take much longer to travel the length of the fetch and pornse= 
quently they are observed for a greater length of time at the 
sourcee The wave system can be thought of as consisting of two 
parts. Within the time interval, O<t<Dg, all spectral components 
are present at x = O with an intensity given by the original 
power spectrum. Then for t greater than Dg; there will be a value 
of ~ such that, at the time of observation, t = D, + 2HF/g, 
those values of 4 which are less than the value of # which 
satisfies this equation will no longer be present in the power 
spectrum at the source. The others will be present with the 
same intensity as before. 
Now that the conditions have been given for x = 0, the con- 
ditions for positive x can be found for the case of the square 
cut-off filter. For any particular uw , under the assumption 
that the sine wave in the partial sum which applies to this parti- 
cular #4 has a forward edge with an amplitude which is either 
one or zero and which travels with the group velocity, the for- 
ward edge of that wave train requires 24x/g seconds to reach 
the point x. The time of arrival of the forward edge of the 
train, t,, is given by equation (7.52). 
Similarly the rear edge of the train starts out D, + 2uF/g 
seconds after the forward edge. The time of passage of the rear 
edge of the train, t,, is consequently given by equation (7.53). 
If the time of observation, Cea lies between te and t, as 
stated by equation (7.54) then that particular spectral component 
will be present. If top is less than tr the train will not have 
arrived, and if top is greater than te the train will already 
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