WAVE SET UP AND SET DOWN DUE TO A NARROW FREQUENCY WAVE SPECTRUM Darwin James McReynolds HOX LIBRARY STGRAOUATE SCHOOL NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS WAVE SET UP AND SET DOWN DUE TO A NARROW FREQUENCY WAVE SPECTRUM by Darwin James McReynolds March 1977 Thesis Advisor E . 3. Thornton Approved for public release; distribution unlimited. T183075 SECURITY CLASSIFICATION OF THIS RACE (Whon Data Kntarad) REPOTT D0OJMENTAT10H PAGE t. REPORT NUMBER READ INSTRUCTIONS BEFORE COMPLETING FORM 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. J\TLE (and SubUlIm) Wave Set Up and Set Down Due to a Narrow Frequency Wave Spectrum S. TYRE OF REPORT ft PERIOD COVERED Master ' s Thesis; March 1977 4 PERFORMING ORG. REPORT NUMBER 7. AuTNORfa> Darwin James McReynolds ft. CONTRACT OR GRANT NUMBER^ t. PERFORMING ORGANIZATION NAME ANO ADDRESS Naval Postgraduate School Monterey, California 93940 10. PROGRAM ELEMENT. PROJECT TASK AREA ft WORK UNIT NUMBERS 11. CONTROLLING OFFICE NAME AND AOORESS Naval Postgraduate School Monterey, California 93940 12. REPORT OATE March 19 7 7 IS. NUMBER OF PAGES 42 14. MONITORING AGENCY NAME ft AOORESS^// Jl/.'aranr /toot Controlling Otllea) Naval Postgraduate School Monterey, California 93940 IS. SECURITY CLASS, 'el tnia rwportj Unclassified IS*. DECLASSIFICATION/ DOWN GRADING SCHEDULE l«. DISTRIBUTION STATEMENT (ol tnia Raporf) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (ol Iho aaafracf antorod In Block 30, It dltioront /toot ffaport IS. SUPPLEMENTARY NOTES I*. KEY WORDS (Conilmto on rarorao aid* II nocaaaory and Idonllfr by block nxammor) 20. ABSTRACT (Contlmto on ravaraa aldo II nocoaaarf and Idontity ky alaak mamkot) A narrow band wave spectrum was appl ied to theoretical relationships previously developed for set down and set up in an attempt to find second order non-steady solutions for these concepts. The initial effort was to aoply this spectrum to the radiation stress tensor using linear wave theory. Another development was attempted by incorporating the spectrum and the solution of the long wave equation int< DO .ER, 1473 (Page 1) EDITION OF I NOV •• IS OBSOLETE S/N 0102-014- 6601 I SECURITY CLASSIFICATION OF THIS PAOE (Whom Data Bntaa-aal) $l:CUt*1TV CLASSIFICATION Of THIS P»GEr*>iw !">»!• EnlmrmJ the Bernoul I i and vertical momentum equations. Results obtained indicate that the solution for mean water level outside the surf zone is composed of a steady component and a periodic unsteady component; the periodic component being of the form of a long wave with a frequency lower than the components of the wave spectrum. The solution for set up is then composed of the same type components. Tne exact relationships depend on the patching process that is made for the solutions through the breaker line. DD Form 1473 1 Jan 73 S/N 0102-014-6601 ICCUHlTV CLASSIFICATION Of THIS * AGei"***" Dmim gnimrrndt Approved for public release; distribution u n I i m i ted 0 Wave Set Up and Set Down Due to a Narrow Frequency Wave Spectrum by Darwin James McReynolds Lieutenant, United States Navy B.S., California Polytechnic State University, 1970 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY f rom The NAVAL POSTGRADUATE SCHOOL March 1977 /vy ;-u $£* ****** ^Ai POSTGRADUATE SCHOOl ABSTRACT A narrow band wave spectrum was appl ied to theoretical relationships previously developed for set down and set up in an attempttofind second order non-steady solutions for these concepts. The initial effort was to apply this spectrum to the radiation stress tensor using linear wave theory. Another development was attempted by incorporating the spectrum and the solution of the iong wave equation in- to the Bernoul I i and vertical momentum equations. Results obtained indicate that the solution for mean water level outside the surf zone is composed of a steady component and a periodic unsteady component; the periodic component being of the form of a long wave with a frequency lower than the components of the wave spectrum. The solution for set up is then composed of the same type components. The exact relationships depend on the patching process that is made for the solutions through the breaker I ine. TABLE OF CONTENTS I. INTRODUCTION ------------------ 9 I I . BACKGROUND -------------------|| A. DEVELOPMENT OF THE RADIATION STRESS