value of t such that -nT<t<nT. Then as x approaches zero through 
positive values the upper limits of integration of the first term 
in G approaches plus infinity. The upper limit of the second term 
approaches minus infinity. The total contribution of the bracket 
is then minus 2, and equation ( 5.20) holds. 
Similarly if t <-nT or t>nT, equation (5.21) holds. The sine 
term approaches - sin 2rt/T. The expression for H can be analyzed 
similarly and H is zero if t is not equal to either nT or -ntT. 
Therefore equation (5.15) reduces to equation (5.1) when x approaches 
zero except possibly at two points, namely t = nT and t = - ni. 
At these two points, the actual behavior of the solution is 
clarified if, now, after passing to the limiting value of x equal 
to zero, t is allowed to approach t = nT and t = - nT from both 
positive and negative values. The free surface approaches the 
value zero as t approaches nT or -nT from either direction, and 
therefore (5.15) can be defined to be equal to zero at t = -nT 
and t = +nT. Therefore (5.15) equals (5.1) everywhere as x ap= 
proaches zero through positive values of x. 
There is a reason for the particular care which must be em=- 
ployed in the study of the solution near these two special points. 
It is that the slope of the original expression, equation (5.1), 
is discontinuous at these points. The effect of this discontinuity 
in slope causes the solution to have a very peculiar appearance 
as a function of time for values of x near x = 0. 
Evaluation 
Now that a solution to the problem has been obtained, the 
properties of the solution will be discussed and graphed as in the 
eG. = 
