18 DYNAMICAL THEOKY OF SOUND 



point, the distance CP being equal to the length of the simple 

 pendulum whose free period is equal to the imposed period 



This example is due to Young*, who applied it to illustrate 

 the dynamical theory of the tides, where the same question of 

 phase arises. It appears from this theory that the tides in an 

 imagined equatorial belt of ocean, of a breadth not exceeding 

 a few degrees of latitude, and of any depth comparable with 

 the actual depth of the sea, would be "inverted," i.e. there 

 would be low water beneath the moon, and high water in 

 longitudes 90 E. and W. from it, the reason being that the 

 period of the disturbing force (viz. 12 lunar hours) is less than 

 the corresponding free period, so that there is opposition of 

 phase. 



The arbitrary constants in the complete solution (4) are 

 determined by the initial conditions. Suppose, for example, 

 that the body starts from rest in the zero position at the instant 

 t = 0. We find 



x -j4 a (cos nt cos pt), ......... . ..... (8) 



as may be immediately verified. 



When the imposed frequency p/2?r is nearly equal to the 

 natural period, the last term in (4) becomes very large, and it 

 may be that the assumption as to the smallness of x on which 

 the equation (1) is usually based (as in the case of the pendulum) 

 is thereby violated. The result expressed by (4) is then not to 

 be accepted without reserve, but we have at all events an indica- 

 tion of the reason why an amplitude of abnormal amount ensues 

 whenever there is approximate agreement between the free and 

 the forced period. 



In the case (p = ri) of exact coincidence between the two 

 periods, the solution (4) becomes altogether unmeaning, but an 

 intelligible result may be obtained if we examine any particular 



* Dr Thomas Young (1773 1829), famous for his researches on light, and 

 other branches of physics. The elementary theory of free and forced oscilla- 

 tions was given by him in an article on " A Theory of the Tides, including the 

 consideration of Resistance," Nicholsons Journal, 1813 ; Miscellaneous Works, 

 London, 1855, vol. n., p. 262. 



