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Some Remarks Regardingthe Solution of Stokesand Rayleigh for SurfaceWaves of Finite AmplitudeJonas Ekman FjeldstadPublished online: 30 Jul 2008.

To cite this article: Jonas Ekman Fjeldstad (1954) Some Remarks Regardingthe Solution of Stokes and Rayleigh for Surface Waves of Finite Amplitude,Norsk Geografisk Tidsskrift - Norwegian Journal of Geography, 14:1-4, 53-60,DOI: 10.1080/00291955308542715

To link to this article: http://dx.doi.org/10.1080/00291955308542715

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SOME REMARKS REGARDINGTHE SOLUTION OF STOKES AND RAYLEIGH

FOR

SURFACE WAVES OF FINITEAMPLITUDE

BY

JONAS EKMAN FJELDSTAD

I.

An exact solution of the hydrodynamic equations was found by Gerstner,JLx. but the waves represented by Gerstner's solution are not irrota-tional such that they cannot be generated from rest by conservativeforces in an ideal fluid. Stokes found an approximate solution for theirrotational case where the surface profile of the waves was representedby the equation

y = PeFv cos kx

but in this case the surface condition of constant pressure was onlysatisfied approximately.

An exact solution was at last found by Levi-Civitta in 1925, butthe complexity of the solution makes the application of his theoryextremely difficult.

Though the solution of Stokes is not exact, it will neverthelessgive a satisfactory representation of surface waves when the amplitudeis not too large.

The theory of Stokes and Rayleigh is given by Lamb1 in hisHydrodynamics and in the following we refer to his exposition ofthe theory.

The theory as given by Lamb contain some inaccuracies andmisunderstandings, and it may therefore be of interest to give a moreelaborate treatment of the subject since it will then be possible toarrive at results which have a general interest.

Rayleigh has treated the subject as one of steady motion by im-posing on the whole water mass a velocity c in the opposite directionof the wave propagation, but no essential simplification of the problem

Lamb: Hydrodynamics, 250.

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— 54 —

is obtained in this manner, and we shall here not make use of thismethod.

An approximate solution corresponding to that of Stokes is foundby introducing a velocity potensial <I> and a stream function y>, by

Then " a* au = — — = — 22 =, c kpeft cos k (x—ct)

dX dyand

v = — — = -^ = ckpeUlJsink(x—ct),dy dx ^

$ and v* are solutions of Laplace's equation

and the pressure is given by

Since .

the pressure equation may also be represented by

and at the surface

The surface profile of the wave is then given by

y == fie^y cos k(x— ct). (3)

Stokes and Rayleigh treat this equation by a method of successiveapproximations and seem to be unaware of the fact that the equationmay be solved simply by an application of Lagrange's formula. Accor-ding to Lagrange, the solution of the equation

y = a + xcp(y)

which tends to a when x tends to zero is given by the series

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— 55 —

and in the present case we get '

^-n"-1cosnA(x—cO. (4)K n = 1 n •

The condition of convergency is then easily found. By taking the

ratio of two consecutive terms • n + , we getUn

pkcosk(x—ct)-ll + -\

such that the condition will be

This condition may also be found by a simple geometric consideration.If we take a fixed value of (x — ct) and put /? cos k (x — ct) = y

the corresponding value of y is found by solving the equationy = ye^O

which is equivalent to the two equations

y = yz>

and y = —lnz.K.

If we represent these two equations in a system of rectangularcoordinates (y,z), the solution is found as the intersection of the straight

line y = yz with the logaritmic curve y = — lnz. The extreme value of yK

which makes a solution possible is obtained when the straight line touchesthe logaritmic curve. If we call the coordinates of this point (y0, z0),we have

dz kzoo r ky0 — kyz0 = 1 . z0 = e ky — e~l.

If the solution had been exact instead of being only approximate,the extreme wave form would be represented by the equation

ky = ehy ~l cos k(x— ct)

or since k is only a scale factor,

y = e'J ~ 1 cos (x— ct) .

