446 ON CONVECTION CURRENTS IN A HORIZONTAL LAYER OF FLUID [412
The question with which we are mainly concerned is the sign of da' - da for the various values of n. When n = l,Jl(z}-- J0' (z) = 0; so that da = da', a result which was to be expected, since the terms in al3 & represent approximately a displacement merely of the circle, without alteration of size or shape. We will now examine the sign of Jn/Jn' when n= 2, and 3.
For this purpose we may employ the sequence equations 2/i
JTI+I — "n J«—1> Jn = 2"n—3 ~~ 2"«+:u
which allow Jn and Jn' to be expressed in terms of Jx and J0, of which the former is here zero. We find
J"2 = - /OJ Js =- 4s-1 Jo, J~4 = (1 - 24^~2) Jo; J7 = J0, JV = 2*-'J0, Jr3/ = (12^-2-l)/0.
rru,,,, ^"l — 0 ~___- i —_?£_-•
111118 // ' J/"" 2' J-/"^2-'^'
whence on introduction of the actual value of z, viz. 3-832, we see that Jg/JV is negative, and that Js/Js is positive.
-• When n > z, it is a general proposition that Jn (z) and Jn' (#) are both positive*. Hence for n = 4s and onwards, Jn/Jn' is positive when # = 3'832. We thus arrive at the curious conclusion that when n = 2y da > da, as happens for all values of n (exceeding unity) when the boundary condition is w = 0, but that when n > 2, da' < da. The existence of the exceptional case n = 2 precludes a completely general statement of the effect of a departure from the truly circular form; but if the terms for which n = 2 are absent, as they would be in the case of any regular polygon with an even number of sides, regarded as a deformed circle, we may say that da' < da. In the physical problems the effect of a departure from the circular form is then to depress the pitch when the area is maintained constant (da' = 0). But for an elliptic deformation the reverse is the case.
At first sight it may appear strange that an elliptic deformation should be capable of raising the pitch. But we must remember that we are here dealing with a vibration such that the phase at both ends of the minor axis is the opposite of that at the centre. A parallel case which admits of complete calculation is that of the rectangle regarded as a deformed square, and vibrating in the gravest symmetrical modef. It is easily shown that a departure from the square form raises the pitch. Of course, the one-dimensional vibration parallel to the longer side has its pitch depressed.
[1918. This problem had already been treated by Aichi (Proc. Tokio Math.-Phys. Sac. 1907).]
* See, for example, Theory of Sound, % 210. t Theory of Sound, § 267 (,p = g = 2).y suppose the nodal system to divideem to one or two particular problems. The special limitation which characterizes them is the neglect of variations of density,