Stokes Waves Benny Lautrup May 11, 2011

Stokes waves Benny Lautrup May 11, 2011

In 1847 Stokes published a seminal paper [1] on nonlinear gravity waves that generalize the linear small-amplitude harmonic waves at constant depth discussed in Section 25.3 at page CM-424. For many years these Stokes waves stood as the model for nonlinear waves, even if they today are known to suffer from small frequency instabilities [Mei 1989].

1 Periodic, permanent line waves

Gravity waves in a liquid (“water”) are essentially only interesting for large Reynolds number, where viscosity plays an insignificant role. Compression plays likewise only a minor role, when the material velocity in a wave is small compared to the sound velocity in the liquid, which it normally is. Consequently, the liquid may be assumed to be an incompressible Euler liquid (see Chapter CM-13). Furthermore, if the waves have been created from water originally at rest without vorticity, the flow may also be assumed to be irrotational at all times. y 6 Incompressible, inviscid, and irrotational water g0 atmosphere Under these conditions, the two-dimensional flow in a line wave may be derived from a ve- ? locity potential ‰.x; y; t/ that satisfies Laplace’s equation (see page CM-220), - x 2 2 vx x‰; vy y ‰; ‰ 0: (1) sea D r D r rx C ry D Any solution to the Laplace equation yields a possible two-dimensional flow. With constant gravity g0 pointing along the negative y-axis, the undisturbed surface of the In the two-dimensional flat-Earth sea corresponds to y 0. Let the surface of the wave be described by a function y h.x; t/. coordinate system, constant grav- A liquid particle sittingD right at the surface (or rather just below) must follow theD motion of ity points along the negative y- axis, and the surface of the undis- the surface. In a small time interval ıt, the particle is displaced horizontally by ıx vxıt D turbed sea is at y 0. The z-axis and vertically ıy vy ıt, so that we must have h.x ıx; t ıt/ h.x; t/ ıy. Expanding points out of the paper.D to first order in theD small quantities, we get the kinematicC surfaceC condition,D C @h y vx xh vy for y h: (2) 6 @t C r D D t ıt ...... C...... The pressure in the liquid is given by Equation (13.34) at page CM-220. At the liquid surface ...... ...... against vacuum (or air) the pressure must be constant (in the absence of surface tension), and ...... bıy ..... ıx ...... we get the dynamic surface condition ...... t ...... r ..... - 1 2 2 @‰ ...... x ...... g0y v v const for y h (3) ...... C 2 x C y C @t D D ...... The constant is rather unimportant, since it can be eliminated by shifting the origin of y. A fluid particle at the surface Finally, we must require that the vy 0 for y d, where d is the constant depth of the must follow the motion of the sur- undisturbed ocean. D D face from time t to time t ıt. C 2 PHYSICS OF CONTINUOUS MATTER

Permanence A line wave is said to be permanent or stationary if it progresses along the x-axis with un- changing shape and constant celerity c, such that all surface features move with the same velocity c. The surface shape can therefore only depend on the combination x ct, and we ex- pect that this is also the case for the velocity potential, that is, ‰ ‰.x ct; y/. We may consequently replace all time derivatives by x-derivatives accordingD to the rule @=@t c@=@x, and afterwards put t 0, so that the velocity potential ‰.x; y/ and the derived! ﬁelds only depend on x and y. TheD surface boundary conditions may thus be written

vy .vx c/h0 for y h; (4) eSWkinematic D D where h dh=dx, and 0 D 1 2 2 g0h cvx v v const; for y h: (5) eSWdynamic D 2 x C y C D

The bottom condition is as before vy 0 for y d. D D Periodicity and symmetry y 6 Stokes also assumed that the waves are periodic with spatial period ,

...... h.x / h.x/; ‰.x ; y/ ‰.x; y/; (6) ...... C D C D ...... -...... x so that the nonlinear waves in this respect are generalizations of the linear ones...... The Laplace equation is trivially invariant under a change of sign in x. Because of its ...... linearity, its solutions may accordingly be classiﬁed as symmetric (even) or antisymmetric ...... (odd) under a change of sign of x. There is, however, no guarantee that a solution satisfying ...... the nonlinear boundary conditions might not be an asymmetric linear combination of odd and even velocity potentials. We shall later return to this question, but for now we assume that Even and odd periodic functions. ‰.x; y/ may be chosen odd in x,

