18 



Theory of Short and Long Waves 



27ih\ 



x y \2n 



2. tanh ?. 



(11.10) 



When the wave length is smaller than double the depth (A < 2h) we can assume 

 that tanh(2^/z/A) = 1, and therefore 



c = vXgZ/lji) or c = gT/271 . (II. 1 1) 



The velocity is independent of the depth but proportional to the square 

 root of the wave length. 



If, on the other hand, the wave length is moderately large compared 

 with h, we have xa.nh(2jth/A) nearly equal to (2nhlX), and we obtain the 

 velocity of long waves (Lagrange 1781) 



c = }(gh). (11.12) 



It is independent of the wave length and proportional to the square root 

 of the depth. 



To determine the orbit of the individual water particles, we can compute with the aid of (11.9) 

 from the velocities it and v the component displacements in the horizontal and vertical directions: 



cosh x(z+ h) 



Xi—x 2 = A cos(xx— at), 



sinhar/j 



sinh x{z-r h) 



z x — z 2 = A — sin(xjt— at) . 



sinh y.h 



(II. 13) 



This gives for each individual particle an elliptic-harmonic orbit. The horizontal and vertical 

 semi-axes are 



cosh x(z-\-h) 

 sinh*/? 



and 



sinh*(.z+/f) 

 sinh*/? 



Both axes decrease from surface to the bottom (r = —h). Only a horizontal movement can 

 exist at the bottom, where the vertical semi-axis vanishes. Figure 8 shows such orbits and the 

 position of small perpendicular water filaments at rest in a shallow water wave (/j/A = 0*2). For 



Fig. 8. Orbital motion and positions of water filaments in a progressive wave travelling to 

 the right in shallow water (h : ). = 0-2). 



deep water waves (h > k) the quantities rj and £ become Ae xz , and each individual particle describes 

 a circular orbit with a constant angular velocity a = i/(2jigjX). The radii of the circle are given by 

 the formula Ae* : and decrease with depth in a geometrical progression. The velocity of the particles 

 is 2(A7i' l T)e >tz . 



