Generation, Growth and Propagation of Waves 79 



less water surface disappear instantaneously the moment the force acting on 

 the water surface subsides. The damping force is the internal friction or 

 viscosity. 



According to p. 17, we have for free oscillatory waves on deep water, 

 and neglecting viscosity a certain velocity potential 



(f = — Ace* z cos x(x—ct) 



belonging to a wave train 



r\ = A sin x{x-ct) . (IV. 18) 



From this it follows that the total velocity is 



q 2 = u 2 + w 2 = x 2 c 2 A 2 e 2 * z . 



The energy used up by the viscosity per unit time is, according to Lamb 

 (1932, p. 677), 



in which ju is the coefficient of viscosity of water and dS is a unit area of 

 the volume under consideration. We select a volume of water which extends 

 vertically from the surface with a width b and with the length A, to the bottom. 

 Then with the above-mentioned value for q, we have 



R„ = 2fix 3 c 2 A 2 ■ Xb . (IV. 19) 



The loss of energy through viscosity per unit area is then 



R, = -2[ix*c 2 A 2 . (IV. 19a) 



The total energy of the wave for the same volume is according to 

 equation (11.12) 



E= lqxc 2 A 2 -Xb . (IV. 20) 



If we equalize the velocity of the decrease of energy with the loss of energy 

 through friction, we have 



j (i qxc 2 A 2 ) = - 2p#<*A* , (IV. 21 ) 



or 



-j- = —Ivy? A 

 dt 



and from this follows: 



A =A e- 2 ™ 2 ', (IV. 22) 



in which A is the amplitude at the time t = and v = p/q is the kinematic 

 coefficient of viscosity. The damping factor t or modulus of decay which 