‰. x; y/ ‰.x; y/: (7) D

The boundary conditions then imply that vx is even and vy odd, and consequently that the surface height h must be even. Notice that the antisymmetry of ‰.x; y/ also implies that all even derivatives at x 0 must vanish. Conversely, all odd derivatives of h.x/ must vanish, implying that x 0 isD an extremum of h.x/. D Choice of units The parameters describing periodic line waves are the celerity c, wavelength (or wavenum- ber k 2=), depth d, amplitude a, and strength of gravity g0. The only dimensionless variablesD are ka, kd, and

g0 G : (8) eSWdispersionG D kc2 For general reasons G must be a function of ka and kd (dispersion relation). In the linear approximation CM-(25.28) we have G coth kd. To simplify the formalism, we shallD choose units of space and time so that,

k 1; c 1: (9) D D

This implies 2, and that Equation (8) simpliﬁes to g0 G. D D STOKES WAVES 3

Fourier expansion

The periodicity and symmetry permit us to expand the surface height into even Fourier components,

X1 h Hn cos nx; (10) D n 1 D

where the Hn may depend on a and d. Although an even function would permit a term H0, such a term is absent here because of mass conservation in the wave shape, which re- R quires that h.x/ dx 0. The series converges provided it is square integrable, that is, R 2 P D 2 h.x/ dx n1 1 Hn < . The periodicityD andD oddness1 of the velocity potential ‰ guarantees likewise that it may be expanded into odd Fourier components of the form ‰ fn.y/ sin nx that must sat- 2 isfy the Laplace equation, which here takes the form f n fn. The general solution is n00 D fn An cosh ny Bn sinh ny, but since each Fourier component of the vertical velocity D C vy f .y/ sin nx must vanish for y d, we must require f . d/ 0, and the solution n0 D n0 D becomes fn cosh n.y d/. In non-dimensional form we may thus write, C

X1 cosh n.y d/ ‰ ‰n C sin nx; (11) D sinh nd n 1 D

where the ‰n may depend on a and d. The denominator is introduced for later convenience (see the linear solution CM-(25.29)). From the velocity potential we derive the velocity ﬁeld components,

X1 cosh n.y d/ vx n‰n C cos nx; (12) D sinh nd n 1 D X1 sinh n.y d/ vy n‰n C sin nx: (13) D sinh nd n 1 D Using trigonometric relations, the y-dependent factors may also be written

cosh n.y d/ fn.y/ C sinh ny Cn cosh ny; (14) Á sinh nd D C sinh n.y d/ gn.y/ C cosh ny Cn sinh ny; (15) Á sinh nd D C

where Cn coth nd contain all the dependence on depth d. Notice that f ngn and D n0 D g nfn. n0 D

Truncation

Having solved the field equations and implemented the bottom boundary condition, we must now apply the surface boundary conditions to determine the unknown coefficients ‰n and Hn. A natural approximation consists in truncating the infinite series at n N , that is, dropping all terms of harmonic order n > N . Here we shall only investigateD the linear first order approximation and the second order nonlinear correction to it. 4 PHYSICS OF CONTINUOUS MATTER

Linear approximation For N 1 we have D

h H1 cos x; (16) D and

vx ‰1f1.y/ cos x; vy ‰1g1.y/ sin x: (17) eSWlinear D D Inserted into the surface boundary condition (4) and (5), we get in the linear approximation vy h0 and Gh vx const for y 0 (because setting y h would generate second orderD harmonics), andD weC arrive at D D

‰1 H1; GH1 ‰1C1; (18) D D

while const 0 in this approximation. Eliminating ‰1 we obtain D 2 g0 g0 G C1; or c tanh kd (19) eSWcelerity D D kG D k which is simply the expression CM-(25.28) for the celerity of linear waves. The first-order coefficient H1 is clearly a free parameter, and usually one defines H1 ‰1 a where a is the linear wave amplitude. D D

Second order approximation For N 2 we have D

h H1 cos x H2 cos 2x; (20) D C and

vx ‰1f1.y/ cos x 2‰2f2.y/ cos 2x; (21) eSWvelocities D C vy ‰1g1.y/ sin x 2‰2g2.y/ sin 2x: (22) D C

Setting y h and expanding fn.h/ and gn.h/ in powers of h, we only need to keep the ﬁrst-orderD harmonic term in the ﬁrst terms and the zeroth-order term (h 0) in the second. In other words, for y h, D D

vx ‰1.C1 H1 cos x/ cos x 2‰2C2 cos 2x; (23) D C C vy ‰1.1 C1H1 cos x/ sin x 2‰2 sin 2x: (24) D C C Inserted into the kinematic boundary condition (4), we get

