Page:EB1911 - Volume 26.djvu/989

Rh Solving (20), we have

{{c|$$\left. \begin{aligned} & (\Sigma b_i x_i)(f^2 - \cos^2 \theta) = \frac{I}{4m} \left [ \frac{d}{d\theta} \Sigma b_i u_i + \frac{s \cos \theta}{f \sin \theta} \Sigma b_i u_i \right ] \\ & (\Sigma b_i y_i) \sin^2 \theta(f^2 - \cos^2 \theta) = -\frac{I}{4m} \left [ \frac{\cos \theta}{f} \frac{d}{d\theta} \Sigma b_i u_i + \frac{s}{\sin \theta} \Sigma b_i u_i \right ] \end{aligned} \right \} \text{(21)}$$}}

Then substituting from (21) in (19) we have

This is almost the same as Laplace's equation for tidal oscillations in an ocean whose depth is only a function of latitude. If indeed we treat bi as unity (thereby neglecting the mutual attraction of the water) and replace Σui and Σei by u and e, we obtain Laplace's equation.

When ui is found from this equation, its value substituted in (21) will give xi and yi.

§ 17. Zonal Oscillations.—We might treat the general harmonic oscillations first, and proceed to the zonal oscillations by putting s=0. These waves are, however, comparatively simple, and it is well to begin with them. The zonal tides are those which Laplace describes as of the first species, and are now more usually called the tides of long period. As we shall only consider the case of an ocean of uniform depth, γ the depth of the sea is constant. Then since in this case s=0, our equation (22), to be satisfied by ui; or hi-ei, becomes

This may be written

where A is a constant.

Let us assume

where Pi denotes the ith zonal harmonic of cos θ. The coefficients Ci are unknown, but the Ei are known because the system oscillates under the action of known forces.

If the term involving the integral in this equation were expressed in terms of differentials of harmonics, we should be able to equate to zero the coefficient of each dPi/dθ in the equation, and thus find the conditions for determining the C's.

The task then is to express P; sin θdθ in differentials of zonal harmonics.

It is well known that Pi satisfies the differential equation

Therefore $$\int P_i \sin \theta d\theta = -\frac{I}{i(i + I)} \sin \theta\frac{dP_i}{d\theta}$$, and

Another well-known property of zonal harmonics is that

If we differentiate (25) and use (24) we have

Multiplying (25) by sin θ, and using (26) twice over,

Therefore

This expression, when multiplied by 4ma/γ and by Ci and summed, is the second term of our equation.

The first term is

In order that the equation may be satisfied, the coefficient of each dPi/dθ must vanish identically. Accordingly we multiply the whole by γ/4ma and equate to zero the coefficient in question, and obtain

This equation (27) is applicable for all values of i from 1 to infinity, provided that we take C0, EO, C-1, E-1 as being zero.

We shall only consider in detail the case of greatest interest, namely that of the most important of the tides generated by the attraction of the sun and moon. We know that in this case the equilibrium tide is expressed by a zonal harmonic of the second order; and therefore all the Ei, excepting E2, are zero. Thus the equation (27) will not involve Ei in any case excepting when i=2.

If we write for brevity

the equation (27) is

Save that when i=2, the right-hand side is b2γE2/4ma, a known quantity ex hypothesi.

The equations naturally separate themselves into two groups in one of which all the suffixes are even and the other odd. Since our task is to evaluate all the C's in terms of E2, it is obvious that all the C's with odd suffixes must be zero, and we are left to consider only the cases where i=2, 4, 6, &c.

We have, said that C0 must be regarded as being zero; if however we take

so that C0 is essentially a known quantity, the equation (28) has complete applicability for all even values of i from 2 upwards.

The equations are

It would seem at first sight as if these equations would suffice to determine all the C's in terms of C2, and that C2 would remain indeterminate; but we shall show that this is not the case.

For very large values of i the general equation of condition (28) tends to assume the form

By writing successively i+2, i+4; i+6 for i in this equation, and taking the differences, we obtain an equation from which we see that, unless Ci/Ci+2 tends to became infinitely small, the equations are satisfied by Ci=Ci+1 in the limit for very large values of i.

Hence, if Ci does not tend to zero, the later portion of the series for h tends to assume the form Ci(Pi+Pi+2+Pi+4...). All the P's are equal to unity at the pole; hence the hypothesis that C, does not tend to zero leads to the conclusion that the tide is of infinite height at the pole. The expansion of the height of tide is essentially convergent, and therefore the hypothesis is negatived. Thus we are entitled to assume that Ci tends to zero for large values of i.

Now writing for brevity

we may put (28) into the form

By successive applications of this formula we may write the right hand side in the form of a continued fraction.

Let

Then we have

or

Thus

If we assume that any of the higher C's, such as C14 or C16, is of negligible smallness, all the continued fractions K2, K4, K6, &c., may be computed; and thus we find all the C's in terms of C0, which is equal to -3b2γE2/4ma. The height of the tide is therefore given by

It is however more instructive to express h as a multiple of the