of time, distance, and depth is given by equation (4.7). 
The pressure given by equation (4.7) is evidently a rather 
complicated function. It is complicated because infinitely deep 
water is a dispersive medium. The various spectral components 
of the pressure are attenuated at different rates with depth 
and the various spectral wave components travel at different speeds 
along the surface. It is therefore to be expected that the shape 
of the wave profile as a function of time at different values of 
x will not be the same as the shape of the wave profile at x 
equal to zero and that the apparent period of the pressure vari- 
ation at a depth zg below the surface will not be the same as the 
apparent period of the disturbance at the surface. 
Solution 
The next step is to integrate equation (4.7). The value of 
the integral is given in Table 269 on page 375 of the table of 
definite integrals compiled by Bierens de Haan [1867]. After © 
some algebraic manipulations the result can be put into the form 
of equation (4.8) of Plate VI. 
The free surface can be found from equation (4.8), with the 
use of equation (2.9), by substituting p = 0 on the left, z = 0 
into the first term on the right, and z =7 into the second term 
on the right. Equation (4.9) then gives the free surface. 
The pressure as a function of time and depth at the point 
x = 0, is also of interest because the expression is simpler. 
By substituting x = 0 into equation (4.8) and clearing fractions, 
the pressure can be found below the original point of observa- 
tion. The pressure is given by equation (4.10). 
The derivation given in Plate V and the results obtained 
SPA 