TENSOR ---------------|| B. WAVE SET DOWN ---------------15 C. SET UP------------------ -2! III. APPLICATION OF A NARROW BAND WAVE SPECTRUM TO THE EXISTING THEORIES ------- 23 A. WAVE SET DOWN --------------- 25 B. WAVE SET DOWN USING THE BERNOULLI INTEGRAL -------------29 C. WAVE SET DOWN ON A SLOPING BEACH ----- -30 D. WAVE SET UP ----------------32 IV. COMPARISONS ------------------34 V. CONCLUSIONS ----------------- -39 LIST OF REFERENCES ------------------40 INITIAL DISTRIBUTION LIST ------------- -41 LIST OF FIGURES 1. Coordinate System - - - 2. Profi le of Mean Sea Leve I 2 37 TABLE OF SYMBOLS AND ABBREVIATIONS c c c E g h hl H H, k m M M M P R i J T. i U U u' u local wave amplitude wave celerity wave group velocity total energy density acceleration due to gravity still water depth still water depth at breaking wave height wave breaking height local wave number slope of beach mass transport of unsteady flow mass transport of mean flow total mass transport pressure friction term radiation stress tensor horizontal force due to slope of free surface mean velocity component total transport velocity deviation from mean velocity x d i rected velocity y d i rected velocity ve rt i ca I velocity a angle between wave crest and beach E p ha se a ng I e t\ wave prof i le H mea n water level H f deviation from mean water level r|. mean water level at breaking b a p water dens i ty a local radial frequency velocity potential NTRODUCTION Most of the observable phenomena along a coastl i ne are the direct result of the action of the incoming waves, waves which begin in most instances as a disorganized con- fused state of the ocean surface, produced by a storm far at sea. In their transit across the vast expanse of the ocean, they begin to sort themselves out and form a somewhat regular oscillation of the ocean surface, the lower frequen- cy oscillations traveling faster and thereby leading the train. This train eventual ly ends by encountering a beach, where its energy is expended in the form of breaking waves. Since it is this aspect of the wave's life cycle that influences man the most, considerable effort has been ex- pended investigating this area. Among other effects, it has been observed that waves, in the process of shoal i ng and eventual breaking, produce a variation in the mean sea level This variation in sea level has been considered as the pri- mary cause for such nearshore currents as rip currents. The sea surface variation consists of: (a) a gradual depression of the mean sea level Deginning offshore and reaching a maxi' mum at the breaker line and (b) inside the surf zone a slooe of mean sea level wnich increases and extends shoreward to a point on the beach higher than the still water line. The depression is termed set down and the slope is called set up It is reasonable to expect that variations in the amount of set down or set up along a beach can provide the head to produce a current. Previous investigations have considered steady state solutions. Experimental results obtained by both Bowen C I 9 6 7 H and Van Dorn [1976] agree quite favorably with the steady state solutions produced by Longuet-H i gg i n s and Stewart [1962] using linear wave theory. This investigation considers an application of a simple wave spectrum to the existing theories in an attempt to ob- tain a non-steady solution for the set down and set up phenomena. An opening chapter on background is provided to ensure the necessary understanding of the existing theories and their development. Included is a section on the develop- ment of the "radiation stress tensor", a concept which was proved useful by Lo ng uet-H i gg i n s and Stewart L I 9 6 2 H in treating the shoaling process of waves. Chapter III deals with the application of a simple spectrum to the derivation of set down and set up. First order linear theory and first order theory including a slop- ing bottom are used to describe the spectral wave components. The final chapters conclude with a comparison of the steady state solutions produced by the earl ier work and the unsteady results obtained here. The numerical results are compared with the results for the steady case given by Bowen C I 9 6 7 ~ . 