1550078481-Ordinary_Differential_Equations__Roberts_

260 Ordinary Differential Equations

a. y(x) = cos x

-1.5

1.5 y

1

0.5

b. y(x) = [u(x -n) -u(x -2n)] - 2^0

2 4 6

-0.5

-1

-1.5

1.5 y

0.5

b. y(x) = [u(x -n) -u(x -2n)] cos x -2^0 2

• 0.5
  -1
  -1.5

Figure 5.5 Graphs of cosx, [u(x - 7r) - u(x - 27r)], and

[u(x - 7r) - u(x - 27r)] cosx

8 x

8 x

Now suppose that two identical sensing devices are placed at points A and
B which are separated from one another by a relative large distance. Further
suppose that at time x = 0 a signal is sent from point A. Let the signal
received by the sensing device at point A be the function f(x) shown in
Figure 5.6a. Assuming that no distortion or attenuation has occurred, the
signal received by the sensing device at point B will be the function

{

0,

g(x) =

f(x - c),

0::::: x < c

c::::: x,

where c > 0 is the time it takes the signal to travel from point A to point B.

The function g(x) is shown in Figure 5.6b. The function g(x) is call ed the
c-time delay function of f(x). That is, the function g(x) is the function
f(x) delayed by c units of time. One encounters time-delayed functions in
many physical circumstances. In most physical situations the function f(x)
is defined only for x ~ 0. We have defined the c-time delay function g(x) of

f(x) so that g(x) = 0 for 0 :::; x < c, since no disturbance at point Bis caused

by the signal sent from point A during the interval of time [O, c). Notice that
g(x) can be defined more concisely by g(x) = u(x - c)f(x - c).