‰1.1 C1H1 cos x/ sin x 2‰2 sin 2x C C . 1 ‰1C1 cos x/H1 sin x 2H2 sin 2x; D C C where all terms of harmonic order 3 and higher have been dropped. Using that sin x cos x 1 D 2 sin 2x, we arrive at the equations,

1 2 ‰1 H1;‰2 H2 C1H ; (25) eSWkinematic1 D D 2 1 by matching the sin x and sin 2x terms on both sides. STOKES WAVES 5

Finally, the dynamic boundary condition (5) becomes to second harmonic order

G .H1 cos x H2 cos 2x/ ‰1.C1 H1 cos x/ cos x 2‰2C2 cos 2x C D C C 1 2 2 .‰1C1 cos x/ .‰1 sin x/ const: 2 C C

2 1 2 1 Using that sin x 2 .1 cos 2x/ and cos x 2 .1 cos 2x/, and matching the harmonic terms, we ﬁnd, D D C

1 1 2 2 GH1 ‰1C1; GH2 2‰2C2 ‰1H1 ‰ .C 1/; (26) D D C 2 4 1 1 where the zeroth-order (constant) terms have been absorbed in the unknown constant. Eliminating the ‰’s by means of (25), the solution becomes

2 2 C1 4C1C2 3 2 3.C1 1/ 2 G C1;H2 C H1 ;‰2 H1 : (27) D D 4.2C2 C1/ D 4.2C2 C1/ 1 1 Using that C2 C1 C , this simpliﬁes to: D 2 C 1

1 2 2 3 2 2 G C1;H2 C1 3C 1 H ;‰2 C1 C 1 H ; (28) D D 4 1 1 D 4 1 1

with the celerity again given by (19), and C1 coth d. Notice that H1 is again a free variable, D so we shall again put H1 ‰1 a where a is the amplitude. The surface shape becomesD inD the extremes, 1 h a cos x a2 cos 2x for d (deep water); (29) D C 2 ! 1 3a2 h a cos x cos 2x for d 0 (shallow water): (30) D C 4d 3 ! In deep water the corrections to the linear wave can be ignored for a 1. In shallow water the condition for ignoring the nonlinear term is that a d 3 (in units of 2). The dimensionless ratio, D a Ur ; (31) D k2d 3

is called the Ursell number[2], and the linear approximation is valid for Ur 1. Example 1 [Tsunami]: A tsunami proceeds with wavelength 500 km in an ocean of 1 1 depth d 4 km. The celerity becomes c 200 m s , or about 700 km h . For the linear wave theory toD be valid we must have a k2d 3 10 m, which is satisﬁed for a 1 m, corresponding to Ur 0:1. One should note that there are,D however, nonlinear secular changesD to the wave shape takingD place over very long distances that may become important.

Higher order approximations In principle these methods may be used to obtain higher order harmonic corrections to the linear waves. As the analysis of second harmonics indicates, the calculation of the higher harmonics are liable to become increasingly complicated. This was also recognized by Stokes, and in a Supplement to his 1847 paper [1], he carried out a much more elegant series expansion of deep-water waves to ﬁfth order. In the following section, we shall extend this calculation to 9’th order. 6 PHYSICS OF CONTINUOUS MATTER

Stokes drift In the strictly linear approximation, that is, to the first order in the amplitude a, the fluid particles move in elliptical orbits (see page CM-428). In higher order approximations, this is no more the case, and we shall see that the fluid particles beside periodic motion also perform a secular motion. We shall calculate this effect in the Lagrange representation, where the intuitive meaning is clearest. A particle starting in .X; Y / at time t 0 and found at .x; y/ at time t is displaced D horizontally by ux x X and vertically by uy y Y . The particle velocity components are then determinedD from D

dux duy vx.X t ux;Y uy /; vy .X t ux;Y uy / (32) eSWdisplacementderivatives dt D C C dt D C C To lowest harmonic order we disregard the displacements on the right hand sides and integrate the velocities (21) to get

ux ‰1f1.Y / sin.t X/; uy ‰1g1.Y / cos.t X/: (33) D D Here the integration constants have been chosen such that the horizontal displacement vanishes for t X and the vertical displacement vanishes at the bottom Y d. Thus, to first harmonic orderD of approximation a fluid particle moves on an ellipse, whichD has horizontal major axis and vertical minor and flattens towards the bottom. Next, we expand to first order in the displacements on the right-hand side of (32) to get

