with happy observation of many British rivers as well as respectful study through the 

 printed page of some great American ones. 



2. Frequency dispersion and amplitude dispersion 



I shall have a lot to say about the effect of friction on river waves, but it is 

 best to begin, I think, with a survey of water-wave motions in the absence of friction, 

 as these are already fairly complicated [19, 15]. Waveforms on water tend to change 

 shape as a result of two kinds of dispersion, which I propose to call frequency dis- 

 persion and amplitude dispersion. The word dispersion is normally used to mean 

 frequency dispersion, according to which components of the wave of different fre- 

 quencies are propagated with different velocities. Long waves, however, suffer also 

 amplitude dispersion, according to which high values of the surface elevation are propa- 

 gated with greater velocity than lower values. This amplitude dispersion, producing 

 steepening of the front of the wave, is familiar also in sound waves. We shall see that 

 there are important kinds of water waves for which the deforming effects of frequency 

 dispersion and amplitude dispersion are in competition. 



3. Gravity waves of small amplitude 



To illustrate frequency dispersion one may draw a graph of wave velocity c 

 against wavelength A, beginning with Cauchy's relation* 



(1) 

 2x 



V 



which is valid for gravity waves of small amplitude if the depth exceeds half a wave- 

 length. (In fact, in the Cauchy solution, the disturbance half a wavelength below the 

 surface is reduced to only 4% of its surface amplitude.) This wave velocity c is the 

 velocity of individual crests, but the energy in the wave is propagated at the group 

 velocity** 



dc 

 U = c - X — , (2) 



d\ 

 which is Vic for these waves. This is of course why the group of waves generated 

 say by throwing a stone into a pond has its energy travelling slower than the individual 

 crests, which therefore have to disappear on reaching the front of the group and passing 

 out of the energy-containing region. Similarly, new crests have to appear at the back 

 of the group. 



When the wavelength exceeds 2h, where h is the depth, the bottom begins to 

 influence the waves, but we still have a simple solution if the depth is uniform, when 

 the wave velocity is 



V 



g\ 2irh 



— tanh (3) 



2tt X 



tending to the constant value \/gh as X/h becomes large (see the righthand part of the 

 lower full-line curve in Fig. 1 ) . The group velocity 



2irh/\ \ 



= c { K + ' (-4) 



sinh 4:Trh/X/ 



* Plotted as the broken line in Fig. 1 below. 



** The most general proof of this statement is that given in the Appendix to Rayleigh's 

 Theory of Sound, which applies to an arbitrary conservative system capable of transmitting 

 sinusoidal waves of all frequencies. 



18 