10 I I . BACKGROUND Changes in mean sea level near a. shorel i ne have been studied both theoretically and experimentally. A theoreti- ca I framework was formulated by Long uet-H i gg i n s and Stewart L~I962, 1963, 19643 which dealt with the excess momentum flux due to the presence of unsteady wave motion and /vnich they termed "radiation stress". Longuet-Higgins and Stewart were able to define many of the shoaling effects of a train of waves including wave set down and set up utiliz- ing this radiation stress concept. This chapter reviews the development of the radiation stress tensor and its re- lationship to the concepts of set down and set up. It also includes a direct approach used by Longuet-Higgins [J 967]] to derive an expression for set down utilizing the vertical momentum equation and the Bernoul I i equation. A. DEVELOPMENT OF THE RADIATION STRESS TENSOR Since the approach of Longuet-Higgins and Stewart is rather lengthy and tends to obscure the concepts involved, the later and more sysTematic approach by Phillips CI9663 is used. The development of the radiation stress tensor and the resulting phenomena of set down and set up is kept as general as possible. The coordinate system is given by Figure (I) where the x-axis is perpendicular to the shore, the y-axis is parallel to the shore, and the z-axis is 11 PLAN VIEW /- \ x \ WAVE CRESTS PROFILE FIGURE 1 Coordinate System I 2 vertical ly upward from the st i II water level . The govern' ing equations are the continuity equation and the hori- zontal momentum equations. The continuity equation is given by a 3pu 3t 3x. 3z U ' i = I, 2 ( I ) where I, 2 refer to the x, y components. The horizontal velocity u. is composed of a mean flow component, U., and a fluctuating component representing the deviation from mean flow, u.', such that u. = U. + u.'. Since there is no mean flow in the vertical, w = w!. Multiplying the con- tinuity equation by u. and adding this result to the hori- zontal momentum equation 3u. 3u 3u p — + puj — + pw TF J J 3d ^ 3x . i i p rod uces (2) 3u . 3pu.u. 3pu.w 3t ! J 3x . J 3z 12. 3x + R (3) Integrating over depth from -h to r\ , using Leibnitz's rule and applying the kinematic free surface and bottom boundary condition yields n . n . n ■s-r / pu . dz + tt— / pu . u . dz + -^— / at J . 3x. J , j 3x. J p d z -n -h 3(-h) t p_h -IT-- = R -h 13 By time averaging this equation term by term and by making the following definitions 1 / M. = / pU.dz = pU. (n + h ) , -h* = j PU p u . ' d z , -Ti M. = M. + M. , (4) M. p(n + h) = u. + p(fT + h ) the following expression is obtained: 9M / n M.M -=- {U.M. + / [pu.'u.' + p5..]dz - — -J 3t 9x. ii / i J I J ,~ un j J J J p c n + h) -h pg (n + h)5 . . } = T. + R. . r J for i = j (5) The first term on the left hand side is the local change in horizontal momentum flux. In the braces, the first term represents the momentum flux produced by the steady state flow. The last three terms in the braces contain al I the unsteady contribution to the momentum flux with the hydro- static effect subtracted out; this is the momentum flux due 14 to the unsteady motion or the excess momentum flux referred to as the radiation stress tensor: S . . = i J /n m.m . [pu 'u «+p6 "Jdz L-J 1 pg(n + h)Z6. . . 1 J IJ p(n+h) z IJ -h Equation (5) is then simplified to (6) |x M. + J- {U.S. + S. .} = T. + R. . at i 3x . i i i j i i (7) T. represents the horizontal force produced by the slope of the free surface and is given by T. = - pg(n>h) |2 . I (8) Outside the surf zone it is assumed n << -h and (8) reduces to T u ^ 'i = - p9h 37. ' (9) R. is the averaged and integrated frictional stress term. These equations are general and apply to all kinds of steady and unsteady motion. The only simplifying assumption is that mean flow is uniform over depth. The advantage of us- ing the "radiation stress" technique to solve physical prob- lems is that the second order effects are obtained using first order theory . B. WAVE SET DOWN To present the concept of set down, consider only the X' component of horizontal momentum flux equation (7) with 15 waves that prooagate shoreward from deep water with their crests making an arbitrary angle a with the shoreline. For simpl icity, the bottom is assumed to be composed of paral lei contours so that gradients in the y-direction are zero. The bottom slope is allowed to vary only gradually so that energy reflection from the shore may be neglected and the shoal ing