14.8 Exercises 413
The general solution of the wave equation that satisfies the boundary conditions is
therefore a superposition of normal modes
(14.103)
(the constantsc
1, 1 c
2, 1 c
3,=in (14.101) have been absorbed in the A’s and B’s in
(14.103)). This solution satisfies the initial conditions when the A’s and B’s are
determined from the equations
(14.104)
The two series in (14.104) are examples of the Fourier series discussed in Chapter 15.
We return to this problem in Section 15.5 to determine the A’s and B’s for typical
initial functionsf(x)andg(x).
0 Exercises 16 –18
14.8 Exercises
Section 14.2
1.Show that the functionf(x, t) 1 = 1 a 1 sin 1 (bx) 1 cos 1 (vbt)(i)satisfies the one-dimensional wave
equation (14.1), (ii)has the formf(x, t) 1 = 1 F(x 1 + 1 vt) 1 + 1 G(x 1 − 1 vt).
2.The diffusion equation
provides a model of, for example, the transfer of heat from a hot region of a system to a
cold region by conduction whenf(x, 1 t)is a temperature field, or the transfer of matter
from a region of high concentration to one of low concentration whenfis the
concentration. Find the functionsV(x)for whichf(x, t) 1 = 1 V(x)e
ctis a solution of the
equation.
- (i) It is shown in Example 14.2 that the functionf(x, t) 1 = 1 a exp[−b(x 1 − 1 vt)
2
]is a
solution of the wave equation (14.1). Sketch graphs off(x, t)as a function of xat times
t 1 = 1 0, t 1 = 122 v, t 1 = 142 v(use, for example,a 1 = 1 b 1 = 11 ), to demonstrate that the function
represents a wave travelling to the right (in the positive x-direction) at constant speed
v. (ii)Verify thatg(x, t) 1 = 1 a exp[−b(x 1 + 1 vt)
2
]is also a solution of the wave equation,
and hence that every superpositionF(x, t) 1 = 1 f(x, t) 1 + 1 g(x, t)is a solution. (iii) Sketch
appropriate graphs off(x, t) 1 + 1 g(x, t)to demonstrate how this function develops
in time.
∂
∂
=
∂
∂
f
t
D
f
x
2
2
tnnny
t
gx B
nx
l
==∂
∂
==
∑
01() sin
∞ω
π
yx f x A
nx
l
nn() (),= = sin
=∑
0
1∞π
yxt
nx
l
AtBt
nnnnn( ),=sin cos +sin
=∑
1∞π
ωω