj T3 Tj TJ 'O T3 T3 Tj Tj Tj TJ tnw ^ c c c c c c C c c C c c G C C c c c 5-1 O rH CM CM CM CM CM CM CM CM CN CM CM CM CM CM CM CM CM CM o o CO 65 m m o r- o o o CN CM o vO o o o o r-l rH 00 • • • • • • • & o o o o o o o St 00 00 ON CO ON ON CN CM r»» 00 1-1 O r* o i-l r-1 co r>» t-H CM CO rJ t-4 r-l r-t r-t o o o O O O O o o o & • • * • • • * RT o o o o o o o CM VO o m m co ^ CM vO u PROFILE 71 , x y z. be valid for 2 ^("TH - 1«° • Substitution into this breaking occurs at the wave crest. Figure 8 shows an unrealistically deep trough in front of the crest. The use of the Bessel function approximation has been shown to relation of the values of the terms used to construct Figure 8 requires x^: 21.89 feet. It is seen, therefore, that the excessively deep trough is predicted in a region where the Bessel function approximation is not valid. The predicted breaking for the specific instance shown in the figure is at x = 33.73 feet, which is in the applicable region for the approximation. For each case investigated, the predicted breaking point occurred well within the region where the approximation is accurate; hence the derived breaking criteria is deemed valid. Ayyar (1970) , utilizing the kinematic breaking criterion, produced the additional shallow water breaking criterion of y^/hb=2.0. Table 1 summarizes the values obtained in the present study for this ratio, all of which are considerably larger than 2.0. Ayyar fs ratio does not account for the variations associated with HQ/L suggested by the observed h^/HQ data. In addition, his theory is limited to waves which have the geometry of a plunging breaker. Ayyar also assumes that breaking occurs at the wave front, which may not in fact take place. The limitations placed upon Ayyar' s formulation and the random values of the ratio displayed in Table 1 prevents this from being considered a valid indication of breaking. The most significant disadvantage associated with the breaking criteria derived from the theory presented in this study is that the ratios require re-calculation for changes in beach slope and deep water wave height to wave length ratio. This objection is similar to that which was raised against Stoker's use of the method of charac- teristics. The complex dependence of the values of a, u and c upon both the beach slope and H /L , however, leaves the investigator with little choice but to resort to a numerical solution. In defense of the approach, the solution technique applied to specific situations is relatively simple and requires minimal computations once the beach slope and H /L are known. Selected data points r o o may be used to construct graphs which approximate the hv,/H ratio for each specific beach slope. This can be used to provide an indication of breaking for varying H0/LQ values. 73 VII. CONCLUSIONS Two shallow water breaking criteria have been formu- lated through the application of the kinematic breaking criterion to a second order solution of the long wave equations. These are the ratios of ^u/Lq and h,/H . Both of these criteria are dependent only upon the beach slope and the ratio of the deep water wave height to the deep water wave length. Each ratio requires re-calculation as these two parameters vary. Comparison with previous theory indicates that the theoretically derived values for h. /H offer significantly improved approximations to the assembled experimental data. The increased accuracy is partially attributed to the use of a higher order solution to the long wave equations. First order solutions obtained indicate that the solution technique applied offers improvement over previous theory. The predicted breaking depths are somewhat less than those observed in field measurements. It is believed that the extension of the theory to higher order solutions of the long wave equations would reduce the error associated with the pre- dicted breaking depth. Inclusion of vertical water particle accelerations would also increase the accuracy of the solution, APPENDIX A 1 DERIVATION OF -"^ Iwagaki and Sakai have obtained a second order solution for the free surface, 7?