1911] HYDRODYNAMICAL NOTES 9
It appears that UfU0 does not differ much from unity between I' — '23 and I' = oo 5 so that the shallowing of the water does not at first produce much effect upon the height of the waves. It must be remembered, however, that the wave-length is diminishing, so that waves, even though they do no more than maintain their height, grow steeper.
Concentrated Initial Disturbance with inclusion of Capillarity.
A simple approximate treatment of the general problem of initial linear disturbance is due to Kelvin*. We have for the elevation tj at any point x and at any time t
77 = — cos lex cos art dk
TTJo
1 f00 1 /""
= s— cos (lex — art) die + ^— cos (lex + at} die, ...... (1)
27rji0 STT.'O
in which o- is a function of Jc, determined by the character of the dispersive medium — expressing that the initial elevation (t = 0) is concentrated at the origin of x. When t is great, the angles whose cosines are to be integrated will in general vary rapidly with k, and the corresponding parts of the integral contribute little to the total result. The most important part of the range of integration is the neighbourhood of places where kx ± crt is stationary with respect to k, i.e. where
•±'a-° .................................. w
In the vast majority of practical applications dcr/dk is positive, so that if x and t are also positive the second integral in (1) makes no sensible contribution. The result then depends upon the first integral, and only upon such parts of that as lie in the neighbourhood of the value, or values, of k which satisfy (2) taken with the lower sign. If k^ be such a value, Kelvin shows that the corresponding term in 77 has an expression equivalent to
_ cos (cr^ — • k-^x — ^TT) ,<j\
^ _._____.__. j ........................ (6)
PI being the value of or corresponding to Ic^.
In the case of deep-water waves where <r = \/(gk), there is only one predominant value of k for given values of x and t, and (2) gives
ki = gPl^x\ o-i-gt/Za;, ..................... (4)
making cr-it — k-ix — l'rr=gtzl^x- ^TT, ........................ (5)
A & n {9*? ""1 /^\
and finally vt = ••.• ••= cos \~. --- rr > ........................ (6)
J
the well-known formula of Cauchy and Poisson.
* Proc. Eoy. Soc. Vol. XMI. p. 80 (1887) ; Math, and Phya. Papers, Vol. iv. p. 303.an earlier date