599 
29 
AS 
AS = ve Ar 
Thus a pulse of height AS is applied at the time 7, while at a time Ar later 
another pulse is applied, and so on. It is desired to find the effect of all 
these pulses at the definite time t. The response of any one of them is pic- 
tured in the lower half of Figure 17, and is given by 
= Ar-A(t — T) 
AT 
There is also the contribution of the pulse whose height is S(0). The corre- 
sponding response is given by the product 
S(0) - A(t) 
The total response is obtained by adding all the steps. This yields 
R(t) = S(0) A(t) + = Alt — 7) Ar 
In the limit as Ar > 0, the summation is replaced by the following expression, 
known as the Duhamel integral 
t 
R(t) = S(0) A(t) + ps2 Mites ade 
0 a 
This formula gives the response for a given input signal S(t) pro- 
vided the response to a unit pulse, i.e., A(t), is known; see Figure 16 and 
the upper half of Figure 17. 
A specific case will be worked out in detail to illustrate the 
method. 
Assume for instance that the response to a unit pulse has been ob- 
tained experimentally and is representable by the empirical equation 
A(t) =1—- e 
t, 18 a measure of "the time of rise" of the amplifier.* Let the input sig- 
nal be that due to an underwater explosion. This may be represented fairly 
well by the form** 
S(t) =« i? 
where T is the duration of the signal. Thus 
The response of actual amplifiers frequently approximates this form. 
For simplicity, S(0) is chosen as unity, and S(t) and R(t) are dimensionless. 
