202 ON THE STABILITY OF THE [377
The fact that n in equation (15) appears only as nz is a simple consequence of the anti-symmetrical character of U. For if in (13) we measure y from the centre and integrate between the limits ± \l> we obtain in that case
in which only n2 occurs. But it does not appear that n2 is necessarily real, as happens in (15).
Apart from such examples as were treated in my former papers in which d? Ujdif vanishes except at certain definite places, there are very few cases in which (3) can be solved analytically. If we suppose that v = sin (Try /I), vanishing when y — 0 and when y = l, and seek what is then admissible for U, we get •
U+n/k = Acos{k* + 7r*/rfy + l]sin{k2 + 7rz/lrfy, ...... (17)
in which A and B are arbitrary and n may as well be supposed to be zero. But since II varies with k, the solution is of no great interest.
In estimating the significance of our results respecting stability, we must of course remember that the disturbance has been assumed to be and to remain infinitely small. Where stability is indicated, the magnitude of the admissible disturbance may be very restricted. It was on these lines that Kelvin proposed to explain the apparent contradiction between theoretical results for an inviscid fluid and observation of what happens in the motion of real fluids which are all more or less viscous. Prof. McF. Orr has carried this explanation further *, Taking the case of a simple shearing motion between two walls, he investigates a composite disturbance, periodic with respect to x but not with respect to t, given initially as
v = B cos Ix cos my, ........................... (18)
and he finds, equation (38); that when m is large the disturbance may increase very much, though ultimately it comes to zero. Stability in the mathematical sense (B infinitely small) may thus be not inconsistent with a practical instability. A complete theoretical proof of instability requires not only a method capable of dealing with finite disturbances but also a definition, not easily given, of what is meant by the term. In the case of stability we are rather better situated, since by absolute stability we may understand complete recovery from disturbances of any kind however large, such as Keynolds showed to occur in the present case when viscosity is paramount f. In the absence of dissipation, stability in this sense is not to be expected.
* Proc. Roy. Irish Academy, Vol. xxvn. Section A, No. 2, 1907. Other related questions are also treated.
t See also Orr, Proc. Hoy. Irish Academy, 1907, p. 124.7r = iu —rr + v -r- + w -T-4 dxdy dz.............(14)