(x,t). The relation derived com- pared favorably with experimental data. Several approxi- mations made in the solution, however, warrant discussion. (2) Combining equations (18) and (19) so that uv is eliminated, gives -D^o^ +D*vA« +U«^uVli . (A-D Iwagaki and Sakai substituted their first order equations (20) and (21) into the right side of (A-l). They then offered approximations for the Bessel and Neumann (Weber) functions of Jr W~ (^wY/2-c o s (vJ - v- V2-- ^A) (A"2) and ,'/: Nr(wV (fi<* (w~ '"V2-- ^ )• (A"3) The authors provide evidence which shows that for |w| >1.0, these asymptotic expansions are accurate. Defining p (x) = (2 3rd Term/lst Term ^ and C{(3T^V^cX2 4th Term/lst Term ~ [^ $ (^T/ZiV )/(g 0 ; C J • The interpretation of these ratios is that the successively higher terms become relatively smaller in proportion to i. Iwagaki and Sakai considered the specific case of i = ~ , h = 20 cm and T = 3 sec. Examination of the first and third terms show that they are in phase. For these particular conditions, the ratio between the third and first terms is less than — . The conclusion is that the difference 10 between the constants for the third terms is negligible. 78 Similarly, the fourth term is compared to the second, which is in phase with it, and seen to be negligible. The use of (A-5) , (A-6) and (A-7) can therefore be used to (2) provide an accurate expression for yn as + pa •£ (vt ys.-3*- co s i>f oon rrcr2" (2) The final solution for 7^ (x,t) offered by Iwagaki and Sakai was r- 2- o ^Wf^-^H- (a-io) 79 APPENDIX B COMPUTER PROGRAMS The results which are summarized in Table 1 were calcu- lated through the use of a series of simple computer pro- grams. They are essentially a series of do-loops designed to perform a number of iterations over an interval. The results are examined to determine the desired solutions. The first two programs used are concerned with estab- lishing a value for the quantity, a, found in equations (56), (57) and (63). As was discussed in Section V.E., this evaluation can be made through the use of (67), (69), (71), (72) and (70). The first step in the process is to employ (72) , %/h! = (H^HoKVL^/Chx/Lo), (72) to find H-^/h . In this expression, h-j^/L is known for the particular case under consideration. Equation (62) gives this term as h, /L = 0.0157 for the limiting condition of h/L 6l 1/20. Entry into the wave energy flux curves of hyperbolic wave theory with the value of h,/L = 0.0157 gives a value of H../H = 1.28. The remaining quantity in the right side of (72) , HQ/L , is a known value for the specific wave conditions. It can be determined from deep water observations of the wave. Therefore, once this value is specified for a certain set of wave conditions, (72) yields a value of H-^/h, . For instance, when HQ/L is equal to 0.001, Hj/1^ = (1.28) ( .001) /. 0157 = 0.0815. Equation (69) provides a relation for the quantity b as b = a| /A^1). Use of (67), which defines A^ and A- , allows this equation to be evaluated as Since ill = 0.0157 and L ■ fiT . (B-l) is equivalent to LQ ° 2-r b- A* :* (b./O^ ?%^7> (b-2) The next value which must be determined is that of f ( & ) c in (70) which is defined in (69) as Use of the identity reduces (69) to f(6£)- c^s C^c - J^'"^- . (B-3) I + to.^ ?