The initial value problem in the y,t plane for a wave train of 
finite width and finite duration 
The problem to be solved now is the logical extension of 
the problem in Chapter 5 to the case of a storm of finite width. 
Equation (8.28) is the starting point and when 7)(0,y,t) is given, 
the problem consists of evaluating the indicated integrations. 
The initial values are given by equation (8.34). Outside of 
a certain range of y at x = O given by plus and minus one half the 
width of the storm, W.,no disturbance is observed at the source. 
Outside of a certain range of time at x = O given by plus and 
minus one half the duration of the waves, Do no disturbance is 
observed at the source. Inside of the indicated rectangle in 
the y,t plane, a disturbance given by A sin((y4°/e)y sin @, - #4t) 
is observed. For y fixed, the disturbance inside the rectangle 
is a disturbance whose record, as a function of time, would look 
very much like the disturbance produced in Chapter 5. Since # 1 
is a fixed number, the apparent period of the disturbance would 
be given by T, = 2 / 1 Within the time interval given in equation 
(8.34). For a fixed 6,, },, and t, as y varies, the disturbance 
is a slowly varying sinusoidal function, if 8, is small. The small- 
er the value of Q1> the more rapidly the crests of the disturbance 
move in the y direction. The crests in the y,t plane do not move 
in the y direction with the speed of gravity waves because they 
are really only a component of the wave as observed at x = 0. 
Given the graph of 1(0,y,t) the use of equation (8.34) determines 
By and 0, uniquely. 
Several integrals must be evaluated in order to obtain the 
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