A THEORY FOR WAVES OF FINITE HEIGHT

A theory for waves of finite height, presented in this paper, is an exact theory, to any order for which it is extended. The theory is represented by a summation liarmdnic series, each term of which is in an unexpanded form. The terms of the series when expanded result in an approximation of the exact theory, and this approximation is identical to Stokes' wave theory extended to the same order. The theory represents irrotational divergenceless flow. The procedure is to select the form of equations for the coordinates of the particles in anticipation of later operations to be performed in the evaluation of the coefficients of the series. The horizontal and vertical components of these coordinates are given respectively by the following; . / tr, . v , u A \ cosh Nk (I + z rj ) >. kx = k(x-C) + I (kA0) cmllKlt> '— SinNk(x-Ct-0 smh Nki

Where <f> is the velocity potential, g acceleration of gravity; P density of fluid} and p the pressure.
The first equation is the usual equation of hydrodynamics, the second specifies irrotational flow, the third specifies no flow across the sea bed, and the fourth specifies no flow across the free wave surface.

Coordinates
The coordinates of the particles of water can be represented by a set of equations around which a theory can be developed.If the equations are selected in anticipation of later operations to be performed, then one might be able to minimize the work envolved.In the presence of wave motion the horizontal and vertical displacements (f, 1 ?) of the water particles from the position of rest or the position of no motion can be represented respectively as follows?
K^|o N (KA 0 ) N

-ct-o
The horizontal and vertical components of particle velocity are also given by: JL-.-±l± = __L Ulf (12) c c ax c az and «L = _ JL ii s+ i d± (i3) c c dz ^ ax Where d) and ty are the velocity potential and the stream function respectively.It is seen from Equations 10 and 11 together with Equations 12 and 13 that the velocity potential and stream function except for arbitrary constants will have the following forms: The above proof is more easily verified by performing the above operation on the equations given in the next section (Table I, for example).

Power Series Equations for Particle Velocity
In the development following it will be convenient to use and and The simultaneous solution of Equations 22 and 23 can be made by the process of resubstitution to as high an order as required where the Mth order will include all terms of U, W, and U W , where p = 0 to M, q = 0 to M, and r + s = 1 to M.
The process of resubstitutions leads to the following terms: It will be seen that a general expression can be written for U/C, having the following power series equation: where

Bernoulli's Equation
The problem of wave motion can be reduced to one of steady state by superimposing a steady current on the wave motion equal to the wave celerity but of opposite direction.This operation, known as the Hayleigh principle, leads to Bernoulli's equation applicable along the free surface, where it is assumed that everywhere along the free surface the pressure is constant or is zero with respect to atmospheric pressure, whence where the subscript s refers to the conditions at the free surface, Equation 28 can be written as follows: (-^--I) 2 + (^) 2 + -^8-=K= constant (29) or solving for k 1 ?s It will be convenient to define the Bernoulli term as Along the free surface equations 20 and 21,Z-V = 0, whence and where X N = '

N tanh Nki
From Table 1, one may obtain the Bernoulli term B which leads to the following termss It will be seen that a general expression can be written for B , having the following pt>wftT series equation: For example, the 8th order terms will have the following combinationsof (r,s) = (8,0); (6,2)} (4,4); (2,6); and (0,8); whence from Equation 35 the 8th order terms for B are: Thus Equation 35 can be used to obtain the Bernoulli term B to as high an order as required.The term B will have an expanded form as follows: s 135)

