due to a Travelling Disturbance. 541 



of k, will be determined by the nature of the medium. For 

 instance, in the case of water-waves we have 



4>(k)=ik/pa, (11) 



where p is the density, whilst 



<r 2 =^, (12) 



if the depth be great. 



It follows from (10) by a theorem of C. Neumann that 

 the effect of a unit impulse concentrated in a point at Q 

 will be 



Since 



_ _L ( e i<rt J (k^{k)kdk, . . . (13) 



. J o(^J = ^( ff e-^™xd X , • . . (M) 

 -'"J _7r 



this may be regarded as made up of rectilinear wave-trains 

 whose directions of propagation are distributed uniformly 

 in azimuth about Q. 



Suppose now that we have a source of disturbance 

 travelling in the negative direction along the axis of a?, with 

 the constant velocity c, and that at the instant under con- 

 sideration it has reached the origin 0. Let Q be its position 

 at any antecedent time t, so that OQ = ct. If x, y be the 



coordinates of any point P, the distance of P from a recti- 

 linear wave-front through Q will be 



(ct — x) cos ty—ysmyjr, 



where yfr denotes the angle which the normal to the wave- 

 front makes with Ox. This expression takes the place of 

 fir cos % in (14), and since the integrand is periodic in respect 

 to ty, with the period 2ir, we may replace % by ^ as 

 independent variable, with the same range of integration. 