■ (B-6) (1) A, is determined from (67) as Equations (B-6) and (B-7) can now be combined to express ,^ (B-7) H^/h-i in terms of the quantity a. The requirement then is that of determining the value of a for which the right side of (70) equals the value of H-^/h-^ as given by (72). The first computer program used defines a function F as the difference of the right hand value of (70), obtained 82 through (B-6) and (B-7) , and the left side evaluated by (72) . The program is designed to plot F as a function of a. The value of a for which F equals zero is the required solution of (70) . Examination of the graph produced by the first program can thus give a first approximation to the quantity a. The second computer program employed is simply an iteration routine used to refine the value of a. It begins with a first estimate of a determined from the graph of F versus a. The value of a is then incremented in steps of 0.001 and the corresponding values of F calculated. The program is designed to determine the value of F closest to zero and to print the value of a for this case. This is the value of a which satisfied (70) . The next series of programs are designed to determine the maximum value of x at which the kinematic breaking criterion is satisfied and the time at which this occurs. This can be accomplished by finding a solution set of (x,t) combinations for which the difference between the right sides of equations (63) and (57) is equal to zero. The technique used is similar to that used in establishing the value of a. The presence of two dependent variables, however, makes the procedure somewhat more complex. 83 The first program in this determination utilizes the method employed for the plot of F as a function of a. A function is defined which equals the difference between (63) and (57), (c-u) . This is plotted as a function of x for a series of times t. Examination of the graphs provides a first guess at the maximum x for which the kinematic breaking condition is satisfied and the time at which it occurs. As was the case for determining a, a second pro- gram is now utilized to obtain a more refined solution. Here, time is varied over one second in steps of 0.05 seconds and x is incremented in intervals of 0.01 feet. The value of the function defined as (c-u) is printed for these specific (x,t) combinations. These results are examined to determine the point at which the kinematic criterion is first satisfied. These first programs have thus determined the maximum x at which the kinematic breaking criterion is satisfied and the time at which it occurs . The remaining programs utilize this result to produce the breaking criteria summarized in Table 1. The fifth program in the complete series simply calculates the value of ti (the free surface) and the depth of the bottom for the point specified for the solution of (xb,t). This is accomplished by using (56) to define 77 and h = i*x,. The ratios of 7j ^/h^ and 84 and h, /L are then calculated. A similar program is also provided which graphs rn as a function of x. Pages 86-91 contain flow charts for the programs employed in these determinations. 85 C st*rt ) READ h.,/L0, H0/L0, SLOPE, PERIOD I CALCULATE H1/h1 DO 1 = 1.1500 DESIRED SOLUTION FOR a WHEN F=0 CALCULATE F FOR 0.1 INCREMENTS OF a PLOT F VS. a ( ST0P ) PROGRAM 1i APPROXIMATION OF a Figure 9 86 C START J Read hx/L , VLo>Hl/Sl Slope, Period i Do 1=1, 10000 Program 1 gives firstt. estimate of a Calculate F for 0.001 increments of a Do J-l, 10000 Determine Smallest F l Write value of a which gives small- est r stop \ PROGRAM 2: REFINEMENT OF a Figure 10 87 '< START J u Read a, W Slope, Period Do J=] incren time ] steps perioc .,20 lent .n ?20 1 i 1 f 1 1 1 r Do 1-1,1500 1 1 — 1 ! 1 1 1 1 1 1 1 1 1 i Calculate c-u for 0. 1 ft. incre- ments of X • 1 1 1 i . tfrite time t, Plot c-u vs. X __i 1 N f STOP I PROGRAM 3: FIRST APPROXIMATION OF x AND t AT BREAKING Figure 11 88 ( START J ! i Read a, Slope, Period Do J-1,20 increment time in steps of 0.05 sees. 1 1 1 First est. of time given by Program 3 i Do 1=1,1500 • r ,"i ll , . First est. of x given by Program 3 I i h 1 1 I ■ . 