Fourier Series and Partial Differential Equations Lecture Notes

Chapter 4
The wave equation
In this chapter we look at the wave equation, concentrating on applications to waves on
strings. We discuss methods for solution and also uniqueness of solutions.
4.1 Derivation in one space dimension
Consider a flexible string stretched to a tension T, with mass density ρ, undergoing small
transverse vibrations. First suppose the string to be at rest along the x-axis in the (x,y)plane.
A point initially at xi is displaced to r(x,t) = xi + y(x,t)j, where y(x,t) is the
transverse displacement and i and j are the usual unit vectors along the coordinate axes.
We will assume that |∂y/∂x| ≪ 1 and ignore gravity and air-resistance.
Figure 3
The vector
τ := ∂r ∂y
= i+ j,
∂x ∂x
(4.1)
is a tangent vector to the string and, since
|τ| = 1+ ∂y
2
= 1+
∂x
1
2 ∂y
−
2 ∂x
1
4 ∂y
+··· ,
8 ∂x
(4.2)
it is approximately a unit tangent. Thus, in the figure:
• +Tτ = force exerted by + on −;
• −Tτ = force exerted by − on +.
The velocity and acceleration vectors are
v = ∂r
∂t
= ∂y
∂t j, a = ∂2 r
∂t 2 = ∂2 y
∂t 2j,
(4.3)
respectively.
Consider the piece of string which occupies the interval [a,a + h], where, at a later
stage in the argument, h → 0:
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