VIBRATION OF A CIRCULAR MEMBRANE

6. VIBRATION OF A CIRCULAR MEMBRANE

A circular membrane (for example, a drumhead) is attached to a rigid support along its circumference. Find the characteristic vibration frequencies and the correspond- ing normal modes of vibration.

Take the (x, y) plane to be the plane of the circular support and take the origin at its center. Let z(x, y, t) be the displacement of the membrane from the (x, y) plane. Then z satisfies the wave equation

z = F (x, y)T (t),

Section 6 Vibration of a Circular Membrane 645

we separate (6.1) into a space equation (Helmholtz) and a time equation (see Prob- lem 3.10 and Section 3). We get the two equations

T+K ¨ 2 v 2 T = 0. Because the membrane is circular we write ∇ 2 in polar coordinates (see Chapter 10, Section 9); then the F equation is

When we put (6.5)

F = R(r)Θ(θ),

(6.4) becomes (5.5), and the separated equations and their solutions are just (5.6), (5.7), and (5.8). The solutions of the time equation (6.3) are sin Kvt and cos Kvt. Thus the solutions for z are z = R(r)Θ(θ)T (t), where R(r) = J n (Kr), Θ(θ) = {sin nθ, cos nθ} and T (t) = {sin Kvt, cos Kvt}. Just as in Section 5, n must be an integer. To find possible values of K, we use the fact that the membrane is attached to a rigid frame at r = a, so we must have z = 0 at r = a for all values of θ and t. Thus J n (Ka) = 0 so the possible values of Ka are the zeros of J n . As in Section 5, let k = Ka, that is, K = k/a. Then the possible values of k for each J n are k mn , the zeros of J n . We can now write the solutions for z as

sin nθ

sin kvt/a

(6.6) z=J n (kr/a) .

cos nθ

coskvt/a

For a given initial displacement or velocity of the membrane, we could find z as

a double series as we found (5.17) in the cylinder temperature problem. However, here we shall do something different, namely investigate the separate normal modes of vibration and their frequencies. Recall that for the vibrating string (Section 4), each n gives a different frequency and a corresponding normal mode of vibration (Figure 4.2). The frequencies of the string are ν = nv/(2l); all frequencies are

integral multiples of the frequency ν 1 = v/(2l) of the fundamental. For the circular membrane, the frequencies are [from (6.6)]

The possible values of k are the zeros k mn of the Bessel functions. Each value of k mn gives a frequency ν mn =k mn v/(2πa), so we have a doubly infinite set of char- acteristic frequencies and the corresponding normal modes of vibration. All these frequencies are different, and they are not integral multiples of the fundamental as is true for the string. This is why a drum is less musical than a violin. From your computer or tables you can find several k mn values (Problem 2) and find the fre-

quencies as (nonintegral) multiples of the fundamental (which corresponds to k 10 , the first zero of J 0 ). Let us sketch a few graphs (Figure 6.1) of the normal vibration modes corresponding to those in Figure 4.2 for the string, and write the correspond- ing formulas (eigenfunctions) for the displacement z given in (6.6). (For simplicity, we have used just the cos nθ cos kvt/a solutions in Figure 6.1.) In the fundamental

mode of vibration corresponding to k 10 , the membrane vibrates as a whole. In the

646 Partial Differential Equations Chapter 13

Figure 6.1

k 20 mode, it vibrates in two parts as shown, the + part vibrating up while the − part vibrates down, and vice versa, with the circle between them at rest. We can show that there is such a circle (called a nodal line) and find its radius. Since

k 20 >k 10 , the circle r = ak 10 /k 20 is a circle of radius less than a. Hence it is a circle on the membrane. For this value of r, J 0 (k 20 r/a) = J 0 (k 20 k 10 /k 20 )) = J 0 (k 10 ) = 0, so points on this circle are at rest. For the k 11 mode, cos θ = 0 when θ = ±π/2 and is positive or negative as shown. Continuing in this way you can sketch any normal mode (Problem 1).

