304 Prof. H. Lamb and Mr. G. Cook on Transmission 



at the free surface (* = ()), the time-factor for the simple- 

 harmonic vibration being assumed to be e iat . If the depth 

 be li, the condition of zero vertical velocity at the bottom 

 (z= —h) is satisfied if we assume that <f> involves z only 

 through a factor of the form cosh &(£-{- A) ; and the condition 

 (2) then gives 



a 2 = gk tanh kh (3) 



It remains to satisfy (1), which now takes the form 



£+$+"*-* ^ 



and the condition that ^</>/^n = at the vertical boundaries. 

 The analysis is now identical with that which applies to the 

 two-dimensional form of the problem of aerial waves, or of 

 electrical waves when the magnetic force is everywhere 

 parallel to z. The conditions stated determine the admissible 

 values of k, and the corresponding frequencies are then given 



by (3). 



Proceeding to the case of the rectangular tank, we take 

 the origin at the centre of the free surlace, and the axis of 

 a: parallel to the length (/). If there were no obstacles, then 

 in the case of the longitudinal oscillations the second term in 

 (4) would disappear, and we should have, in the anti- 

 symmetrical modes, 



(j> — A sin hv, (5) 



the factors which involve z and t being omitted. The con- 

 dition that 90/d# = O for x= +J/ then gives cos^/=0, the 

 lowest root of which is kl—ir. The period is accordingly 

 that of water-waves of length 2/, viz.*: 



?=V(f c <) (6) 



The horizontal dimensions of the obstacles being supposed 

 small compared with Z, the transverse component (v) of the 

 velocity will be sensible only in their immediate neighbour- 

 hood. We may imagine tw r o planes #= +#' to be drawn, 



!Oi 



such that x' is moderately large compared with the dimensions 

 in question, whilst still small in comparison with I. Outside 



* The verification of this formula was at one time a favourite lecture 

 experiment of the late Sir George Stokes, 