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For

Atmination

/ =

theof

= 0 we have the

y

crest x — 0 y =the tangents we

— 56 —

equation

= e!^~1 cos x

1 the angledifferentiate

.

would be 90°.the equation

For the deter-giving

-— = — e'J ~1 sin x •dx

from which we see that the value o

Differentiating once more we get

** —: QD — i CQC v — 2 &u sin x — -dx" dA:

Putting y = 0 y = 1 we get

ax

-T—I is indeterminate.

dxor

which shows that the tangents intersect at right angles.In general the surface profile of the wave is given by an equation

where y = y>(x—ct,y), satisfy the equation of Laplace

dx" ay-

Let the maximum value of y be ?;, such that when t= 0, x= 0, y

correspond to the wave crest.

We then get

dx dx dy dx

and the condition that this shall be a singular point is that

dyl c + ^

ay

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— 57 —

shall be indeterminate, and therefore

dJL = 0 c + ^ = 0dx dy

when x — 0 , y = »;.Differentiating once more we get

dry 3> 9d^p_dy_ ^v(dy\2 3y d*ydx2 + a*9 "*" S.v3y dx "*" 3y2 ^ J "*" 3y ' d.v2

Assuming symmetry about the point AT = O, we get for.v=0, y =

If —^ £ 0 we shall have3A- 2

dy

3V 32Vsuch that the tangents intersect at right angles. If —v = •£ = 0dx' ^y~

when x = 0 y = >;, we differentiate once more, giving

3;c3y dx2 . 3y2 dx dx2 3y dxs ~

On the same assumptions we get

or

such that the tangents intersect at an angle of 120°. This is the neces-sary condition found by Stokes.

It is then easily seen that the successive approximations used byRayleigh cannot be used for the limiting case of the wave.

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— 58 —

II. Orbital motion of the particles.

If we use the solution of Stokes, the orbital motion of the particlesmay be found by integrating the system of equations

^^kcfie^ cos k(x— ct)

%L = kcpe1® sin k (x—ct). (2,1)at

From these equations we find

dC ~ck df

or

andy + h = fie^Jcoskix— ct).

If we put k(ct—x) = 95we have according to Lagrange's formula

+ h = 1 2 ^kep" n" ~ * cos" y . (2,2)Further we get

dx•— = —cotgy.dy

and from equation (2, 2) we get

dy = 5^ — nn

k*-1 n\where a = fike~kh

such that we get

1 x an

dx = — 2 J —: nn c o s n <pd<p

and

x = const. + - ^ ^7 n" I c o s" y ^ (2. 3>

1 sr\ annn Cct = x + cp = const. + q> + ~2J —— cos"q>dq>. (2, 4)

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• !

— 59 —

To perform the integrations, it will be convenient to replace the powerseries of cos q> by series containing multipla-of the angle q>. We have

™s2n<P = [n)-rtR + rtZ=l2j I )cos2p9>

and

cos 2 n + V=^2H ZJ I lcos<^ p=o'

Introducing these expressions and changing the order of summations,we get

Bo ' ^ " — - (2,5)

and

where

1/2n + l\2" + 1

V 2 j

From the formula (2, 7) we see that when <p increases by 2JI thecorresponding increase of t is

2TI

TI= —kc

or if we put ko — —

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— 60 —

The period of the orbital motion will consequently depend on thedepth.

These formulae cannot be used in the extreme case, when a = e~l

because the series

diverges.The extreme case

dx _dt ~

rfy _dt

cei>~:

cev-i

.!en)2

lcos(x—ct)

sin \X— ci)

has the singular solution x=ct, y = l , indicating that the particle,which is exactly at the crest will describe the straight line x = ct,

This result is general and independent on the special choice of thestream function introduced in this case.

"The annexed drawing shows the surface profile and the orbitalmotion of the water particles calculated on the assumption

(ik = 0,25.

As will be seen, there is a water transport at the surface whichis not negligible, but this transport dimipishes very rapidly with depth.

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