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DERIVATION OF THE WAVE EQUATION
WAVE MOTION ON AN INTERVAL WAVE
CHAPTER 16 MOTION IN AN INFINITE MEDIUM

The Wave

Equation

16.1 Derivation of the Wave Equation

Vibrations in a membrane or steel plate, or oscillations along a guitar string, are all modeled
by the wave equation and appropriate initial and boundary conditions. We will begin with a
derivation of the one-dimensional wave equation.
Suppose an elastic string has its ends fastened by two pegs. The string is displaced, released
and allowed to vibrate in a plane.
Place the string along the x - axis from 0 to L and assume that it vibrates in the x, y - plane.
We want a function y(x,t) such that, at time t, the graph of y(x,t) is the shape of the string at
that time. We call y(x,t) the position function for the string.
To take a simple case, neglect damping forces such as the weight of the string and assume
that the tension T(x,t) acts tangent to the string, and that individual particles of the string move
only vertically. Also assume that the mass ρ per unit length is constant. Consider a segment of
string between x and x + x. By Newton’s second law of motion, the net force on this segment
due to the tension is equal to the acceleration of the center of mass of this segment, multiplied
by its mass. This is a vector equation. Its vertical component (Figure 16.1) gives us
2
∂ y
T (x + x,t)sin(θ + θ) − T (x,t)sin(θ) = ρ x (x,t),
∂t 2
where x is the center of mass of this segment and T (x,t) = T(x,t) . Then

2
T (x + x,t)sin(θ + θ) − T (x,t)sin(θ) ∂ y
= ρ (x,t).
x ∂t 2
Now v(x,t) = T (x,t)sin(θ) is the vertical component of the tension, so this equation becomes

2
v(x + x,t) − v(x,t) ∂ y
= ρ (x,t).
x ∂t 2
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October 14, 2010 15:23 THM/NEIL Page-565 27410_16_ch16_p563-610