442 ON CONVECTION CURRENTS IN A [412
expressing n in terms of a-. To find a- we have to eliminate n between (48) and (49). The result is
<T*KV (K - v}1 4- cr4/3V (* + ^)2 - <?* • 2jQV (*2 + v2) - /9V*4 = °> • • -(50) from which, in particular cases, cr could be found by numerical computation. From (50) we fall back on (23) by supposing v — 0, and again on a similar equation if we suppose K = 0.
But the case of a nearly evanescent n is probably the more practical. In an experiment the temperature gradient could not be established all at once and we may suppose the progress to be very slow. In the earlier stages the equilibrium would be stable, so that no disturbance of importance would occur until n passed through zero to the positive side, corresponding to (44) or (46). The breakdown thus occurs for s = TT/ £ and by (42) P 4- m2 = 7r2/2£2. And since the evanescence of n is equivalent to the omission of djdt in the original equations, the motion thus determined has the character of a steady motion. The constant multiplier is, however, arbitrary; and there is nothing to determine it so long as the squares of u, v, w, 9 are neglected.
In a particular solution where w as a function of x and y has the simplest form, say
w = 2 cos x. cos y, ...........................(51)
the particular coefficients of oo and y which enter have relation to the particular axes of reference employed. If we rotate these axes through an angle $, we have
w = 2 cos {#' cos 0 — y' sin <£j. cos {a/ sin <j) + y' cos c/>} = cos [of (cos <£ - sin <£)} . cos {y' (cos c/> + sin <£)} + sin JV (cos </> - sin $)}. sin [yr (cos 0 + sin <£)} + cos \at (cos (/> 4- sin </>)}. cos {y' (cos <£ — sin $)}
— sin {#' (cos <f> + sin 0)}. sin {y' (cos $ — sin </>)}..........(52)
For example, if </> = |TT, (52) becomes
w = cos (y' V2) + cos (of \/2)......................(53)
It is to be observed that with the general value of 0, if we call the coefficients of x', y'} I and m respectively, we have in every part P + m* — 2, unaltered from the original value in (51).
The character of w, under the condition that all the elementary terms of which it is composed are subject to £2 4- m2 = constant (&}, is the same as for the transverse displacement of an infinite stretched membrane, vibrating with one definite frequency. The limitation upon w is, in fact, merely that it satisfies
(dzldxz + d2(dyz + kz)w = 0......................(54)
The character of w in particular solutions of the membrane problem is naturally associated with the nodal system (w = 0), where the .membrane may be regarded as held fast; and we may 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,