10 



General Remarks on Waves 



of the generating force, when the period of the generating force of the 

 forced oscillations is greater than the period of the free oscillations, viz. when 

 t > T. If, on the contrary, the period of the generating force is smaller than 

 that of the free oscillations (T > t), the phase of the forced waves is opposite 

 to the phase of the generating force. These properties of the forced oscilla- 

 tions pertain to a general dynamic principle of the wave theory and will be 

 frequently referred to in subsequent paragraphs. 



(c) Short and long waves. We can divide the wave-like motion of water- 

 masses into two different classes, taking as a measure the ratio of the wave- 

 length A to the depth to the bottom h. If the bottom depth h is smaller than 

 half a wave length, it can be considered as "shallow"; if, on the contrary, 

 the depth exceeds about half a wave length, the water is called "deep". 



In deep water waves, the orbital motion of the water particles decreases 

 fairly rapidly with increasing depth. Close to the surface the individual water 

 particles move in circular orbits. The radii of these orbits decrease with depth 

 and in intermediate depths of approximately half a wave length below the 

 surface, the diameter of the orbits is theoretically one twenty-third of the 

 orbit diameter at the surface. The movement of the wave is restricted to 

 a realtively thin surface layer. Hence the denomination of surface waves. 

 Scott Russel (1837, p. 417) named them secondary waves, and in the 

 English literature they are called quite often oscillatory waves. Let us imagine 

 the water-mass is divided into horizontal and vertical filaments of water 

 in a state of rest. The positions of the corresponding filaments during the 

 passage of a wave are shown in Fig. 4. Their distortion is greatest in the 



Fig. 4. Behaviour of vertical line of particles during passage of a short wave (great depth). 



upper layers and decreases rapidly with depth. In greater depth it becomes 

 imperceptible, and finally it disappears entirely. Thus the comparison made 

 by Leonardo da Vinci between a waving cornfield and such water waves 

 does not apply only to the oscillatory motion of the single particles and to 



