Circular Membranes, Bessel Functions, and Radial-Angular Modes

Circular Membranes, Bessel Functions, and Radial-Angular Modes

M06-17 solved the fixed rectangular membrane by separating the wave equation in Cartesian coordinates.

A circular membrane has the same local membrane physics but different geometry.

The wave equation is still

(1)

with

(2)

where 𝒯 is membrane tension per unit length and σ is surface mass density.

For a circular membrane, polar coordinates are natural:

(3)

The two-dimensional Laplacian is

(4)

Therefore the circular-membrane equation is

(5)

For a membrane of radius a fixed at its rim,

(6)

The normal modes are built from Bessel functions rather than ordinary sines.

Their frequencies are

(7)

where α_mn is the nth positive zero of the Bessel function J_m.

Equivalently,

(8)

Here:

• m = 0, 1, 2,… is the angular index,
• n = 1, 2, 3,… is the radial-zero index.

1 Membrane wave speed

The same local force balance used in M06-17 gives

(9)

Thus

(10)

Changing from a rectangle to a circle changes the boundary-value problem, not the local wave speed.

2 Polar-coordinate membrane equation

For a scalar field in two-dimensional polar coordinates,

(11)

Expanding the radial derivative,

(12)

Therefore

(13)

3 Separation of variables

Seek a normal mode in the form

(14)

Substitution gives

RΘQ

= c2.

(15)

Divide by

(16)

Then

(17)

The left side depends only on time.

Set both sides equal to

(18)

The time equation becomes

(19)

Thus

(20)

4 Separating radial and angular dependence

After separating time,

(21)

Multiply by

(22)

(23)

The left side depends only on r, while the right side depends only on 𝜃.

Therefore both sides equal a constant.

For periodic angular solutions, write the constant as

(24)

Then

(25)

and

(26)

5 Angular equation

The angular equation has solution

(27)

Physical displacement must be single-valued:

(28)

This periodicity requires

(29)

The integer m is the angular mode index.

6 Why m must be an integer

Suppose

(30)

After one full revolution,

(31)

For this to equal

(32)

for every angle, the phase increase must be an integer multiple of

(33)

Thus

(34)

must be an integer.

This is angular quantization produced by periodic boundary conditions.

7 Radial equation and Bessel’s equation

The radial equation is

(35)

Introduce the dimensionless variable

(36)

Then

(37)

and

(38)

Substitution gives

(39)

This is Bessel’s differential equation of integer order m.

8 Bessel functions of the first and second kinds

The general radial solution for

(40)

is

(41)

where:

• J_m is the Bessel function of the first kind,
• Y _m is the Bessel function of the second kind.

For a membrane that includes the center

(42)

the displacement must remain finite.

The functions

(43)

are singular at the origin.

Therefore regularity requires

(44)

The physical radial factor is

(45)

9 Behavior at the center

For the first-kind Bessel functions,

(46)

while

(47)

Thus:

• axisymmetric modes with m = 0 can have nonzero center displacement,
• all modes with m ≥ 1 vanish at the center.

Near the origin,

(48)

This behavior is regular and finite.

10 Fixed-rim boundary condition

For a circular membrane of radius a fixed at the rim,

(49)

For a nonzero mode,

(50)

Since

(51)

we require

(52)

Thus

(53)

must equal a positive zero of

(54)

Let

(55)

denote the nth positive zero of

(56)

Then

(57)

11 Circular-membrane frequencies

Since

(58)

(59)

Therefore

(60)

The corresponding spatial modes are

(61)

and, for

(62)

(63)

12 Important low-order Bessel zeros

The first few positive zeros are approximately

α01	= 2.40483,	(64)
α11	= 3.83171,	(65)
α21	= 5.13562,	(66)
α02	= 5.52008,	(67)
α31	= 6.38016,	(68)
α12	= 7.01559.	(69)

Unlike a rectangular membrane, the spectrum is not generated by a simple square root of two integer squares.

The zero structure of Bessel functions determines the circular spectrum.

13 Fundamental mode

The smallest positive Bessel zero is

(70)

Therefore the fundamental mode is

(71)

Its spatial shape is

(72)

It is axisymmetric and has no interior nodal diameter or interior nodal circle.