It is difficult experimentally to obtain pure normal modes of a vibrating object. However, a complicated vibration will have nodal lines of some kind and it is easy to observe these. Fine sand sprinkled on the vibrating object will collect along the nodal lines (where there is no vibration) so that you can see them clearly—but see Am. J. Phys. 72, 1345–1346, (2004). [For experimental work on the vibrating circular membrane, see Am. J. Phys. 35, 1029–1031, (1967); Am. J. Phys. 40, 186–188, (1972); Am. J. Phys. 59, 376–377, (1991). Also see Problem 1(b).]

PROBLEMS, SECTION 6

1. (a) Continue Figure 6.1 to show the fundamental modes of vibration of a circular membrane for n = 0, 1, 2, and m = 1, 2, 3. As in Figure 6.1, write the formula for the displacement z under each sketch.

(b) Use a computer to set up animations of the various modes of vibration of a circular membrane. [This has been discussed in a number of places. See, for example, Am. J. Phys. 67, 534–537, (1999).]

2. Find, from computer or tables, the first three zeros k mn of each of the Bessel func- tions J 0 ,J 1 ,J 2 , and J 3 . Find the first six frequencies of a vibrating circular mem- brane as (non-integral) multiples of the fundamental frequency.

3. Separate the wave equation in two-dimensional rectangular coordinates x, y. Con- sider a rectangular membrane as shown, rigidly attached to supports along its sides.

Section 7 Steady-state Temperature in a Sphere 647

Show that its characteristic frequencies are

ν nm = (v/2) p(n/a) 2 + (m/b) 2 ,

where n and m are positive integers, and sketch the normal modes of vibration corresponding to the first few frequen- cies. That is, indicate the nodal lines as we did for the circular membrane in Figure 6.1 and Problem 1.

Next suppose the membrane is square. Show that in this case there may be two or more normal modes of vibration corresponding to a single frequency. (Hint for one example: 7 2 +1 1 =1 2 +7 2 =5 2 +5 2 .) This is an example of what is called degeneracy; we say that there is degeneracy when several different solutions of the wave equation (eigenfunctions) correspond to the same frequency (eigenvalue). Sketch several normal modes giving rise to the same frequency. Comment: Compare Chapter 3, Section 11, where an eigenvalue of a matrix is called degenerate if several eigenvectors correspond to it.

4. Find the characteristic frequencies for sound vibration in a rectangular box (say a room) of sides a, b, c. Hint: Separate the wave equation in three dimensions in rectangular coordinates. This problem is like Problem 3 but for three dimensions instead of two. Discuss degeneracy (see Problem 3).

5. A square membrane of side l is distorted into the shape

f (x, y) = xy(l − x)(l − y)

and released. Express its shape at subsequent times as an infinite series. Hint: Use a double Fourier series as in Problem 5.9.

6. Let V = 0 in the Schr¨ odinger equation (3.22) and separate variables in 2-dimensional rectangular coordinates. Solve the problem of a particle in a 2-dimensional square box, 0 < x < l, 0 < y < l. This means to find solutions of the Schr¨ odinger equation which are 0 for x = 0, x = l, y = 0, y = l, that is, on the boundary of the box, and to find the corresponding energy eigenvalues. Comments: If we extend the idea of

a “particle in a box” (see Section 3, Example 3) to two or three dimensions, the box in 2D might be a square (as in this problem) or a circle (Problem 8); in 3D it might be a cube (Problem 7.17) or a sphere (Problem 7.19). In all cases, the mathematical problem is to find solutions of the Schr¨ odinger equation with V = 0 inside the box and Ψ = 0 on the boundary of the box, and to find the corresponding energy eigenvalues. In quantum mechanics, Ψ describes a particle trapped inside the box and the energy eigenvalues are the possible values of the energy of the particle.

7. In your Problem 6 solutions, find some examples of degeneracy. (See Problem 3. Degeneracy means that several eigenfunctions correspond to the same energy eigen- value.)

8. Do Problem 6 in polar coordinates to find the eigenfunctions and energy eigenvalues of a particle in a circular box r < a. You want Ψ = 0 when r = a.