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4.4 Vibrating Drumhead

The two-dimensional version of a string is an elastic, flexible, homogeneous drumhead, that
is, a membrane stretched over a frame. Say the frame lies in the xy-plane (see Figure 4.4),
u(x, y, t) is the vertical displacement and there is no horizontal motion. The horizontal
components of Newton’s law again give constant tension T. Let D be any domain in the
xy-plane, say a circle or a rectangle. Let G be its boundary curve. We use reasoning
similar to the one-dimensional case. The vertical component gives (approximately)
Z ZZ
∂u
F = T ds = ρu tt dxdy = ma,
G ∂n D
where the left side is the total force acting on the piece D of the membrane, and where
∂u/∂n = n · ∇u is the directional derivative in the outward normal direction, n being the
unit outward normal vector on G. By Green’s theorem, this can be rewritten as
ZZ ZZ
∇ · (T∇u)ds = ρu tt dxdy.
D D

u

0 y

D ~n
~n

~n
x

Figure 4.4: Frame in xy-plane

Since D is arbitrary, we deduce from the second vanishing theorem that ρu tt = ∇ ·
(T∇u). Since T is constant, we get

2
2
u tt = c ∇ · (∇u) = c (u xx + u yy ), (4.3)
p
where c = T/ρ as before and ∇ · (∇u) = div grad u = u xx + u yy is known as the
two-dimensional laplacian. Equation 4.3 is the two-dimensional wave equation.
The pattern is now clear. Simple three-dimensional vibrations obey the equation

2
u tt = c (u xx + u yy + u zz ). (4.4)
2
2
2
2
2
2
The operator L = ∂ /∂x +∂ /∂y +∂ /∂z is called the three-dimensional laplacian op-
2
erator, usually denoted by ∆ or ∇ . Physical examples described by the three-dimensional
wave equation or a variation of it include the vibrations of an elastic solid, sound waves in
air, electromagnetic waves (light, radar, etc.), linearized supersonic airflow, free mesons
in nuclear physics, and seismic waves propagating through the earth.
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