Vibrations of a circular drum

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One of the possible modes of vibration of an idealized circular drum.

The vibrations of an idealized circular drum, essentially an elastic membrane attached to a rigid circular frame, are solutions of the wave equation with zero boundary conditions.

There exist infinitely many ways in which a drum can vibrate, depending on the shape of the drum at some initial time and the rate of change of the shape of the drum at the initial time.

Using separation of variables, it is possible to find a collection of "simple" vibration modes, and it can be proved that any arbitrarily complex vibration of a drum can be decomposed as a linear combination of the simpler vibrations.

1 The problem
2 The radially symmetric case
3 See also
4 References

[edit] The problem

Consider an open disk $Ω$ of radius $a$ centered at the origin, which will represent the "still" drum shape. At any time $t,$ the height of the drum shape at a point $(x, y)$ in $Ω$ measured from the "still" drum shape will be denoted by $u (x, y, t),$ which can take both positive and negative values. Let $\partial \Omega$ denote the boundary of $Ω,$ that is, the circle of radius $a$ centered at the origin, which represents the rigid frame to which the drum is attached.

The mathematical equation that governs the vibration of the drum is the wave equation with zero boundary conditions,

$\frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right) \text{ for }(x, y) \in \Omega \,$

$u = 0\text{ on }\partial \Omega.\,$

Here, $c$ is a positive constant, which gives the "speed" of vibration.

Due to the circular geometry, it will be convenient to use polar coordinatess, $r$ and $θ.$ Then, the above equations are written as

$\frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial r^2}+\frac {1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2}\right) \text{ for } 0 \le r < a, 0 \le \theta \le 2\pi\,$

$u = 0\text{ for } r=a.\,$

[edit] The radially symmetric case

We will first study the possible modes of vibration of a drum that are radially symmetric. Then, the function $u$ does not depend on the angle $θ,$ and the wave equation simplifies to

$\frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial r^2}+\frac {1}{r}\frac{\partial u}{\partial r}\right) .$

We will look for solutions in separated variables, $u (r, t) = R (r) T (t).$ Substituting this in the equation above and dividing both sides by $c 2 R (r) T (t)$ yields

$\frac{T''(t)}{c^2T(t)} = \frac{1}{R(r)}\left(R''(r) + \frac{1}{r}R'(r)\right).$

The left-hand side of this equality does not depend on $r,$ and the right-hand side does not depend on $t,$ it follows that both sides must equal to some constant $K .$ We get separate equations for $T (t)$ and $R (r)$ :

$T''(t) = Kc^2T(t) \,$

$rR''(r)+R'(r)-KrR(r)=0.\,$

The equation for $T (t)$ has solutions which exponentially grow or decay for $K > 0,$ are linear or constant for $K = 0,$ and are periodic for $K < 0.$ Physically it is expected that a solution to the problem of a vibrating drum will be oscillatory in time, and this leaves only the third case, $K < 0,$ when $K = - λ 2$ for some number $λ > 0.$ Then, $T (t)$ is a linear combination of sine and cosine functions,

$T(t)=A\cos c\lambda t + B\sin c \lambda t.\,$

Turning to the equation for $R (r),$ with the observation that $K = - λ 2,$ all solutions of this second-order differential equation are a linear combination of Bessel functions of order 0,

$R(r) = c_1 J_0(\lambda r)+ c_2 Y_0(\lambda r).\,$

The Bessel function $Y 0$ is unbounded for $r\to 0,$ which results in an unphysical solution to the vibrating drum problem, so the constant $c 2$ must be null. We will also assume $c 1 = 1,$ as otherwise this constant can be absorbed later into the constants $A$ and $B$ coming from $T (t).$ It follows that

R (r) = J 0 (λ r).

The requirement that height $u$ be zero on the boundary of the drum results in the condition

R (a) = J 0 (λ a) = 0.

The Bessel function $J 0$ has an infinite number of positive roots,

$0< \alpha_1 < \alpha_2 < \cdots$

It follows that $λ a = α n,$ for $n=1, 2, \dots,$ so

$R(r) = J_0\left(\frac{\alpha_n}{a}r\right).$

Therefore, the radially symmetric solutions $u$ of the vibrating drum problem that can be represented in separated variables are

$u(r, t) = \left(A\cos c\lambda_n t + B\sin c\lambda_n t\right)J_0\left(\lambda_n r\right)$ for $n=1, 2, \dots, \,$

where $λ n = α n / a .$

[edit] See also

Hearing the shape of a drum

[edit] References

H. Asmar, Nakhle (2005). Partial differential equations with Fourier series and boundary value problems. Upper Saddle River, N.J.: Pearson Prentice Hall, page 198. ISBN 0-13-148096-0.