dux @vx @vx vx.X t; Y / ux uy ; (34) dt D C @X C @Y duy @vy @vy vy .X t; Y / ux uy : (35) dt D C @X C @Y The leading terms are determined by (21) and the remainder from the lowest order displacements. To second harmonic order we ﬁnd

dux ‰1f1.y/ cos.t X/ 2‰2f2.Y / cos 2.t X/ dt D C 2 2 2 2 2 2 ‰ f1.Y / sin .t X/ ‰ g1.Y / cos .t X/: (36) C 1 C 1 duy ‰1g1.Y / sin.t X/ 2‰2g2.Y / sin 2.t X/: (37) dt D C Finally, integrating with respect to t and imposing the boundary conditions mentioned above, we get

1 2 2 2 ux ‰1f1.Y / sin.t X/ ‰2f2.y/ ‰ f .Y / g .Y / sin 2.t X/ D C 4 1 1 1 1 ‰2 f 2.y/ g2.y/ .t X/; (38) C 2 1 1 C 1 uy ‰1g1.Y / cos.t X/ 2‰2g2.y/ cos 2.t X/: (39) D C The term linear in t X shows that on top of its second order harmonic motion, the particle drifts along the x-axis with the Stokes drift velocity, which may be written:

cosh 2k.Y d/ U 1 c.ka/2 C : (40) D 2 sinh2 kd

where we have put back the dimensional constants. For shallow-water waves, this becomes 1 2 U 2 c.a=d/ . In the tsunami example above, the drift velocity becomes a measly two centimeter per hour! STOKES WAVES 7

2 Stokes expansion

Instead of working in “laboratory” coordinates where the waves move with celerity c 1 along the positive x-axis, it is most convenient to use comoving coordinates in whichD the wave surface is static and the flow below is steady. Transforming the laboratory velocity field to a dimensionless comoving velocity field,

ux vx 1; uy vy ; (41) D D the surface boundary conditions (4) and (5) become

uy uxh0 for y h; (42) eSWtop1 D D and Gh 1 u2 u2 C for y h: (43) eSWtop2 D 2 x C y C D where C is a constant and G is gravity when c k 1; see Equation (8). The bottom conditions become D D

ux 1; uy 0 for y (44) ! ! ! 1 The ﬁrst condition simply expresses that the laboratory moves with velocity 1 relative to the comoving frame.

Conformal transformation In comoving coordinates the dimensionless velocity potential is denoted .x; y/, and the conjugate stream function .x; y/. These functions satisfy the relations (see page CM-221) @ @ @ @ ux ; uy : (45) eNLconjugatepotentials D @x D @y D @y D @x Combining the derivatives, we get the Laplace equations, @2 @2 @2 @2 0; 0; (46) @x2 C @y2 D @x2 C @y2 D expressing, respectively, incompressibility and irrotationality. Rather than viewing x and y as independent variables, and .x; y/ and .x; y/ as dependent, Stokes proposed to view and as independentD variables, and x D x.; / and y y.; / as dependent. To carry out the transformation from Cartesian coordinatesD .x; y/ toD the curvilinear coordinates .; /, we ﬁrst solve the differential forms,

d uxdx uy dy; d uxdy uy dx; (47) D C D and ﬁnd,

uxd uy d uy d uxd dx ; dy C : (48) D u2 u2 D u2 u2 x C y x C y From these we read off

@x @y ux @y @x uy ; : (49) ePWderivs @ D @ D u2 u2 @ D @ D u2 u2 x C y x C y Combining the derivatives, we arrive at @2x @2x @2y @2y 0; 0: (50) @2 C @ 2 D @2 C @ 2 D Thus, the Cartesian coordinates x and y are also solutions to the Laplace equation in the curvilinear coordinates and . 8 PHYSICS OF CONTINUOUS MATTER

Periodicity and symmetry For periodic line waves with k 1 and period 2, the dimensionless velocity ﬁeld must D D .. 6 obey ...... ux.x 2; y/ ux.x; y/; uy .x 2; y/ uy .x; y/: (51) ...... C D C D ...... - x ...... But since ux 1 for y , it follows from (45) that x and y in this ...... ! ! 1 ! ! ...... limit, so that horizontal periodicity along x can only be imposed on x and : ...... C ...... .x 2; y/ .x; y/ 2; .x 2; y/ .x; y/: (52) C D C D Solving these equations for x and y, it follows that, x.; / and y.; / are also periodic Periodicity in and x. No- in with period 2. C tice that both and x con- We shall as before require that .x; y/ is odd in x and .x; y/ is even, tinue into the diagonally adjacent boxes while x is truly peri- . x; y/ .x; y/; . x; y/ .x; y/; (53) odic (dashed) inCx. D D which translates into x.; / being odd in and y.; / even. Using the periodicity and oddness we may expand x into a sum of harmonic Fourier C 2 components of the form Fn. / sin n. The Laplace equation requires Fn00. / n , and n D using that for y it follows that Fn Ane where the coefﬁcients An are constants.! Using C1 the relations! 1@x=@ @y=@ andD @x=@ @y=@ to determine y, we arrive at D D

