38(> Prof. H. Lamb on Waves due 



Summary. 



Evidence is given that the active deposit o£ thorium emits 

 a smdl number of alpha particles of greater velocity than 

 any previously observed. These alpha particles are believed 

 to have their origin in the transformation of thorium C, 

 and appear to be divided into two homogeneous groups of 

 maximum range 10*2 and 11*3 cm. The number of these 

 alpha particles is about 1/10000 of the total number emitted 

 from thorium C, two-thirds of the number having a range 

 J 1-3 cm. 



The results indicate that the atoms of thorium can break 

 up in three and probably four distinct ways with the emission 

 of four characteristic groups of alpha particles of ranges 5'0, 

 8'6, 10-2, and 11*3 cm. 



University of Manchester, 

 February 1916. 



XLIV. On Waves due to a Travelling Disturbance , with an 

 application to Waves in Superposed Fluids. By Horace 

 Lamb, F.R.S.* 



1. TN any case of wave-propagation in one dimension 

 JL a distribution of impulse of amount cos kx per unit 

 length gives rise to an oscillation of the type 



7j = <p(k) coskxe i(T \ (1) 



where a is a function of k determined by the theory of free 

 waves. The effect of a concentrated unit impulse at the 

 origin is accordingly given by a Fourier expression 



1 f °° 

 = 2kO y 



cos kxe wt dk 



u 



To avoid possible indeterminateness in the sequel we may 

 introduce a factor e~^ to represent the effect of slight 

 dissipative action. If it is required in order to make the 

 integrals convergent, we may suppose ^ to be a suitable 

 function of k, but it will serve our purpose if we take //, to 

 be constant, and in the end infinitesimal. 



To find the wave-system due to a concentrated disturbing- 

 force of unit magnitude travelling with the velocity c in the 

 direction of. ^-negative, we may take the origin at the 

 instantaneous position of the disturbing influence. The 

 result of an impulse 8t delivered at an antecedent time t is 

 * Communicated by the Author. 



