is composed of infinitesimal waves added together by means of 
Fourier Integrals. If the potential energy can be accounted 
for, the kinetic energy can be accounted for; and the total 
energy is therefore accounted for. For additional information, 
see Lamb [1932]. 
The problem, then, is to construct a balance sheet for all 
the energy in the system at x = O and show that that energy is 
finally observed at x = xy without loss. The potential energy 
will be traced, and since an equal amount of kinetic energy must 
be present, the total energy will be accounted for. 
Equation (6.19) can be applied to an ordinary sinusoidal 
wave for an introductory elementary example. This is done in 
equation (6.20) where 7 equals -A sin(4r°x/gT* - 2rt/T). The 
last expression in equation (6.20) is a funetion of T, t*, and 
x. For any x and t* as T becomes large, P.E. is given by equation 
(6.21). The two sine terms can be at most two in absolute value 
for a fixed x, t*, and T, and if T is of the order of ten times 
the period, P.E. differs from its limiting value by less than 
two per cent. The result shows that the same average potential 
energy is present at any point in space at any time. 
Ee ee a ne ——--— 
The infinite periodic train of wave groups (equation (6.3)) 
can be written as an infinite sum of sinusoidal waves as in 
equation (6.8) and (6.13). If the waves which travel in the nega- 
tive x direction are omitted because they are negligible, the 
a, would be defined by the top part of equation (6.14). (See 
the discussion at the beginning of the section on the finite 
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