[ 303 ] 



XX Y. ^i Hydrodynamkal Illustration of the Theory of tie 

 Transmission of Aerial and Electrical Waves hy a Grating. 

 By Horace Lamb, F.R.S., and Gilbert Cook, M.Sc* 



npHE theory of the scattering of aerial and electrical waves 

 1 by isolated obstacles whose breadth is small compared 

 with the wave-length has been discussed in a series of papers 

 by Lord Rayleigh f. A direct verification of the results is 

 hardly to be looked for, but the case of a grating, which has 

 been investigated by one of the present writers J, would 

 appear to be more promising in this respect ; and in fact the 

 transmission of Hertzian waves by a metallic grating has 

 been studied experimentally, and compared with the theory 

 by Schaefer and Langwitz §, and by G. H. Thomson ||, and a 

 satisfactory agreement has been found. 



A confirmation of the mathematical formulae may, how- 

 ever, be sought in another direction. It is known 1" that in 

 the case of a cylindrical obstacle, or system of obstacles, ihe 

 problem is identical with that of waves on a sheet of water 

 of uniform depth, as modified by cylindrical obstacles whose 

 generating lines are vertical. In particular, in the longi- 

 tudinal oscillations of water in a long and narrow rectangular 

 tank, having one or more such obstacles near its centre, we 

 have an exact analogue of aerial waves incident on a grating, 

 provided the obstacles be disposed with the proper degree of 

 symmetry. The effect of the obstacles in altering the period 

 of the gravest mode of oscillation can in certain cases be 

 calculated, and the comparison with experiment is of course 

 a very simple matter. 



The mathematical theory** may, for the purpose in hand, 

 be briefly recapitulated. The origin being taken in the 

 undisturbed level of the water-surface, and the axis of z being- 

 directed vertically upwards, we have to satisfy the equation 



V 2 <£ = ....... (1) 



subject to the condition that the normal derivative "d<j>fon 

 shall vanish at the rigid boundaries, and that 



^=4! - ' ™ 



* Communicated bv the Authors. 



f Phil. Mag. [5] vol. xlii. p. 259 (1897), and vol. xliv. p. 28 ^1897) ; 

 Sc. Papers, vol. iv. pp. 283, 305. 



X H. Lamb, Proc. Lond. Math. Soc. vol. xxix. p. 523 (1898) ; Hvdro 

 dynamics, 3rd ed., §§ 300, 301. 

 " § Ann. d. Phys. vol. xxi. p. 587 (1906). 



|| Ann. d. Phys. vol. xxii. p. 365 (1907). 



f Kavleigh, Phil. Mag. [5] vol. i. p. 257 (187,6) ; Sc. Tapers, vol. i. 

 p. 265. 



** Hydrodynamics, $§ 226, 251. 