effects caused by changes in the bottom can be con- sidered in a step-like fashion. By neglecting the frictiona effects and assuming that any current gradient in the x- direction is small, (7) can be written for outside the surf zone as 3M, pgh In 3x 3S xx 3x ( I 0) Since (10) is an equation in two unknowns (M and r\) , a second equation is required for a solution. To provide it, the continuity equation, (I), is vertically integrated over depth by the use of Leibnitz's rule. This produces the conservation of mass flux equation as given by Phillips C I 96 6 H where gradients in the y-direction are zero. 3ri x dt dx = 0 . (II) Utilizing linear wave theory, the radiation stress tensor is proportional to the square of the local wave amplitude, a , or to the total energy. Therefore, fol lowing the method of Long ue t-H i gg i n s and Stewart d I 9 62]], the applied force of the 16 system travels with the speed of the energy, i.e., at the group velocity, c . Hence the transformation 9/3t + c 3/9x = 0 can be applied. Equations (10) and (II) become 9 8M - c g 9x - + pgh jl - 3S xx 3x (12) 3M, d7 c |n = o , g 3x (13) for which the solution is in 1 3x p (gh - 8S xx 2. 3x c ) (14) or on i n teg rat i on n = P (gh - c 2) xx g (15) Since c ■*■ g h i n shallow water, equation (15) is a non- steady state solution which impl ies that the mean water level increases negatively without bound as the wave moves into shallow water. The explanation offered by Longuet- Higgins and Stewart [J 962]] for this apparent resonant condi- tion is that its effect takes time to build and the energy involved is dissipated prior to it reaching any significance by the breaking of the wave. By imposing steady state conditions and describing the unsteady motion using linear wave theory, Longuet-H i gg i ns and Stewart [J 9623 subsequently developed the following situ- ation for n '• 1 a2|< 2 s i nh 2kh * ( I 6) 17 This is a second order equation in terms of the local depth, amplitude, and wave number. It is apparent that as the wave train approaches the point of breaking, r\ decreases and there is a set down of the mean water level. The set down in- creases to the point where the wave breaks and other assump- tions regarding r\ must be applied. Longuet-H i gg i ns C I 9 6 7 J also derived this solution in another manner without referring to the radiation stress term. By integrating the vertical momentum equation over depth and time averaging, the total average vertical momentum is ob ta i ned : (p - pw )z = Q - pgff = 0 . (17) A second equation used is Bernoulli's integra P + ^ P ( u 2 x + u 2 + w2) + pgz + p |i = 0 , y 9t ( 18) with the restriction that the flow be i r ro ta t i ona I . By set' ting z = 0 and time averaging, (18) becomes - 2 - 2 -2 s u + u + w z=0 2 x y z=0 p+Tp(u + u + w ) + C = 0 , (19) where C is at most a constant. From (17) and (19), p ~ ' r z = 0 can be eliminated giving H-.^i = - 2_ , — _ sin(k.x. -at) , i = I, 2 Y p=r , cosh kh ii /2 c k (31 ) where k. is used to indicate direction, i.e. k /k = cosa i x and k /k = s i na, then in general u 3<(> 3x or ag /I i cos h k ( h + z ) k cos h kh cos ( k . x . - at ) . (32) Since u. = u. + u. in this case, u. becomes ag /2 f cosh k , ( h+z ) cosh k , h cos(k. x. - a.t) k. + i 7 cosh k? ( h + z ) cos h k?h cos ( k . x . - a0t ) . i ? i 2 (33) Since k, - k« = Ak and a. - a~ ~ ^o , the subscripts may be dropped and (28) can be written as u . = /2 k. i k cosh kh ag r i cos h k( h + z ) cos ( k . x . - at ) + ( i i. coshQ( k+Ak) ( h+z ) ] r,. ... . . ,. ,,11 i — m — } \ (i .a i \ -cos[ k.+Ak. )x . -(a+Aa tj} k +Ak cosh(k+Ak)h i i i (34) 24 This simplifies to 2ag i cosh k ( h + z ) , i u. = — - — - — r — rr cos ( 0 x.- -~- t ) cos ( k . x . -at ) . i nr k cosh kh 2 i 2 /2 c Aa (35) Similarly, the particle velocity in the z-direction, w, is obta i ned and is 2aq s i n h k ( h + z ) , i W = * — — COS(— rr- cos h kh 2 Aa t ) s i n ( k . x . -at ) . 