404 REPORTS ON THE STATE OF SCIENCE. — 1919. 



an aquarium, say 2 feet 6 inches long by 3 feet deep, to make the water 

 beat time with a half-second pendulum, taken as 10 inches long or 

 25 centimetres in round numbers, in a wa^e 5 feet long, double the 

 length of the tank. 



Placed on a lecture table this will be visible to the audience, to illus- 

 trate Lord Kelvin's definition of Wave Motion, as the passage of periodic 

 motion through matter, in this case water or other liquid. 



On a smaller scale, for mere illustrative action, any small vessel will 

 serve, such as a basin, dish, tumbler, saucer, or water bottle of moderate 

 size. 



Colonel Russo, of the Italian Navy, has constructed an experimental 

 tank where the motion of the water is kneaded by appropriate flexible 

 diaphragms, and he shows the effect in a series of moving pictures, such 

 as those he brought before the Institution of Naval Architects, 1916. 



In our tank the walls are rigid, and, the water being initially at rest, 

 the wave motion may be set up by a slight tilt of the vessel, or by a paddle 

 passed through the water. 



In the theoretical discussion the motion is supposed started by a 

 variable impulse pressure applied suddenly over the surface ; no motion 

 would be set up if the pressure was uniform ; and assuming the water 

 incompressible and sound velocity through it practically infinite (really 

 about 1,450 metres per second), this impulse pressure will be propagated 

 instantaneously and uniformly down inside the water, and taken up with 

 no reduction by the bottom and sides of the tank. 



We may suppose this pressure applied as in the Humphrey pump by 

 the explosion of a charge of petrol gas. 



But any variation of the impulsive pressure over the free surface will 

 set up an internal commotion, resulting in a wave motion of the water. 



1. Take an origin in the middle of the rectangular free surface of 

 the water, and coordinate axes of reference, Ox, Oy, Oz ; Oa; vertically 

 downward, Ox and Oy in the surface, Ox parallel to the length of the 

 tank. 



Suppose a variation of the applied mean uniform impulsive pressure 

 A is given by the term B sin mx, so the total impulse over the surface is 



A + B sin mx, 



with a wave length L, such that 7?iL=27r, but using m in the work for 

 economy of printing. The term A is retained, so as to avoid the idea of 

 a negative pressure, practically impossible. 



When the tank length is half the wave length, ^L, a pure stationary 

 half wave will be set up, as in Colonel Russo's experiments ; and this is 

 the longest wave in the tank, say 5 feet long. 



The coefficient B will not remain uniform down in the water ; it will 

 be found on the theory of the Equation of Continuity that at a depth z it 

 will have diminished to b, some function of z ; and if the tank is deep 

 enough for the motion to be insensible at the bottom, b dies out at 

 compound discount with the depth z according to the law 



&=Be-"'^ 



This with our tank 3 feet deep, and z—S, L=5, mz=§Tr, say 3"7, 

 g-«u_o-o25, say 2-5 per cent, at the bottom, which we take as insensible. 