X1 n X1 n x Ane sin n; y Ane cos n; (54) D D C n 1 n 1 D D

where we have deﬁned An 0 for n < 1 to avoid writing n 1 in the sums. D Boundary conditions At the surface of the water, y h.x/, the stream function .x; h.x// is constant: D @ @ d .x; h.x// h0.x/ uy uxh0.x/ 0; (55) D @x C @y D C D because of the boundary condition (42). Since a constant value of can be absorbed into the An, we are free to choose 0 at the surface, and we obtain a parametric representation of the surface, D

X1 X1 x./ An sin n; y./ An cos n: (56) eSWimplicit D D n 1 n 1 D D

The wave height h.x/ must satisfy the relation

h.x.// y./: (57) D for all . From (49) we obtain, ˇ ˇ Â @x Ã2 Â @y Ã2ˇ 1 ˇ S./ ˇ ˇ ; (58) Á @ C @ ˇ D u2 u2 ˇ ˇ 0 x y ˇy h.x/ D C D STOKES WAVES 9

so that the dynamic condition (43) may be written:

F .F K 2y.//S./; (59) ePWtop2a D C

where F 1=G and K .2C 1/F are “constants” (that may depend on the amplitude). The constantD F determinesD the dispersion relation, g c2 0 F .ka/; (60) D k expressed in proper dimensional parameters. The implicit representation (56) of y h.x/ through does not guarantee that the average value of h vanishes. A quick calculationD yields the average

Z Z 1 1 dx./ 1 X1 2 h0 h.x/ dx y./ d nA : (61) eSWheight D 2 D 2 d D 2 n n 1 D This will be used to plot the shape of the wave.

Truncation

What remains is to determine the coefficients An. As in the previous section, the boundary conditions permit us to determine all the An for n 2 in terms of the first A1 b. We shall later find the relation between b and the amplitudea of the first-order harmonicD in x. Since

2 cos n cos m cos.n m/ cos.n m/; (62) D C C 2 sin n sin m cos.n m/ cos.n m/; (63) D C

it follows that the product AnAm will be associated with harmonic functions up to order n m. n C Consequently, An can only depend on b and higher powers, so that we may write

X1 m An Anmb : (64) D m n D Similarly, we shall put

X1 n X1 n F 1 Fnb ;K Knb ; (65) D C D n 1 n 1 D D where zeroth order values F 1 and K 0 have been chosen so that Equation (59) is fulfilled in zeroth order (with SD 1). It is nowD “merely” a question of matching powers and trigonometric functions on both sidesD of this equation to determine the unknown coefficients. Truncating the series by dropping all powers higher than bN , we obtain from (59) an iterative scheme that successively determines the highest order coefficients (FN , KN , and An;N with n 1; ;N ) from the lower order coefficients already known. To simplify the D analysis we shall assume that the coefficients Fn and Kn are only nonzero for even n whereas Anm are only nonzero for n and m being either both even or both odd. These assumptions may be written

Fodd Kodd Aeven;odd Aodd;even 0; (66) D D D D and will be justiﬁed by the solution. 10 PHYSICS OF CONTINUOUS MATTER

First order (N 1) D Defining A1 b, that is, A11 1 and A1;n 0 for n > 1, we have D D D x b sin O .b/ ; y b cos O .b/ (67) D C D C Equation (59) us automatically fulfilled (since by assumption F1 K1 0). From Equation (61), we find the height average D D 1 2 3 h0 b O b (68) D 2 C Notice that this is of second order, even if the truncation is to first order.

