546 Prof. H. Lamb on Wave-Patterns 



Since the group-velocity is always one-half the wave- 

 velocity, we have 



U=iccos<9, (41) 



130 that the formula (27) applies. The proviso 



x cos 6 +y sin 6>0 



shows moreover that the wave-system lies to the right of 0, 

 i. e. it follows the disturbing agent. The formula (40) leads 

 to the well-known wave-pattern * of transverse and diverging 

 waves. 



Again, from (35) and (39) 



/•"(6O = 2n7r(l-2tan 2 0) (42) 



Taking the value of <f>(k) from (11), it appears that along 

 an isophasal line the elevation due to a travelling pressure of 

 integral amount unity is 



ig sec 4 { 2n ±\)** 



r= 



Vnirpc 



wn- 



tan 2 6 



(43) 



of which the real part is of course to be retained. The 

 upper sign in the exponential relates to that branch of the 

 -curve (40) for which tan 2 #<^, and the lower to that for 

 which tan 2 #>i. There is accordingly a difference of phase 

 of a quarter- period between the transverse and diverging 

 waves in the neighbourhood of the cusps. This was first 

 remarked by Kelvin, in 1905 f . The law of height which 

 he gave in the paper referred to differs, however, somewhat 

 from (43), owing to the fact that his source of disturbance 

 was constructed by superposition of lines or narrow hands 

 of pressure distributed uniformly in azimuth about a point. 

 This does not give a strictly localized pressure. Mathe- 

 matically the effect is equivalent to the omission of the 

 factor k in (13) and subsequent formulae. 



The formula (43) makes f infinite for tan 2 = J, i. e. at 

 the cusps where the two systems of waves coalesce, and 

 again for#=+^7r. The former infinity is a mathematical 

 accident, due to the failure of our approximation through 

 the vanishing of f*(0) in {26). The calculation might be 



* * Hydrodynamics,' Art. 256. The pattern was first investigated by 

 Kelvin in 1887 ; see his ' Popular Lectures,' vol. iii. p. 482. 

 f Math, and Phys. Papers, vol. iv. p. 412. i 



