181] CANAL OF VARIABLE SECTION. 293 



Since the equation (4) is unaltered when we reverse the sign of t, the 

 complete solution, subject to the above restrictions, is 



V = b-*h-*{F(6-t)+f(6+t)} (vi), 



when F and f are arbitrary functions. 



The first term in this represents a wave travelling in the direction of 

 ^-positive j the velocity of propagation is determined by the consideration 

 that any particular phase is recovered when dti and 8t have equal values, and 

 is therefore equal to (gh)*, by (i), exactly as in the case of a uniform section. 

 In like manner the second term in (vi) represents a wave travelling in the 

 direction of .^-negative. In each case the elevation of any particular part of 

 the wave alters, as it proceeds, according to the law b~^ K~*. 



The reflection of a progressive wave at a point where the 

 section of a canal suddenly changes has been considered in Art. 

 173. The formulae there given shew, as we should expect, that 

 the smaller the change in the dimensions of the section, the 

 smaller will be the amplitude of the reflected wave. The case 

 where the transition from one section to the other is continuous, 

 instead of abrupt, comes under a general investigation of Lord 

 Rayleigh's*. It appears that if the space within which the 

 transition is completed be a moderate multiple of a wave-length 

 there is practically no reflection ; whilst in the opposite extreme 

 the results agree with those of Art. 173. 



If we assume, on the basis of these results, that when the 

 change of section within a wave-length may be neglected a pro- 

 gressive wave suffers no disintegration by reflection, the law of 

 amplitude easily follows from the principle of energy)-. It 

 appears from Art. 17 1 that the energy of the wave varies as the 

 length, the breadth, and the square of the height, and it is easily 

 seen that the length of the wave, in different parts of the canal, 

 varies as the corresponding velocity of propagation]:, and therefore 

 as the square root of the mean depth. Hence, in the above notation, 

 r) z bht is constant, or 



77 oc 6-to-*, 



which is Green's law above found. 



* " On Reflection of Vibrations at the Confines of two Media between which the 

 Transition is gradual," Proc. Lond. Math. Soc., t. xi. p. 51 (1880) ; Theory of Sound, 

 2nd ed., London, 1894, Art. 1486. 



t Lord Eayleigh, 1. c. ante p. 279. 



J For if P, Q be any two points of a wave, and P', Q' the corresponding points 

 when it has reached another part of the canal, the time from P to P' is the same 

 as from Q, to Q', and therefore the time from P to Q is equal to that from P' to Q'. 

 Hence the distances PQ, P'Q' are proportional to the corresponding wave- velocities. 



