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Chapter 6

Wave Equation

6.1 Solved Problems

Exercise 113. Consider the small transverse vibrations of a composite string of
infinite extent, made up of two homogeneous strings of different densities joined at

x = 0. In each region 1) x < 0, 2) x > 0 we have

2
u − c u = 0 j = 1, 2 c 6= c ,
tt
2
1
j xx
and we require continuity of u and u at x = 0. Suppose for t < 0 a wave
x
approaches the junction x = 0 from the left, i.e. as t approaches 0 from negative
values:
(
F(x − c t) x < 0, t ≤ 0
1
u(x, t) =
0 x > 0, t ≤ 0
As t increases further, the wave reaches x = 0 and gives rise to reflected and
transmitted waves.

(a) Formulate the appropriate initial values for u at t = 0.

(b) Solve the initial-value problem for −∞ < x < ∞ , t > 0.

(c) Identify the incident, reflected and transmitted waves in your solution and
determine the reflection and transmission coefficients for the junction in terms

of c and c . Comment also on their values in the limit c → c .
2
1
1
2
Solution:
(a)

(
F(x), x < 0
u(x, 0) =
0, x > 0
(
0
−c F (x), x < 0
1
u (x, 0) =
t
0, x > 0

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