as it passes the line x = 0. Before a certain time its ampli- 
tude would be zero and after, say, ten hours had elapsed it would 
again be zero. In addition, outside of a certain range of y at 
x = 0 it would never be observed. 
The immediate problem is to find out how this disturbance 
behaves at other values of xe In anticipation of what is to 
follow, though, think of a storm at sea as a sum of many elemental 
sine waves traveling in various directions but bounded by the 
Closed curve described above. The disturbance is, by the principle 
of superposition, and due to the linearity of the system, equal 
to the sum of the individual disturbances, no matter how they 
differ in direction, amplitude, phase, and period. 
The function, 7(0,y,t), has now been obtained. What is 
the function 1(x,y,t)? Strictly speaking, 7 (0,y,t) does not 
determine 7 (x,y,t) because there is an ambiguity in the possible 
directions of the individual spectral components. For a com- 
pletely general problen, 1) (0095) would also have to be measured. 
However waves in a storm at sea travel in the direction of the 
wind and if the reasonable assumption that each spectral component 
has a component of direction of travel in the positive x direction 
is made, then a solution can be obtained. 
Equation (8.14) postulates that the free surface is composed 
of spectral sine waves of special frequency pm which travel in 
the spectral direction x cos 6 + y sin 9 (see equation (2.29)). 
Note that the limits of integration are over only half a circle 
in the p , © polar coordinate system, and that mw is always posi- 
tive since the integration is from zero to infinity. With this 
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