GRE Physics Companion: Circular Membranes, Bessel Functions, and Radial-Angular Modes

GRE Physics Companion: Circular Membranes, Bessel Functions, and Radial-Angular Modes

For a circular membrane of radius a,

(1)

where α_mn is the nth positive zero of J_m.

The angular index m counts nodal diameters, while the radial index n gives n − 1 interior nodal circles.

1 High-value GRE facts

1. The polar Laplacian contains both (1∕r)u_r and (1∕r²)u _𝜃𝜃.
2. Angular periodicity requires m = 0, 1, 2,….
3. The regular radial solution is J_m(kr).
4. The singular Y _m solution is excluded at the center.
5. The fixed rim requires J_m(ka) = 0.
6. α_mn is the nth positive zero of J_m.
7. f_mn = cα_mn∕(2πa).
8. A mode has m nodal diameters.
9. A mode has n − 1 interior nodal circles.
10. Modes with m ≥ 1 have sine-cosine angular degeneracy.

Part I: Original GRE-style problems

Problem 1: polar wave equation

Which radial term must appear in the two-dimensional polar Laplacian?

1. 2u_r∕r
2. u_r∕r
3. u_r∕r²
4. ru_r
5. no first radial derivative

Problem 2: angular quantization

The requirement

(2)

implies that the angular separation index is

1. any real number
2. any negative real number
3. a nonnegative integer
4. only zero
5. only an odd integer

Problem 3: regular radial solution

For a circular membrane that includes r = 0, the physical radial solution is proportional to

1. Y _m(kr)
2. J_m(kr)
3. exp(kr)
4. sin(kr) for every m
5. 1∕r

Problem 4: fixed-rim condition

For membrane radius a, the allowed wave numbers satisfy

1. J_m(k) = 0
2. Y _m(ka) = 0
3. J_m(ka) = 0
4. sin(ka) = 0 for every mode
5. ka = m + n

Problem 5: fundamental mode

The fundamental fixed circular-membrane mode is

1. (0, 0)
2. (0, 1)
3. (1, 0)
4. (1, 1)
5. (2, 1)

Problem 6: nodal diameters

A mode with angular index m = 3 has

1. no nodal diameters
2. one nodal diameter
3. two nodal diameters
4. three nodal diameters
5. six nodal diameters

Problem 7: nodal circles

A mode with radial index n = 3 has how many interior nodal circles?

1. 0
2. 1
3. 2
4. 3
5. 4

Problem 8: degeneracy

For an ideal circular membrane, the pair

(3)

and

(4)

is degenerate when

1. only m = 0
2. m ≥ 1
3. only n = 1
4. only m = n
5. never

Problem 9: radius scaling

If the membrane radius doubles with all material parameters unchanged, every mode frequency becomes

1. four times as large
2. twice as large
3. unchanged
4. one half as large
5. one fourth as large

Problem 10: fundamental frequency

A membrane has

(5)

and

(6)

The fundamental frequency is closest to

1. 38.7 Hz
2. 63.3 Hz
3. 96.8 Hz
4. 154 Hz
5. 241 Hz

Problem 11: radial orthogonality

The radial orthogonality integral contains which Weight?

1. 1
2. r
3. r²
4. 1∕r
5. 𝜃

Problem 12: overtone structure

Compared with an ideal fixed string, circular-membrane overtones are generally

1. exact integer multiples of the fundamental
2. controlled by Bessel zeros and not exact integer multiples
3. all degenerate
4. independent of radius
5. independent of tension

Part II: Complete worked solutions

Solution 1

The two-dimensional polar Laplacian is

(7)

Answer: (B).

Solution 2

Single-valued periodic displacement requires an integer number of angular oscillations in one full revolution. Answer: (C).

Solution 3

Y _m is singular at the origin, while J_m is regular. Answer: (B).

Solution 4

The fixed rim requires

(8)

Answer: (C).

Solution 5

The smallest positive Bessel zero is α₀₁, so the fundamental is (0, 1). Answer: (B).

Solution 6

The angular index directly counts nodal diameters. Answer: (D).

Solution 7

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

(9)

Answer: (C).

Solution 8

For every m ≥ 1, the sine and cosine angular functions are linearly independent but have the same eigenvalue. Answer: (B).

Solution 9

Since

(10)

doubling radius halves the frequencies. Answer: (D).

Solution 10

f01	=	(11)
	≈ 96.8 Hz.	(12)

Answer: (C).

Solution 11

The polar area element is

(13)

so radial orthogonality uses the weight r. Answer: (B).

Solution 12

Circular-membrane frequencies are proportional to Bessel zeros, which are not integer multiples of the fundamental zero. Answer: (B).

2 GRE checklist

1. Write the polar Laplacian correctly.
2. Keep the angular and radial indices conceptually separate.
3. Exclude Y _m at the center.
4. Use zeros of J_m for a fixed rim.
5. Count nodal diameters with m.
6. Count interior nodal circles with n − 1.
7. Remember the twofold angular degeneracy for m ≥ 1.
8. Use f ∝ 1∕a for radius scaling.

References

References