Second order (N 2) D Evaluating (59) it becomes 2 3 2 2 2 3 1 b F2 O b 1 b F2 K2 sin 3 cos 2A2;2 cos 2 O b ; C C D C C C C C and using trigonometric relations it becomes 2 3 2 2 3 1 b F2 O b 1 b .F2 K2 1/ 2b .A2;2 1/ cos 2 O b : (69) C C D C C C C From this we read off K2 A22 1, whereas F2 is not determined (it will be determined in third order). D D Consequently, we have x b sin b2 sin 2 O b3 ; y b cos b2 cos 2 O b3 ; (70) D C D C C and 1 2 4 h0 b O b : (71) D 2 C Evidently, all odd powers must vanish: hodd 0. D Ninth order (N 9) D The above procedure can be continued to higher orders. In each order of truncation, N , we only need to determine the constants AN;N , AN 2;N ;:::, until it breaks off. Expressed in 10 terms of the An’s, the result is to 9’th order (leaving out the explicit O b ),

A1 b; D 2 1 4 29 6 1123 8 A2 b b b b ; D C 2 C 12 C 72 3 3 19 5 1183 7 475367 9 A3 b b b b ; D 2 C 12 C 144 C 8640 8 4 313 6 103727 8 A4 b b b ; D 3 C 72 C 4320 125 5 16603 7 5824751 9 A5 b b b ; (72) D 24 C 1440 C 86400 54 6 54473 8 A6 b b ; D 5 C 1800 16807 7 23954003 9 A7 b b ; D 720 C 302400 16384 8 A8 b ; D 315 531441 9 A9 b : D 4480 Stokes himself carried by hand the expansion to 5’th order. The constants become in the same approximation F 1 b2 7 b4 229 b6 6175 b8; (73) D C C 2 C 12 C 48 K b2 2b4 35 b6 1903 b8; (74) D C C 4 C 36 1 2 4 35 6 1903 8 h0 b b b b : (75) D 2 C C 8 C 72 Notice that K 2h0. D STOKES WAVES 11

kHhh0L 0.6 0.4 0.2 x

1.5 1.0 0.5 0.2 0.5 1.0 1.5 Λ 0.4

Figure 1. Three periods of the truncated wave for N 9 with ka 0:43 together with the ﬁrst-order wave (dashed). One notices that the nonlinear effectsD make the waveD crests sharper and taller, and the troughs wider and shallower.

Explicit wave form

For general reasons the wave must take the symmetric form

X1 h h0 Hn cos nx; (76) D C n 1 D as a function of x. The coefﬁcients Hn must like the An be of the form

X1 m Hn Hnmb (77) D m n D with Heven;odd Hodd;even 0. These coefﬁcientsD are obtainedD successively in the same iterative procedure that led to the An, with the result

9 3 769 5 201457 7 325514563 9 H1 b b b b b ; D C 8 C 192 C 9216 C 2211840 1 2 11 4 463 6 1259 8 H2 b b b b ; D 2 C 6 C 48 C 20 3 3 315 5 85563 7 5101251 9 H3 b b b b ; D 8 C 128 C 5120 C 40960 1 4 577 6 57703 8 H4 b b b ; D 3 C 180 C 2160 125 5 38269 7 318219347 9 H5 b b b ; (78) D 384 C 9216 C 7741440 27 6 30141 8 H6 b b ; D 80 C 5600 16807 7 51557203 9 H7 b b ; D 46080 C 7372800 128 8 H8 b ; D 315 531441 9 H9 b : D 1146880

Finally we want to express b in terms of the lowest order harmonic amplitude a H1. D 12 PHYSICS OF CONTINUOUS MATTER

Solving for b we ﬁnd 3 5 26713 25971763 b a a3 a5 a7 a9; (79) D 8 24 9216 2211840 which when inserted into the H ’s yield

H1 a; D 1 2 17 4 233 6 348851 8 H2 a a a a ; D 2 C 24 C 128 C 46080 3 3 153 5 10389 7 747697 9 H3 a a a a ; D 8 C 128 C 2560 C 40960 1 4 307 6 31667 8 H4 a a a ; D 3 C 180 C 4320 125 5 10697 7 47169331 9 H5 a a a ; D 384 C 4608 C 3870720 (80) 27 6 34767 8 H6 a a ; D 80 C 11200 16807 7 30380383 9 H7 a a ; D 46080 C =7372800

128 8 H8 a ; D 315 531441 9 H9 a : D 1146880 The 9’th order wave shape is shown in Figure 1 for a 0:43. D References

[1] G. Stokes, On the Theory of Oscillatory Waves, Transactions of the Cambridge Philo- sophical Society VIII (1847) 197–229, and Supplement 314–326. [2] F. Ursell, The long-wave paradox in the theory of gravity waves, Proceedings of the Cambridge Philosophical Society 49 (1953) 685–694.