General Remarks on Waves 9 



oscillations), due to the ever present internal and external frictional forces, 

 and finally stops entirely. A damped wave can then be represented in a certain 

 locality by 



7] = A e~P' cos at . (1.6) 



If t] n is the «th point of reversal of the oscillation on the positive side at 

 the time nT, the amplitude at this time will be 



A n = A e-"P T = A e-«y . (1.7) 



The quantity y = (IT is designated as logarithmic decrement of the damped 

 oscillation. It is equal to the natural logarithm of the ratio of two consecutive 

 values of the amplitudes taken in the same direction of displacement, or it is 



!nA„-lnA m 



7= (1.8) 



m — n ' 



if A n and A m represent the nth and /nth amplitude of the oscillation in the 

 same direction (Kohlrausch, 1935). The friction of a free oscillating system 

 can be characterized by y. 



The period of the free oscillations depend exclusively on the dimensions 

 of the system, which for a closed basin bay are the length, the depth and the 

 width of the basin. Dissipating forces can increase the period of the free 

 oscillations (see p. 155). 



Forced waves are generated in a system capable of oscillation by the 

 continuous action of a periodical external force. The tidal forces of moon 

 and sun subject the water-masses of the oceans to periodical displacements 

 in a horizontal direction, which displacements in turn contribute to the 

 formation of waves. The period of forced waves is always identical to the 

 period of the generating force. However, the amplitude and the phase are 

 not free as is the case with free waves, but they essentially depend on the 

 ratio between forced and the free oscillations of the system. The amplitude 

 and phase of the forced waves depend not only on the generating force, 

 but also on the dimensions of the oscillating system. The amplitude A of 

 the forced waves increases as the period of generating force x approaches 

 the period of the free oscillations T. Here applies the relation 



(1.9) 



oV 



in which A k is the amplitude and a k the angular velocity of the generating 

 force. If a = a k or t and T are equal, the amplitude of the forced waves 

 increases steadily: the oscillating system then is in resonance with the 

 periodical force. 



The phase of the forced oscillations is only then the same as the phase 