1 Calculate c-u for OJOI ft. incre- ments' of X i i i | i i 1 I- — _ l , * 1 Do 1= 1,1500 1 [ i i 1 1 ] ' 1 1 i Calculate smallest c-u a t 1 L Write time, smlst c-u and corres- ponding X i PROGRAM 4: ( STOP DETERMINATION OF x AND t AT BREAKING Figure 12 89 c START Read a, VL0' Slope, Period time and x Calculate ^b» hb> Lo> ^b/hb,hb/LQ Write hb/Lo C STOP PROGRAM 5: CALCULATION OF Tu/h. AND h,/L D D DO Figure 13 90 f START J Read a, W Slope, Period, time Do 1=1,1500 Calculate zf for 0.1 ft increments of x Plot 7\ vs. x ( STOP J PROGRAM 6: PLOT OF r\ AT TIME OF BREAKING Figure 14 LIST OF REFERENCES Ayyar, H. R. , "Periodic Waves Shoaling in Waters over Steeply Sloping Bottoms," in Proceedings of the Twelfth Coastal Engineering Conference, v. 1, p. 363- 369, 1970. Benney, D. J., "Long Non-Linear Waves in Fluid Flows," Journal of Mathematics and Physics, v. 45, p. 52-63, 1966. Boussinesq, J., "Theorie de 1' intumescence liquide appelee onde solitaire ou de translation se propageant dans un canal rectangulaire," Institut de France, Academie des Sciences, Comptes Rendus , p. 755, June 19, 1871. Boussinesq, J., "Theories des ondes et de remous qui se porpagent le long d'un canal rectangulaire horizontal, en communiquant ou liquide contenu dans ce canal des vitesses sensiblement paralleles de la surface au fond," Journal Mathematical Pures Appliquess, v. 17, p. 55-108, 1872. Burger, W. , "A Note on the Breaking of Waves on Non- Uniformly Sloping Beaches," Journal of Mathematics and Mechanics, v. 16, p. 1131-1142, 1967. Carrier, G. G. and Greenspan, H. P., "Water Waves of Finite Amplitude on a Sloping Beach," Journal of Fluid Mechanics, v. 4, p. 97-109, 1958. Chappelear, J. E. , "On the Theory of the Highest Waves," Beach Erosion Board Technical Memo No. 116, July, 1959, Davies, T. V., "The Theory of Symmetrical Gravity Waves of Finite Amplitude," in U. S. National Bureau of Standards Circular No. 521, p. 55-60, 1952. De, S. C, "Contributions to the Theory of Stokes Waves," Proceedings on the Cambridge Philosophical Society, v. 51, p. 713-736, 1955. 92 Dean, R. G. and Eagleson, P. S., "Finite Amplitude Waves," in Estuary and Coastline Hydrodynamics, Ippen, A. T. , editor, McGraw-Hill, 1966. Dean, R. G., "Breaking Wave Criteria: A Study Employing Numerical Wave Theory," in Proceedings of the Eleventh Conference on Coastal Engineering, v. 1, p. 108-123, 1968. Freidrichs, K. 0., "On the Derivation of the Shallow Water Theory," Appendix to "The Formation of Breaker and Bores," Stoker, J. J., Communications on Pure and Applied Mathematics, v. 1, p. 81-85, 1948. Gaughan, M. K. , Komar, P. D., and Nath, J. H. , Breaking Waves: A Review of Theory and Measurements, Oregon State University, 1973. Goda, Y., "A Synthesis of Breaker Indices," Transactions of the Japanese Society of Civil Engineers, v. 2, p. 227-230, 1970. Greenspan, H. P. , "On the Breaking of Water Waves of Finite Amplitude on a Sloping Beach," Journal of Fluid Mechanics, v. 4, p. 330-334, 1958. Gwythery, R. F., "The Classes of Long Progressive Waves," Philosophical Magazine, v. 50, p. 213-216, 1900. Havelock, E. T. , "Periodic Irrotational Waves of Finite Height," Proceedings of the Royal Society, v. 95, p. 38-51, 1918. Iwagaki, Y. and Sakai, T. , "Shoaling of Finite Amplitude Long Waves on a Beach of Constant Slope," in Proceedings of the Thirteenth Coastal Engineering Conference, v. 1, p. 347-364, July, 1972. Keller, J. B., "The Solitary Wave and Periodic Waves in Shallow Water," Communications on Pure and Applied Mathematics, v. 1, p. 323-339, 1948. Kinsman, B., Wind Waves , Prentice-Hall, 1965. 