358
8 Waves
P ′ ( x , y )
Fig. 8.4
The general form of the displacement at any point x and time t is given by the
Fourier expansion
y =
∞
n=1
a n sin
nπ x
L
cos
nπvt
L
(1)
The coefficient a n is obtained from
a n =
2
L
L
0
y 0 sin
nπ x
L
dx
(2)
where y 0 = y(x, 0).
We break the integral into two parts, one from 0 to d and the other from d to L.
In the interval from 0 to d the equation of the initial configuration of the string
for a typical point p(x, y) is
y
x
=
h
d
or y =
hx
d
for o < x < d
and in the interval d to L, the equation for P (x, y) is
y
L − x
=
h
L − d
or y =
h(L − x)
L − d
for d < x < L
so that by substituting (1) into (2) with t = 0
a n =
2
L
d
0
hx
d
sin
π nx
L
dx +
L
d
h(L − x)
L − d
sin
π nx
L
dx
(3)