2 i i /2 c (36) These expressions state that the particle velocities are sinusoidal with a periodically varying amplitude. The modu' lating wave is of low frequency and with a wavelength that is much longer than those of the original two base waves. The unsteady velocities specified by (35) and (36) are substituted into (6) and the radiation stress tensor deter- mined. The time averaging process is carried out over the shorter periods of the base waves. Since the modulating wave is of such low frequency, its effects remain in the solution. Specifically, equation (6) becomes term by term / -h n c k.k. Ak.x. . Ak.x. A pu.'u.'dz = E — *■ **• cos ( s - -=- t ) cos ( — —^- - -=- t , I J c2 2 2 2 2' (37) 2 where E = i pga is the total energy density, and c and c are the group velocity and phase velocity respectively; j,.. ........... ..-. ...... -h F c ? Ak. dz = pg(n + h)" + | - E(^2- - 2)cos i—^- Aa . . x,- -^ t) ; (38) 25 and the remaining unspecified term becomes .2 k. k M.M . i J _ I I _[_ - - 2 2 p(n+h) p(n+h) c k Ak. Ag 2 Ak *■ COS ( a X.- -=- t ) COS ( 0 J W.- -w- t) Aa 2 "j 2 (39) Substituting (37), (38), and (39) into (6), the radiation stress tensor becomes: c k. k . Ak. . Ak . . _ r _ g i , i Aa ,N , j Aa , , , E S. . = E — * ***- COS(-rr X.- -s- t)COS(-rr- x- n~ + > + ^7 i j c k 2 i 2 2 j 2 2 c . Ak EC-S. - 2) cos (-^ c 2 Aa , , X . - -s- t ) I 2 (40) c2 k. k . Ak I E i , —7T K"*" COS(-^- p ( ri + h ) c k a Ak . -y t )COS(— j^ X • Aa t) Since only second order effects are being considered and 2 4 E = 0(a ) the last term may be neglected. It is obvious that the proper choice of assumptions will simplify (40) immensely, e.g. consider only the x-direction components in shallow water, k 2 Ak. S = [E (■—■) + E] cosz( xx k ClO .. E 2 x i ~T 2 ' and if the waves approach the shoreline such that the angle a = 0, this expression further reduces to S = E[ I + cos( Ak . x XX I - Aat) + (41 ) 26 This last expression states that the radiation stress, in this circumstance, is composed of a steady stress term and of one that is periodic. This derived form of the radiation stress is now used to find a solution for r\ . Recalling equations (10) and (II), and as before neglecting the friction term and assum- ing that the current gradients in the x-direction are small, the x-component of these expressions become: 3Mn 3S xx 3x 3r| " pgh 17 and 3M .- x dn — + p 3T = 0 By cross differentiating, the M term can be el iminated and 2 H 3 ru 3ri-. xx dt dX (42) is obtained. Assuming a flat bottom so that h is considered constant, (42) becomes 2- n —7 8t 3x 3x n 3 n + - D j + pgh 3 n _ 2 3 S xx (43) This is obviously the long wave equation which is being forced by the radiation stress term. It has a solution in the homogeneous case of n, = a cos(Ak x - Aat ) . C X 27 For a particular solution, make the assumption that led to equation (41), i.e., consider shallow water and a. = 0. Then assume a particular solution of the form n = A cos(Ak x - Aat)+ B sin(Ak x - Aat) . p x x By substituting this into equation (43), the solution for A and B are obta i ned EAk 2 2 p(Aa - ghAk ) B = 0 . The solution for n, is then n = Ca - 1 2 2 a 9 ( - gh ) Ak2 -] cos(Ak x - Aat) , (44) x which is a wave with an amplitude that is proportional to the steady state solution of Long uet-H i gg i ns and Stewart, equation (15). If the same assumptions as before are made except that the bottom is allowed to vary as h = mx where m is some constant slope, a more difficult problem is encountered. Equation (42) becomes a2- .- .2- 82S 9 n 9n 3 n xx P j pgm Tx" pgmx 2 = 2 3t 8x dx (45) This equation again has the characteristics of the long wave equation that is being forced by the radiation stress. An 28 analytical solution of this equation would provide a general form for n; however, since the depth is al lowed to vary with x, the expression for the radiation stress term Pecomes com- plicated and oPtaining a particular solution for (45) Pecomes difficult. B. WAVE SET DOWN USING THE BERNOULLI INTEGRAL Recalling that the second method of Long uet-H i gg i n s to determine a solution for n, resulted in the equation I r - 2 -2 -2, -.1 An, = o— L ( u +u - w nj0 , 2g x y z=0 2 ' a second unsteady solution may Pe obtained by applying th _ 2 simple wave spectrum. From equations (35) and (36) u , - 2 -2 u . andw become 2 2k 2 A k -2 a g , xN2 cosh k(h + z) 2, x Aa _^x u = — *— (- — ) s cos C 0' x - -=- t) , x 2 k ,2, , 2 2 ' c cosh kh - 2 a2g2 ,ky,2 cosh2k(h+z) 2,Aky Ac ., u = —a- (-X) _ cos (-^ y - t) , ' c cosh kh o 22 • u2 , , , , < ~ Ak Ak . -2 a g sinh k(h+z) 2, x y Aa ., w = — —- = cos (— rr— x +——- v - -=- t) . ^ l. 2 , , 2 2 2 c cosh kh , I .- 2 - 2 -2, and - -x—( u +u -w ) . 2g x y z=0 2 2k „ Ak a g r x ,. 