Chapter 4. THE PROPAGATION OF A FINITE WAVE 
GROUP IN INFINITELY DEEP WATER 
Introduction 
Some interesting results can be obtained by the application 
of Fourier Integral Theory to the problem of a finite wave group. 
In this chapter, a special wave group will be studied in order 
to show some of the properties of dispersion in infinitely deep 
water. The particular form of the wave group studied in this 
chapter will be employed in studies of various not-too-realistic 
models of the sea surface. This particular form of the wave group 
is too specific for reality, but if it is imagined that the steps 
taken with reference to the specific modulation envelope employ=- 
ed are taken with reference to arbitrary forms for the envelope, 
then it is possible to see how some of the properties of ocean 
waves can be studied. There are a great many possible forms for 
the finite wave groups discussed in the previous chapter. A 
very special one will be picked for this chapter. 
Formulation 
Suppose then that the origin of the space coordinate system 
is located at the point x = 0, y = 0, z = O on the surface of 
an ocean of very great depth. At this point the height of the 
free surface as a function of time is measured and it is found 
that the equation for the observed free surface is given by equa- 
tion (4.1) of Plate V. 
Equation (4.1) has three parameters. The parameter, 4A, 
determines the amplitude of the disturbance which is greatest 
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