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592 CHAPTER 16 The Wave Equation

1.6

1.2

0.8

0.4

0
0 0.2 0.4 0.6 0.8 1
x

FIGURE 16.9 Profiles of the elastic bar at dif-
ferent times with f (t) = K.

g(t)

(0, 2L/c)

t
(4L /c, 0) (8L /c, 0)

FIGURE 16.10 Sawtooth wave.

The interesting thing about this is that this function has a simple inverse transform. Let g be
periodic of fundamental period 4L/c, with g(t) for 0 ≤ t ≤ 4L/c defined by

t for 0 ≤ t ≤ 2L/c,
g(t) =
4L/c for 2L/c ≤ t ≤ 4L/c.
A graph of this sawtooth wave is shown in Figure 16.10.
Thus the right end of the bar moves according to the graph of g, exhibiting an up-and-down
oscillation.

Case 2 Another case in which we can do a fairly complete analysis is that the end is hit with
an impulse of magnitude I at time zero. Suppose f (t) = Iδ(t), with δ the delta function. The
analysis proceeds as in case 1, except now we obtain

cI sinh(sx/c)
Y(x,s) = ,
E s cosh(sL/c)
differing from case 1 in the power of s in the denominator. This occurs because the Laplace trans-
form of K is K/s, while the transform of Iδ(t) is just I, the delta function having transform 1.
Since

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October 14, 2010 15:23 THM/NEIL Page-592 27410_16_ch16_p563-610