666 



BARBER AND TUCKER 



[chap. 19 



Theory suggests some slight increase in velocity and consequent increase in 

 length for waves of a given period if the waves are very steep (Lamb, 1945, 

 Art. 250). The velocity is expected to increase by a factor {1 +Tr^H^I2L^); 

 ^ = height, iy = wavelength. Formulae (2) and (3) are good enough, however, 

 for most purposes. 



In some cases one is concerned not with the speed of advance of the wave 

 crests but with the speed of advance of the larger region of wave-disturbance, 

 as, for instance, when one attempts to predict the arrival of large "swell" from 

 a distant storm. The two speeds are not the same. This can be seen from the 

 rather artificial example given in Fig. 2. If two wave trains of slightly different 

 wavelengths are present at the same time, there are some regions in which the 

 crests of one coincide with the troughs of the other so that the combined waves 

 are small, and other regions where the waves agree in position and the combined 

 waves are large. Exact agreement may occur at two crests such as those 



TWO WAVE TRAINS 



B 



6L 



M 



TRAVEL 



COMBINED WAVES 



|*_L -M 



Fig. 2. Illustrating the "group" behaviour. Wave trains having a slightly different wave- 

 length add to give a system of "groups", or regions where the waves are high. In- 

 dividual waves run forward through these groups, becoming high as they enter a 

 group and losing height again as they leave. The groups themselves advance at a 

 slower speed. 



labelled A. Immediately in the rear of these, the wave crests are out of step by 

 a small distance BL, the difference in wavelength, but the longer wave is over- 

 taking the shorter one because of the difference in velocities of the wave trains, 

 SO. After a time interval hLjhC these two crests will coincide. During this 

 interval the wave crests will, of course, have advanced a distance C hLfhC but 

 relative to the waves ; the point of exact agreement between the two trains, that 

 is the region of greatest combined wave activity, will have lost ground by one 

 whole wavelength, L. So the group advances at a speed smaller than that of the 

 wave crests themselves, in fact at a speed G, where 



G = C-LdC/dL. 



Equation (2) shows the connexion between C and L for the kind of waves being 

 discussed : gravity waves in deep water. On differentiating this formula the 

 group velocity given above is found to be just half the velocity of the waves ; 



G = hC = gTJ^rr. (4) 