Procedure for the Evaluation of Coefficients
The coefficients a , a , a ...... a must be evaluated such that the surface boundary conditions are satisfied.The surface profile elevation with respect to the undisturbed water level is given by Equation 6.
The surface boundary conditions are satisfied when Equation 6is made identical to Equation 30.To whatever order is required Equation 30 is a means by which the solution is obtained.Incidentally, such a solution is similar to a least squares solution in statistical theory.
It will be convenient to use an expanded form of Equation 30 as follows: k % = Z °N ( kA o) N cos uQ\ where (37) The wave height H = 2A is obtained from the difference between V^ at © * 0 and 17 at 0 = ir, a8d since A will always be equal to unity as long as H = 2A S , whence from equation 37, 0 = (A 13 +A 33 ) (kA 0 ) 2 + (A 15 +A 35 + A 55 ) (kA 0 ) 4 + (A l7 +A 37 +A 57 +A 77 ) (kA 0 ) 6 N Equating to zero terms of (kA ) , one obtains the following!
The procedure is to expand each of" the^individual equations and then equate to zero like terms of (KA ) .It will be convenient to present the higher order terms of the A's and the F 's in terms including the B's terms and the lower order term of A's and F's.Using Equations 41 (and also those of Equation 39) the results are summarized in Table III.Eq 41 Eq 39 Eq 41 Eq 41 Eq 41 Eq 41 Eq 41 Eq 39 Eq 41 Eq 41 Eq 41 Eq 41 Order The above scheme can be carried to as high an order as required, merely by writing down the additional terms.For example, the seventh order terms are obtained from Equation 41 as follows: or F from the last equation using B = 1/F is as follows: Similarily the eighthorder terms can be written down directly as follows: Thus all expressions presented (Tables I, II, and III) can be carried to as high an order as required, with no difficulty whatsoever.These relations are convenient working parameters for the actual solution to a particular order.

TABLE TZI
A" = 1 F, = 1 /X, = tonh k£ The constant in Bernoullis' equation is obtained from the first column of Table V t as follows: The above presentation of consecutive equations are in a convenient form for computing the A-terms and the F-terms for any selected value of TnJt t either by the long hand method or by use of a high speed computer.For example, consider lnJl^ 2ir (deep water), then one obtains tanh k/ = 1; in fact, for kJi * 2rr, tanh Nki « 1, whence X , • X • X * X • X « 1.Now cos Nk£ and sin Nk £ can be expanded by series to as high an order as required.Por example, the fifth order expansion for equations 6 and 7 are as follows;

It will follow in turn
3k£ s l (kA 0 ) 3 sin 3kx + a 4 (kA 0 ) 4 cos 4kx + a 4 [4k£ s J (kA 0 ) 4 sin 4kx 3k£ s 1 (kA 0 ) 3 cos 3kx + a 4 X 4 (kA 0 ) 4 sin 4kx -a 4 X 4 [4kC s ] (kA 0 ) 4 cos 4kx •I-a 6 X 5 (kA 0 ) 5 sin 5kx In the above equations X N = --, .,, The procedure for solution is first to eliminate k £ from the right hand side of Equation 45.This is done by the process of resubstitution: the first order is obtained as k£ « Q X kA sin kx and is substituted into equation 45 to obtain the s seeond ordSr* which in turn is again substituted into equation 45 to obtain the third order, etc, until the desired order is obtained.The resulting expression is then substituted into equation 44 to eliminate k£ from the right hand side, obtaining an expression for kl? independent of k£ .Finally, this equation for ki? is substituted into s equation 5 and the s integration results is an expression for d/L as a function of x/L, It will be convenient to write equation 5 as: *o It was found to the fourth order (also fifth order) that: Where all other terms vanished by integration,.Based on.equation47, the sixth order term was predicted to be 3/2 a X_ (kA ) , and was then verified by the detailed process of resubstitution and°integration.Based on the above findings one can suppose the following power series equation: •"6= T I Na N 2 *N (kA 0 ) 2N  (48) where Returning now to the fifth order solution, and from Table IV a, = I + A l3 (kA 0 ) 2 + A| 5 (kA 0 ) 4 In the above [K J and[K J are given respectively by equations 25 and 27.' ' dU dW Now Qf and Q f can be obtained from equations 20 and 21 respectively as follows: In addition, one obtains the following:

Procedure for Computation
The first operation to be performed is the evaluation of the coefficients a , for example, the fifth order solution as outlined in Table V.This is done by selecting H, ji , and L, and perform computations to obtain the required a coefficients, water depth d and wave period T.These evaluations are then used to obtain expressions for the surface profile and the velocity potential.The next step is to select k(x -£) and k(z -17), coordinates of the undisturbed particle positions, and from equations 3 and 4 compute kx and kz the coordinates of the particles.The surface profile is given for z -17= 0.

