GRE Physics Companion: Separation of Variables, Membranes, and Two-Dimensional Normal Modes

GRE Physics Companion: Separation of Variables, Membranes, and Two-Dimensional Normal Modes

For a uniform membrane,

(1)

For a rectangle fixed on all four edges,

(2)

1 High-value GRE facts

1. membrane tension per unit length has units N∕m.
2. surface mass density has units kg∕m².
3. c = .
4. Both fixed-edge indices begin at one.
5. Φ_mn = sin(mπx∕a) sin(nπy∕b).
6. f_mn = (c∕2).
7. There are m − 1 vertical and n − 1 horizontal interior nodal lines.
8. On a square, (m,n) and (n,m) are degenerate.
9. Distinct modes are orthogonal.
10. Every modal coordinate obeys simple harmonic motion.

Part I: Original GRE-style problems

Problem 1: membrane wave speed

A membrane has

(3)

The wave speed is closest to

1. 4.0 m/s
2. 20.0 m/s
3. 40.0 m/s
4. 63.2 m/s
5. 120 m/s

Problem 2: lowest mode

The lowest nontrivial mode of a rectangle fixed on all four edges is

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

Problem 3: fundamental frequency

For

(4)

the fundamental frequency is closest to

1. 39.5 Hz
2. 52.7 Hz
3. 65.9 Hz
4. 79.1 Hz
5. 105 Hz

Problem 4: nodal lines

A (4, 3) mode has how many interior nodal lines in total?

1. 3
2. 4
3. 5
4. 7
5. 12

Problem 5: square degeneracy

Which pair is necessarily degenerate on a square membrane?

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

Problem 6: tension scaling

If 𝒯 is multiplied by four with everything else fixed, every frequency is multiplied by

1. 1∕4
2. 1∕2
3. 1
4. 2
5. 4

Problem 7: density scaling

If σ is multiplied by four with everything else fixed, every frequency becomes

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

Problem 8: fixed-edge mode shape

Which mode shape satisfies all four fixed boundaries?

1. cos(mπx∕a) cos(nπy∕b)
2. sin(mπx∕a) sin(nπy∕b)
3. exp(mx + ny)
4. x + y
5. cos(mπx∕a) sin(nπy∕b)

Problem 9: orthogonality

The membrane integral of Φ_mnΦ_pq is zero when

1. m = p and n = q
2. m + p = n + q
3. either m≠p or n≠q
4. m = n
5. a = b

Problem 10: nonsquare rectangle

For a≠b, the frequencies f₂₁ and f₁₂

1. must be equal
2. are generally different
3. are both zero
4. differ only if tension changes
5. are independent of geometry

Problem 11: initial displacement coefficient

An A_mn coefficient is obtained by

1. differentiating the mode twice in time
2. projecting the initial displacement onto the mode
3. adding all eigenfrequencies
4. dividing by surface density only
5. evaluating displacement at one corner

Problem 12: modal equation

Each ideal modal coordinate satisfies

1. q_mn = 0
2. q_mn + ω_mn²q _mn = 0
3. q_mn − ω_mn²q _mn = 0
4. q_mn = t²
5. ∇²q _mn = 0

Part II: Complete worked solutions

Solution 1

c	=	(5)
	≈ 63.25 m∕s.	(6)

Answer: (D).

Solution 2

A zero index makes one sine factor vanish everywhere, so the first nontrivial mode is (1, 1). Answer: (D).

Solution 3

f11	=	(7)
	≈ 65.9 Hz.	(8)

Answer: (C).

Solution 4

A (4, 3) mode has 3 vertical and 2 horizontal interior nodal lines, for a total of 5. Answer: (C).

Solution 5

For a square, f_mn depends on m² + n², so interchanging m and n leaves the frequency unchanged. Answer: (B).

Solution 6

Since f_mn ∝, multiplying tension by four multiplies frequency by two. Answer: (D).

Solution 7

Since f_mn ∝ 1∕, multiplying surface density by four halves the frequency. Answer: (B).

Solution 8

The fixed boundaries require sine factors in both coordinates. Answer: (B).

Solution 9

Orthogonality gives

(9)

The integral vanishes whenever either index differs. Answer: (C).

Solution 10

For a≠b, interchanging m and n generally changes

(10)

Answer: (B).

Solution 11

Orthogonality extracts the coefficient by projection:

(11)

Answer: (B).

Solution 12

Each normal coordinate is an independent simple harmonic oscillator:

(12)

Answer: (B).

2 GRE checklist

1. Use surface density, not linear density.
2. Use membrane tension per unit length.
3. Both fixed-edge indices begin at one.
4. Count interior nodal lines as m − 1 and n − 1.
5. On a square, (m,n) and (n,m) are degenerate.
6. Use orthogonality to project initial conditions.
7. Each modal coordinate evolves independently.

References

References