2 2 ," 2c { (■! — ) cos ( , 2 k 2 x *. t) z. k _ „ Ak . ,, y,2 2 , y Aa , , + (7-^-) COS V—tr1- y~ T" t) k 2 2 ? o Ak Ak , - tanh kh cosZ(--^ x + —^- y - =j- t)} . (46) 29 Aqain assuminq a = 0 so that k /k = I, and k /k = 0 and a z> x y 2 reca I I ing that c = g/k tanh kh, (46) becomes n = - 1 a2k 4- • , \, r D + cos(Ak x - Aat)] . (47) 2 smh 2kh x This solution is identical to the solution of Longuet-Hig- gins (25) except for the unsteady term. The set down fluctuates at Aa = a. - a~; which is the difference of the radial frequencies of the two component wave spectrum. This low frequency oscillation is often described as surf beat. The amplitude of r\ varies from twice the amplitude o^ (25) when the two base waves are in phase, to zero when the base waves are out of phase. Hence the set down can be con- sidered the superposition of a steady component and an unsteady component. C. WAVE SET DOWN ON A SLOPING BEACH In an effort to more accurately represent shoaling wave transformations, Iwagaki C I 9 7 2 U implicitly considers con- stant bottom slope resulting in solutions to the long wave equation involving the Bessel functions. He oPtained first order solutions for n an^ u in terms of the asymptotic form of the Bessel and Neumann functions. It seems reasonable to assume that by incorporating these solutions into the Bernoulli integral, a second order solution for n, can be ob- tained which includes a more realistic representation of the bottom effects. Assuming shallow water conditions with the depth given by h = mx and progressive waves, /2 ^— L~J,(x)sin at + N. (y)cos at] mx I A I A (48) 30 a nd n = — [J (y)cos at + N (Y)sin at] , I I- 0 A 0 A > (49) 2 ax where Y =4 — — . The asymptotic expansions of the Bessel A gm and Neumann functions are: J (co) ~ n — cos (to IT oo N (co) n -V- f IT CO s i n (co - nfT 2 JUL 2 2L) 4 ' 4; ' (50) Substituting these expressions into (48) and simplifying yields Vf J>TT % = — W~ Csin(x - - - at)] , (5 1) and from the kinematic free surface boundary condition w- = n 3n a J/gmx r . , tt , , -, —I = — g\ a Lsin(v - t " at) J . dt r- j TTax A 4 (52) Introducing the wave spectrum as before, squaring and time averaging the result, produces -2 a g/gmx r, , 2Aax , . . . -i u = — ^-^ LI + cos (- + Aat)] , z = 0 2mx TTa /gmx (53) and 2 2 -2 w a a /gmx r . , , 2Aax . , . -, — ~ -f LI + cosC — — + Aat)] . z = ri 2x7ra /gmx (54) 3 I By assuming that r\? is in infinitely deep water as before, the second order solution for n is - a /qmx p q 2-ir, , l. H = - o — * — L-2- - o JLI+cos( + Aat)J . (55) 2Aax 2g7rax mx gmx D. WAVE SET UP Because the nature of the solution for n, using linear theory was the superposition of a steady state component and an unsteady component, it is reasonable to assume that this condition persists across the breaker line and the set up resulting in the surf zone from the wave spectrum has a simi lar makeup. The steady state component is then derived separately in the same manner as in Chapter I I with the radiation stress being now defined as the steady state por- tion of equation (41), or S = ^ E , xx 2 ' ( 56) and again n = - kh + C , (57) where this time the proportionality constant is again k = 3 2 q Y Since n = -— yH, , (57) becomes b lob n = k(h, - h) + nK s b b (58) 32 The unsteady component is assumed to be periodic in character; however, unlike the higher frequency waves of the train, observations suggest that it is not attenuated as it approaches the shoreline but is reflected to some extent. The formulation is then that of the long wave equation as given by Stoker E'966j, ^ flh ^h nh 3h a^ n - gh » - gh -*— -s— ■ - 0 dX dx at 3x 2 (59) and the solution is given by Guza and Bowen C I 9 7 7 U as n = a{J (y)sin Aat + N (y)cos Aat + o A o A r[j (x)s in (Acft-e) -N ( x ) cos ( Aat-e ) ]} (60) where a is the amplitude of the incoming wave (in this case a = ri h ) , r is the reflection coefficient, and e a phase shift in the reflected wave. The frequency of this wave must be the same as that of the modulating wave derived out- side the surf zone or Aa. By substituting the asymptotic form of the Bessel and Neumann functions, u b V^X Tr {cos (x-Aat - — )+ r cos ( x-Aat-8 ) } , (61 ) where 9 = — +e. Since the set up shoreward was assumed to be a composite of the two solutions (58) and (61), k(hh-h)+nh +^/~-Ccos (x-Aat- 1) ^rcos ( x-Aat-9 ) (62) 33 I V. COMPAR ISONS The results obtained here are compared with the labora- tory experiments performed by Bowen, et al.Cl968H; Figure 2. They obtained experimental results from a control led wave tank experiment, and in each case only a monochromatic wave was considered. Their results compare quite favorably to the steady state solutions of Lon guet-H i gg i n s and Stewart [I 962]. Since it was assumed that the variance of the simple wave spectrum was equal to the variance of the monochromatic wave, the derived expressions differ from existing theory by that amount which is contributed by the modulating wave of the unsteady portion. Comparing Long ue t-H i gg i n s ' solution (25) I a2k 2 s i nh 2kh with the unsteady solution (47) 1 a2k n = - -x — : — ! — =rrrC I + cos(Ak x - Aat)] , 2 s I n h 2kh x it is found that, by averaging the effect of the unsteady component, they are the identical. As mentioned earlier, the total effect of the unsteady motion is, in this case, to produce a set down that oscillates periodically from twice the steady solution to zero times it, i.e., no set 34 down at all. Thus a group of waves which are close in fre- quency and wavelength may be said to produce in the near shore region a fluctuation in the mean sea level seaward of the breaker I ine. This is also demonstrated by the unsteady solution obtained by allowing for a sloping bottom, eq uat i on C55). It too consists of a steady component and an oscillat- i ng componen t . The initial solution, equation (44), on the other hand is comprised of only an oscillating component. However, this solution was derived in a simplistic manner by assuming that the depth remained constant and by solving the resulting differential equation. While this demonstrates that the mean sea level oscillates in a wave-like fashion when driven by a group of waves, it does little to provide insight into the near shore region. The set down can be expressed in shal low water in terms of deep water conditions by using conservation of energy flux, E c = -~ pqa c = constant , g 2 a g ' whe re I 2 g -* pga tt- 2 . a o 2a 2 pgas /g~h~ or o 9 2 a /gb 35 The subscript o refers to deep water and the subscript s refers to shal low water. The set down in shal low water by equation (25) is then given by 2 1/2, ,-3/2 a g (mx ) o3 I 4a (63) where in shallow water sinh 2kh -*■ 2kh and h = mx for a con- stant sloping bottom. The set down derived from linear theory can be compared to that derived from I inear theory for a sloping bottom, equation (55), by also expressing the shallow water amplitude for sloping bottom solutions in terms of deep water condi- tions. This is accomplished by patching Stokes solution at the off shore to the shal low ater solution. For a smooth match it is required that, from Fried rich [1948^], 2 n a tt 2 o 2m The steady state solution of (55) can now be expressed in terms of deep water conditions 2 . 2 1/2, ,-3/2 ao a , .-1/2 g (mx) + —A (gmx) 4a 2 a a , = n, + -t— (gmx)~l/z 4 (64 Hence, the sloping bottom solution results in slightly greater set down as compared with linear theory. Numerica 36 Wave Period 1.14 sec H0 = 6.45 cm m = 0.082 a a Observed Data 0 0 Equation (63) • • Equation (64 2.5 2.0 1 .5 1 .0 0.5 0.5 -1 6 Z cm 0 400 300 X (cm) — »-4 200 100 Figure 2: Profile ot Mean Sea Level 37 results of these equations, (63) and (64), are plotted in Figure 2 . Close to the break point, most theoretical results fail to compare favorably with the experimental results. Theory increases set down rapidly near the break point while ex- perimental results tend to flatten out. This is due most likely to the failure of linear theory to adequately describe breaking waves. Although simplistic in form, the application of the two frequency wave spectrum demonstrated that fluctuating values are obtained for the set down and set up phenomena, i.e., time dependent solutions were obtained for these con- cepts. Hence, a group of waves, simi iar in frequency can be expected to produce a periodic variation in the mean sea level which becomes most apparent when they encounter a beach . 