Transformation of equations to the form of Stokes'
The previous development resulted in equations in an unexpanded form.These equations can be expanded, using suitable approximations, and it will be shown that the expanded forms are identical to those obtained as outlined in Stokes' solution.The procedure is to expand the following identities.The first order solution for k£and k 17 for substitution in the above are obtained from equations 1 and 2, respectively as follows: 1TTTT 1 (kA 0 ) 2 cosh 2k (d+z) cos 2k(x-Ct) sinh Kd j f = <" kA ° S,n sinh (d kd +Z) ""Mx-Ct) + [smh a 2kd ~ sm^kd ] (kA 0 ) 2 smh 2k (d + z ) s.n2k(x-Ct) Using a = A«" and a = A =1, as given before, equations 83 and 84 becomi: The procedure applied to the second order solution has been extended to the fifth order, using also the expanded relationship of tanh Nk/ .The results of this expansion leads to the following relations for the coefficients: TABLE 301 /, = t = tanh kd ( 92)  When the exact solution of the wave problem to a particular Mth order is expanded to obtain the Stokes' solution to the same Mth order there will be a loss of accuracy.The greatest errors will be with the higher order terms.The first term will have minimum error.The reason for the errors arises from the fact that the coefficients a of the series (either the expanded or the unexpanded form) are evaluated on the basis of the unexpandec form.The above statement appears somewhat difficult to understand if one inadvertently considers Stokes' solution to be in an exact form to the Mth order.If this is the case, then Stokes' form must be expanded along the free surface (which results in the unexpanded form) prior to substitution into Bernoullis' equation.This operation results in an evaluation of the corresponding coefficients based on the unexpanded form, but are then applied incorrectly to the Stokes' or the expanded form.For evaluation of the coefficients of the Mth or last term, the expansion of cosh Mk (d +17 ) will be cosh Mkd which is the same idea asz= 1 7= 7 ?= 0. Any consideration of finite 7] s for the Mth term of Stokes' Mth order results in M + I , M + 2, etc. order terms, which are neglected by the mechanics of the solution.
It then follows that the error in the Mth term of Stokes' solution will be in proportion to: cosh Mk (d + z ) cosh Mk (d + z -1) Along the free surface the error will be cosh Mk (d +%) cosh Mkd and along the sea bottom there will be no error since the above ratio reduces to unity.
If one considers the last term of the third order wave theory, M <= 3, and for example, the wave H = 35 ft., T = 12 sec.and d = 85 ft., then one obtains L = 581 feet, 1) « 22.1 feet at the crest and 17 -H = -12.9feet at the trough and from ?he above ratio: cosh Mk (d +%) " je^ = cosh Mkd 7.869 and y- § § §-= 665 at the trough The deviations of the above ratio from unity reflects considerable error.For the unexpanded form the above ratio is always unity.
For the M-l or next to the last term of the Mth order, the percent error will be less since the expansion of this term for Stokes' solution will be cosh f(M-l) k (d +1) )] = cosh [(M-l) kd] + (M -1) Tk 1 ?sink (m-l) kd] L s J L J L s In view of the above considerations it appears that the use of Stokes* higher order solutions should be limited to low wave steepness, i.e. 1 ?small compared with d. s With the aid of electronic computors, the unexpanded form given in the present paper can be utilized easily for computing wave properties and thereby obtain greater accuracy theoretically than by utilizing Stokes' equations.

For
example the velocity potential component for the Mth or last term of the Mth order, for the unexpanded form and Stokes' form are respectively as follows:-JL&L.= o M (kA 0 ) M cosh M ^hZ ' m sin Mk(x-Ct-

the above expressions, together with Table II, it will be convenient to summarize the results in Table TV
X, J (kA 0 ) s cos 5 0'