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16.2 Wave Motion on an Interval 567

If the string is released from rest (zero initial velocity), then g(x) is identically zero. Further,
if the string has its ends fixed at the same level, then the initial position function f must satisfy
the compatibility condition f (0) = f (L) = 0.
Variations on this problem can allow for moving ends with conditions such as
y(0,t) = α(t) and y(L,t) = β(t).

We can also have a forcing term in which an external force drives the motion of the string. In this
case the wave equation is
2
2
∂ y ∂ y
= c 2 + F(x,t).
∂t 2 ∂x 2
The wave equation in two space dimensions is
2
2
2
∂ z ∂ z ∂ z
= c 2 + (16.2)
∂t 2 ∂x 2 ∂y 2
in which z(x, y,t) is the displacement function.
It is a routine exercise in chain rule differentiation to convert this two-dimensional wave
equation to polar coordinates. If the displacement function is u(r,θ,t), this wave equation is
2
2
2
∂ u ∂ u 1 ∂u 1 ∂ u
= c 2 + + . (16.3)
2
∂t 2 ∂r 2 r ∂r r ∂θ 2
SECTION 16.1 PROBLEMS

1. Let y(x,t)= sin(nπx/L)cos(nπct/L) for 0≤ x ≤ L. and the initial conditions
Show that y satisfies the one-dimensional wave equa-
tion for any positive integer n. ∂y
√ y(x,0) = sin(x), (x,0) = cos(x) for 0 < x <π.
2
2
2. Show that z(x, y,t)=sin(nx)cos(my)cos n + m t ∂t
satisfies the two-dimensional wave equation for any
positive integers n and m. 5. Formulate an initial-boundary value problem for
3. Let f be a twice-differentiable function of one variable. vibrations of a rectangular membrane occupying 0 ≤
Show that x ≤ a,0 ≤ y ≤ b if the initial position is the graph of
1 z = f (x, y) and the initial velocity is g(x, y). The mem-
y(x,t) = [ f (x + ct) + f (x − ct)]
2 brane is fastened to a still frame along the rectangular
satisfies the one-dimensional wave equation. boundary of the region.
4. Show that 6. Formulate an initial-boundary value problem for the
1 motion of an elastic string of length L fastened at
y(x,t) = sin(x)cos(ct) + cos(x)sin(ct)
c both ends and released from rest with an initial posi-
satisfies the one-dimensional wave equation together tion given by f (x). The motion is opposed by air
with the boundary conditions resistance, which has a force at each point of magni-
1 tude proportional to the square of the velocity at that
y(0,t) = y(2π,t) = sin(ct) for t > 0 point.
c

16.2 Wave Motion on an Interval

We will solve initial-boundary value problems for the wave equation on a closed interval [0, L],
starting with special cases we can solve and building toward more complex wave motion.

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October 14, 2010 15:23 THM/NEIL Page-567 27410_16_ch16_p563-610