200 ON THE STABILITY OF THE [377
whence, on integration and replacement of v,
' "'vdy, ......... (10)
J Q ~ 0
H denoting a second arbitrary constant.
In (10) we may suppose y measured from the first wall, where v=0.
Hence, unless V vanish with y, H=Q. Also from (8) when y = 0,
(11)
Let us now trace the course of v as a function of y, starting from the wall where y = 0, v = 0 ; and let us suppose first that U' is everywhere positive. By (11) K has the same sign as (dv/dy)0, that is the same sign as the early values of v. Whether this sign be positive or negative, v as determined by (10) cannot again come to zero. If, for example, the initial values of v are positive, both (remaining) terms in (10) necessarily continue positive; while if v begins by being negative, it must remain finitely negative. Similarly, if U' be everywhere negative, so that K has the opposite sign to that of the early values of v, it follows that v cannot again come to zero. No solution can be found unless U' somewhere vanishes, that is unless n coincides with some value of — kU,
In the above argument U', and therefore also n, is supposed to be real, but the formula (10) itself applies whether n be real or complex. It is of special value when k is very small, that is when the wave-length along x of the disturbance is very great ; for it then gives v explicitly in the form
When k is small, but not so small as to justify (12), a second approximation might be found by substituting from (12) in the last term of (10).
If we suppose in (12) that the second wall is situated at y — l, n is determined by
The integrals (12), (13) must not be taken through a place where U + n/k = 0, as appears from (8). We have already seen that any value of n for which this can occur is admissible. But (13) shows that no other real value of n is admissible ; and it serves to determine any complex values of n.
In (13) suppose (as before) that n/k=p + iq; then separating the real and imaginary parts, we get