38 V. CONCLUS I ONS In this study, unsteady solutions for set down and set up were derived using a simple two component wave spectrum. The spectral wave components were described using I in ear wave theory for both horizontal and sloping bottoms. The sloping bottom solution involving the Bessel functions gave a slightly greater set down compared with that of the linear wave theory solution. However, both unsteady solu- tions for set down showed, that to at least a first approxi- mation, a steady and a fluctuating component, the steady component being identical to the earlier steady state solutions. The set down at the breaker I i ne acts as the boundary condition driving the set up inside the surf zone. The set down solution showed that the steady and unsteady components are simply additive. Hence, inside the surf zone a solu- tion is composed of the stead/ set up and a long wave, i.e., surf beat, which is driven by the fluctuating condition at the brea ker line. 39 LIST OF REFERENCES Bowen, A. J., Rip Currents, Ph. D. Thesis, University of California, San Diego, 1967. Bowen, A. J., Innman, D. L., and Simmons, V. P., "Wave 'Set Down' and Set Up", Journal of Geophysical Research, v. 73, no. 8, p. 2569-2577, 1968. Guza, R. T. and Bowen, A. J., "Resonant Interaction for Waves Breaking on a Beach", Journal of Geophysical Research, (in press), 1977. Iwagaki, Y. and Sakai, T., "Shoaling of Finite Amplitude Long Waves on a Beach of Constant Slope", Proceed i ng s of the 13th Conference on Coastal Engineering, v . I , p. 347-364, Copenhagen, Denmark, 1972. Longuet-H i gg i n s, M. S., "On the Wave-Induced Difference in Mean Sea Level Between the Two Sides of a Submerged Breakwater", Journal of Marine Research, v. 25, no. 2, p. 148-153, 1967. Longuet-Higgins, M. S. and Stewart, R. W., "Radiation Stress and Mass Transport in Gravity Waves with Appli- cation to 'Surf Beats'", Journal of Fluid Mechanics, no. 13, p. 481-504, 1962. Longuet-Higgins, M. S. and Stewart, R. W., "A Note on Wave Set Up", Journal of Marine Research, v. 21, no. I, p. 4-10, 1963. Longuet-Higgins, M. S. and Stewart, R. W., "Radiation Stress in Water Waves; a Physical Discussion with Applications", Deep Sea Research, v. II, p. 529-562, I 964. Phillips, 0. M., The Dynamics of the Upper Ocean, p. 44-56, Cambridge Press, 1966. Stoker, J. J., Water Waves , p. 22-25, I nter sc i ence , 1966. Van Dorn, W. G., Set Up and Run Up in Shoaling Breakers, paper presented at the Conference on Coastal Engineer- ing, 14th, Honolulu, Hawaii, July 1976. 40 INITIAL Dl STRI BUTI ON LI ST I. Defense Documentation Center Came ron Station Alexandria, Virginia 223 14 20 Li brary, Code 0 I 42 Naval Postgraduate School Monterey, California 93940 3. Assoc. Professor E. B. Thornton, Code 68 Department of Oceanography Naval Postgraduate School Monterey, California 93940 4. Assoc. Professor J. J. von Schwind, Code 61 Department of Oceanography Naval Postgraduate School Monterey, California 93940 5. LT. D. J. McReynolds, USN ANTARCTICDEVRON SIX NAS Pt . Mugu , Ca I i f orn i a 6. Ocea nog ra p he r of the Navy Hoffman Building No. 2 200 Stova I I Street Alexandria, Virginia 22332 7. Dr. Robert E. Stevenson Scientific Liaison Office, ONR Scripps Institution of Oceanography La Jo I I a, Ca I i forn i a 92037 8. Department of Oceanography, Code 68 Naval Postgraduate School Monterey, California 93940 9. Office of Naval Research Code 4 I 0 N0RDA, NSTL Bay St. Louis, Mississippi 39520 0. Li brary, Code 3330 Naval Ocea nog ra p h i c Office Washington, D. C. 20373 No . Cop i es 41 12. S I 0 Li brary University of California, San Diego P. 0. Box 2367 La Jo I I a, Ca I i forn i a 92037 Department of Oceanography Library University of Washington Seattle, Washington 98 105 13. Department of Oceanography Library Oregon State University Corvallis, Oregon 9733 1 14. Commanding Officer Fleet Numerical Weather Central Monterey, California 93940 15. Commanding Officer Naval Environmental Prediction Research Fac i I i ty Monterey, California 93940 16. Department of the Navy Commander Oceanog rap h i c System Pacific Box 1390 FPO San Francisco 96610 17. D i recto r Naval Oceanography and Meteorology National Space Technology Laboratories Bay St. Louis, Mississippi 39520 18. N0RDA, Techn i ca I D i rector Bay St. Louis, Mississippi 39520 42 Thesis M2697 c.l ITISCj McReynolds Wave set up and set down due to a narrow frequency wave spec- trum. thesM2697 Wave set up and set down due to a narrow 3 2768 002 04431 5 DUDLEY KNOX LIBRARY