93 Korteweg, D. J. and de Vries, G., "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves," Philosophical Magazine, v. 5, p. 442-443, 1895. Laitone, E. V., "The Second Approximation to Cnoidal and Solitary Waves," Journal of Fluid Mechanics, v. 9, p. 430-444, 1960. Laitone, E. V., "Higher Approximations to Non-Linear Water Waves and the Limiting Heights of Cnoidal, Solitary and Stokes' Waves," Beach Erosion Board Technical Manual No. 133. February, 1963. LeMehaute', B., "Mass Transport in Cnoidal Waves," Journal of Geophysical Research, v. 73, p. 5973-5979, 15 September 1968. LeMehaute", B., "Water Wave Theories," Environmental Science Service Administration Technical Report ERL 118-P0L-3-2 An Introduction to Hydrodynamics and Water Waves Vol. II, July, 1969. Lenau, C. W. , "The Solitary Wave of Maximum Amplitude," Journal of Fluid Mechanics, v. 26, p. 309-320, 1966. McCowan, J., "On the Highest Wave of Permanent Type," Philosophical Magazine, v. 38, p. 351-357, 1894. Mei, C. D. , "On the Propagation of Periodic Water Waves Over Beaches of Small Slope," Coastal Engineering Research Center Technical Note No. 12, October, 1966. Michell, J. H. , "On the Highest Waves in Water," Philosophical Magazine, v. 38, p. 351-357, 1893. Packham, B. A., "The Theory of Symmetrical Gravity Waves of Finite Amplitude, Part 2 The Solitary Wave," Pro- ceedings of the Royal Society, v. 213, p. 238-249, 1952. Peregrine, D. H. , "Equations for Water Waves and the Approximations Behind Them," in Waves on Beaches and Resulting Sediment Transport, Meyer, R. Ed. , editor, p. 95-122, Academic Press, 1972. 94 Rankine, W. J., "Summary of Properties of Certain Stream- lines," Philosophical Magazine, v. 28, p. 282-288, 1894. Rayleigh, Lord, "On Waves," Philosophical Magazine, v. 1, p. 257-279, 1876. Riabouchinsky, D. , "Sur l'Analogie Hydraulique des Mouvements d'un Fluide Compressible," Institut de France, Academie des Sciences-Comptes Rendus, v. 195, p. 998, 1932. Stoker, J. J., "The Formation of Breakers and Bores," Communications on Applied Mathematics, v. 1, p. 1-87, 1948. Stoker, J. J., Water Waves , Interscience Publishers, 1957. Stokes, G. G., "On the Theory of Oscillatory Waves," Transactions of the Cambridge Philosophical Society, v. 8, p. 441-455, 1 March 1847. Stokes, G. G., Mathematical and Physical Papers, Cambridge Univeristy Press, 1880. Tlapa, G., Mei, C. C, and Eagleson, P. S., "The First Order Theory for Infinitesimal Waves on a Mildly Sloping Beach," Appendix A to "On the Propagation of Periodic Water Waves Over Beaches of Small Slope, MIT Hydrodynamics Laboratory Technical Note No. 12, October 1966. Ursell, F., "The Long -Wave Paradox in the Theory of Gravity Waves," Proceedings of the Cambridge Philosophical ' Society, v. 39, p. 685-694, 1953. Yamada, J., "On the Highest Solitary Wave," Report of Research Institute for Applied Mechanics, v. 5, p. 53- 155, 1957. Yamada, H. , Kimura, G. , and Okabe, J., "Precise Determina- tion of the Solitary Wave of Extreme Height on Waters of a Uniform Depth," Report of Research for Applied Mechanics, v. 16, p. 15-32, 1968. 95 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 2 Cameron Station Alexandria, Virginia 22314 2. Library, (Code 0212) 2 Naval Postgraduate School Monterey, California 93940 3. Department Chairman, Code 58 3 Department of Oceanography Naval Postgraduate School Monterey, California 93940 4. Assoc. Professor E. B. Thornton, Code 58 5 Department of Oceanography Naval Postgraduate School Monterey, California 93940 5. Assoc. Professor J. J. von Schwind, Code 58 1 Department of Oceanography Naval Postgraduate School Monterey, California 93940 6. Oceanographer of the Navy 1 Hoffman Building No. 2 200 Stovall Street Alexandria, Virginia 22332 7. Office of Naval Research 1 Code 462 Arlington, Virginia 22217 8. Dr. Robert E. 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Coastal Studies Institute Louisiana State University Baton Rouge, Louisiana 70803 19. School of Marine and Atmospheric Science University of Miami Miami, Florida 33149 20. LT V. S. Hulstrand SMC Box 2489 Naval Postgraduate School Monterey, California 93940 97 Thesis c.l Smith Breaking wave criter- ion on a s 1 op ij*g beach. thesS59824 Breaking wave criterion on a sloping bea 3 2768 002 00775 9 DUDLEY KNOX LIBRARY