The center has maximum magnitude because

(73)

14 Angular nodal structure

For a cosine angular factor,

(74)

when

(75)

These zeros form straight radial lines through the center.

Opposite radial lines belong to the same geometric diameter.

Therefore a mode with angular index

(76)

has

(77)

For

(78)

there are no nodal diameters.

15 Radial nodal structure

For a mode using the nth positive zero

(79)

the radial factor is

(80)

The fixed rim is one zero at

(81)

Before reaching that rim zero, the nth radial mode contains

(82)

interior nodal circles.

Thus:

• n = 1: no interior nodal circles,
• n = 2: one interior nodal circle,
• n = 3: two interior nodal circles.

16 The (1,1) mode

For

(83)

the radial factor is

(84)

and the angular factors are

(85)

or

(86)

There is one nodal diameter.

There are no interior nodal circles.

The two sides of the nodal diameter oscillate with opposite sign.

17 The (2,1) mode

For

(87)

there are two nodal diameters.

The membrane is divided into four angular sectors of alternating sign.

There are no interior nodal circles because

(88)

18 The (0,2) mode

For

(89)

there are no nodal diameters.

There is one interior nodal circle.

The inner disk and outer annular region oscillate with opposite signs.

19 Angular degeneracy

For every

(90)

the two angular functions

(91)

and

(92)

have the same radial equation and the same frequency.

Therefore each

(93)

circular-membrane mode is at least twofold degenerate in the ideal symmetric membrane.

The two independent mode shapes can be written

(94)

and

(95)

20 Rotating the nodal pattern

A linear combination

(96)

can be rewritten as

(97)

Thus changing the coefficients does not change the eigenfrequency.

It rotates the angular nodal pattern.

This is the geometric meaning of the sine-cosine degeneracy.

21 Axisymmetric modes are nondegenerate

For

(98)

the sine solution vanishes identically:

(99)

The only nontrivial angular factor is constant:

(100)

Therefore the

(101)

axisymmetric modes do not have the same twofold angular degeneracy.

22 Example 1: wave speed

Take

(102)

Then

c	=	(103)
	≈ 63.25 m∕s.	(104)

23 Example 2: circular-membrane fundamental

Let the radius be

(105)

For the fundamental,

(106)

Therefore

f01	=	(107)
	≈ 96.83 Hz.	(108)

24 Example 3: the first angular mode

For

(109)

f11	=	(110)
	≈ 154.28 Hz.	(111)

The frequency ratio to the fundamental is

(112)

25 Example 4: first axisymmetric overtone

For

(113)

(114)

Therefore

f02	=	(115)
	≈ 222.26 Hz.	(116)

This mode has one interior nodal circle.

26 Why the circular spectrum is not harmonic

A one-dimensional string fixed at both ends has frequencies proportional to

(117)

Its overtones form exact integer multiples of the fundamental.

For a circular membrane,

(118)

The Bessel zeros are not integer multiples of

(119)

Therefore an ideal drumhead does not have a simple harmonic overtone series.

This is one reason a drumhead spectrum differs strongly from a stretched string spectrum.

27 Low-order spectrum

For a fixed radius and wave speed, the ordering of low modes begins:

(120)

then

(121)

then

(122)

then

(123)

for the first few distinct Bessel zeros.

The actual frequency spacing follows the Bessel-zero spacing.

28 Radial orthogonality

For a fixed angular order

(124)

the radial eigenfunctions obey a weighted orthogonality relation.

For

(125)

(126)

The factor

(127)

is essential.

It comes from the polar area element

(128)

29 Radial normalization

For a zero

(129)

of

(130)

the normalization integral is

(131)

This relation is the circular analogue of the sine-function normalization used for a rectangular membrane.

30 Angular orthogonality

For positive angular indices,

(132)

and

(133)

Also,

(134)

For

(135)

the constant angular factor has

(136)

31 Modal expansion

A general fixed circular-membrane displacement can be expanded in the circular eigenfunctions.

