Vibrating Drumhead

Hit a drum and the skin ripples. Every point accelerates toward the average of its neighbors, and that one rule creates both the patterns you see and the pitch you hear.

click the drumhead to strike it

How it works

The 2D wave equation

In one sentence

A membrane's displacement $u$ satisfies $u_{tt} = c^2(u_{xx} + u_{yy})$ ; on a clamped round drum its modes are $J_m(j_{mn}r/R)\cos m\theta$ , ringing at $c\,j_{mn}/R$ .

General formula

{\htmlData{sym=0}{\frac{\partial^2 u}{\partial t^2}}} = {\htmlData{sym=2}{c}}^2\left({\htmlData{sym=1}{\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}}}\right)

The left side is a patch of skin's acceleration; the right side is its curvature. Bulges get pulled back and dips pushed up, harder for a tighter skin (bigger $c$ ).

What the symbols mean

$\frac{\partial^2 u}{\partial t^2}$
Vertical acceleration
u is how far the drumhead is pushed up or down at each point. This is how quickly its up-and-down speed changes.
$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$
Curvature
How bent the surface is at a point. Bumps get pulled down and dips pushed up.
$c$
Wave speed
How fast ripples cross the drumhead. A tighter drum has a higher c.

The Laplacian $\nabla^2 u$ measures how far a point sits from the average of its neighbors.
On a round drum of radius $R$ with a clamped rim, separation of variables in polar coordinates gives modes $J_m(j_{mn} r/R)\cos(m\theta)\cos(\omega_{mn}t)$ , where $J_m$ is a Bessel function.
Mode $(m, n)$ rings at $\omega_{mn} = c\,j_{mn}/R$ , with $j_{mn}$ the $n$ th zero of $J_m$ . The ratios $1, 1.59, 2.14, 2.30, \dots$ are not whole numbers, which is why drums sound less tonal than strings.
Nodal lines are where a mode never moves: mode $(m, n)$ has $m$ nodal diameters and $n - 1$ nodal circles.

Learn the math

Series SolutionsDiff Eq · Lesson 37, Series Solutions Eigenvalues and EigenfunctionsDiff Eq · Lesson 42, Boundary Value Problems and Fourier Series What Is a PDE?PDE · Lesson 1, Foundations The Wave EquationPDE · Lesson 5, Foundations The Vibrating StringPDE · Lesson 10, Separation of Variables Vibrating Drums and Bessel FunctionsPDE · Lesson 16, Beyond the Basics

Learn the math

Series SolutionsDiff Eq · Lesson 37, Series Solutions

Eigenvalues and EigenfunctionsDiff Eq · Lesson 42, Boundary Value Problems and Fourier Series

What Is a PDE?PDE · Lesson 1, Foundations

The Wave EquationPDE · Lesson 5, Foundations

The Vibrating StringPDE · Lesson 10, Separation of Variables

Vibrating Drums and Bessel FunctionsPDE · Lesson 16, Beyond the Basics