end. Conditions would be the same at all points. 
Equation (6.2) has been solved in Chapter 5. It is far too 
regular to represent a storm at sea. However it does start and 
stop at the origin, and the wave train actually travels and dis- 
perses so that it is observed at different times at different 
values of x. If 2nT is of the order of several hours, at each 
point, x, there is a time interval of several hours where the dis- 
turbance can be thought of as having the properties of equation 
(6.1). Outside of this time interval the waves have either not 
arrived at a point x, or they have passed the point x, and the sea 
surface is essentially undisturbed. 
An infinite periodic train of wave groups 
Equation (6.3) has not been treated before. It has the faults 
of equation (6.1), but it also has some other interesting features 
which make it worth studying. The function represented by equa- 
tion (6.3) is periodic with a period,t . Therefore it can be ex- 
panded into a Fourier Series of simple sine waves with discrete 
spectral components. Thus equation (6.7) shows that it is possible 
to represent the infinitely long periodic train by a sum such as 
the one given by the last expression in equation (6.8). 
Each side of equation (6.8) is multiplied by sin 2rpt/r in 
equation (6.9). The function is odd and there will be no cosine 
terms. Integration of both sides of the equation from - t™/2 to 
t/2, as shown by equations (6.10) and (6.11) yields the values 
of a. Equation (6.12) is then another representation for n port). 
Equation (6.12) is much more informative than equation (6.3) 
because it is a sum of simple sine terms, and the classical theory 
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