Schematically,

u(r,𝜃,t)	= ∑ n=1∞q 0n(t)J0	(137)
	+ ∑ m=1∞∑ n=1∞	(138)
	× Jm.	(139)

Each modal coordinate satisfies

(140)

Thus the circular membrane is again an infinite collection of independent harmonic oscillators.

32 Axisymmetric initial displacement

If the initial displacement depends only on radius,

(141)

then no angular variation is present.

Only

(142)

modes are required.

The expansion becomes

(143)

This is a Fourier-Bessel expansion.

The rectangular sine series of M06-17 has become a Bessel-function series.

33 Effect of membrane radius

Since

(144)

all mode frequencies scale as

(145)

Doubling the radius halves every ideal circular-membrane frequency.

The dimensionless mode pattern as a function of

(146)

is unchanged.

34 Effect of tension and surface density

Because

(147)

(148)

and

(149)

The same scaling found for the rectangular membrane applies here.

The geometry determines the dimensionless eigenvalues; the material parameters set the speed scale.

35 Breaking circular symmetry

The sine-cosine degeneracy for

(150)

depends on rotational symmetry.

A perturbation such as:

• nonuniform Tension,
• elliptical boundary,
• added mass,
• asymmetric support,

can split the degenerate frequencies.

This is the continuous-system analogue of degeneracy splitting in symmetric finite-dimensional eigenvalue problems.

36 Common mistakes

1. Using the Cartesian Laplacian instead of the polar Laplacian for a circular membrane.
2. Forgetting the (1∕r)u_r term.
3. Forgetting the (1∕r²)u _𝜃𝜃 term.
4. Treating the angular separation constant as an arbitrary real number despite 2π periodicity.
5. Confusing the angular index m with the radial-zero index n.
6. Keeping the singular Y _m solution for a membrane that includes the origin.
7. Setting the fixed-rim condition to J_m(k) = 0 instead of J_m(ka) = 0.
8. Assuming Bessel zeros are integer multiples.
9. Calling α_mn the mth zero of J_n instead of the nth zero of J_m.
10. Counting 2m nodal diameters instead of m diameters.
11. Counting the fixed rim as an interior nodal circle.
12. Forgetting that an nth radial mode has n − 1 interior nodal circles.
13. Assuming m = 0 modes have the same sine-cosine degeneracy as m ≥ 1 modes.
14. Forgetting the Weight factor r in radial orthogonality integrals.
15. Expecting drumhead frequencies to form integer harmonics like an ideal string.

37 Practice exercises

1. Write the two-dimensional Laplacian in polar coordinates and derive the circular-membrane PDE.
2. Substitute u = RΘQ and derive the separated time, angular, and radial equations.
3. Explain why angular periodicity requires integer m.
4. Transform the radial equation using x = kr and identify Bessel’s equation.
5. Explain why regularity at the center excludes Y _m.
6. Starting from u(a,𝜃,t) = 0, derive k_mn = α_mn∕a.
7. Derive the frequency formula f_mn = cα_mn∕(2πa).
8. Determine the number of nodal diameters and interior nodal circles for a (3, 2) mode.
9. Reproduce the fundamental and first-angular-mode numerical frequencies.
10. Calculate the frequency ratio f₁₁∕f₀₁.
11. Explain why the (1, 1) cosine and sine modes are degenerate.
12. Show how a linear combination of the two degenerate angular functions rotates the nodal pattern.
13. Derive the radius scaling of the entire frequency spectrum.
14. Explain the origin of the weight factor r in radial orthogonality.
15. Describe how an axisymmetric initial displacement reduces the expansion to m = 0 Bessel modes.

38 Summary

The circular membrane obeys

(151)

with

(152)

Separation of variables gives

(153)

and

(154)

The regular radial solution is

(155)

The fixed rim requires

(156)

If α_mn is the nth positive zero of J_m, then

(157)

and

(158)

The angular index m gives m nodal diameters.

The radial index n gives n − 1 interior nodal circles.

For m ≥ 1, the sine and cosine angular modes are degenerate.

Circular-membrane normal modes therefore combine Bessel radial structure, periodic angular structure, and harmonic time evolution.

References

References