GRE Physics Companion: Forced Wave Equations, Resonant Response, and Modal Green Functions

GRE Physics Companion: Forced Wave Equations, Resonant Response, and Modal Green Functions

A forced wave field expanded in spatial eigenfunctions obeys

(1)

with

(2)

1 High-value GRE facts

1. Modal coupling is a spatial overlap integral.
2. A point source at a modal node does not excite that ideal mode.
3. Undamped harmonic response contains the denominator ω_n² − Ω².
4. Exact undamped resonance produces amplitude proportional to time.
5. Damping gives a finite resonance peak.
6. For the convention used here, Q = ω_n∕(2γ_n).
7. The retarded oscillator Green function contains H(t − t′).
8. The impulse response contains sin[ω_n(t − t′)]∕ω_n.
9. Frequency-domain resonances appear as poles of the modal response.
10. Source location and observation location both enter the full modal Green function.

Part I: Original GRE-style problems

Problem 1: modal projection

For orthogonal modes with norm N_n, the modal force is

1. F_n = N_n ∫ SΦ_n dV
2. F_n = ∫ S dV
3. F_n = (1∕N_n) ∫ SΦ_n dV
4. F_n = ω_nS
5. F_n = ∇²S

Problem 2: point force at a node

A point source is placed exactly at a node of mode n. In the ideal linear model, the coupling to that mode is

1. maximal
2. zero
3. infinite
4. independent of source location
5. proportional only to ω_n

Problem 3: centered fixed-string drive

A point force at the center of a fixed string excites

1. only even modes
2. only odd modes
3. only mode 2
4. all modes equally
5. no modes

Problem 4: nonresonant harmonic response

For

(3)

the steady particular amplitude away from resonance is proportional to

1. 1∕(ω₀ + Ω)
2. 1∕(ω₀² − Ω²)
3. ω₀² − Ω²
4. Ω²
5. t

Problem 5: exact undamped resonance

At exact resonance, the ideal particular solution contains

1. constant amplitude
2. amplitude proportional to t
3. amplitude proportional to 1∕t
4. no oscillation
5. exponential decay

Problem 6: quality factor

For

(4)

the weak-damping quality factor is

1. γ∕ω₀
2. 2γ∕ω₀
3. ω₀∕(2γ)
4. ω₀∕γ²
5. 1∕ω₀

Problem 7: resonance bandwidth

If ω₀ = 1000 rad∕s and Q = 50, the weak-damping full width at half Power is approximately

1. 5 rad/s
2. 10 rad/s
3. 20 rad/s
4. 50 rad/s
5. 100 rad/s

Problem 8: impulse response

The retarded Green function of

(5)

is

1. H(t − t′) cos[ω₀(t − t′)]
2. H(t − t′) sin[ω₀(t − t′)]∕ω₀
3. sin(ω₀t′)
4. 1∕ω₀²
5. exp(+ω₀t)

Problem 9: causality

A retarded Green function must vanish when

1. t > t′
2. t = t′
3. t < t′
4. ω = 0
5. the source and receiver differ

Problem 10: frequency-domain pole

The undamped modal frequency response has poles at

1. Ω = 0 only
2. Ω = ±ω_n
3. Ω = ±iω_n
4. Ω = ω_n∕2
5. no finite frequency

Problem 11: source and receiver factors

In the full modal Green function, one mode contributes a spatial factor proportional to

1. Φ_n(r) + Φ_n(r′)
2. Φ_n(r)Φ_n(r′)
3. ∇Φ_n only
4. λ_n²
5. 1∕N_n only

Problem 12: zero mode

For ω₀ = 0,

(6)

is interpreted by the limit

1. 0
2. 1
3. τ
4. 1∕τ
5. τ²

Part II: Complete worked solutions

Solution 1

Projection and orthogonality give

(7)

Answer: (C).

Solution 2

For a point source,

(8)

At a node this vanishes. Answer: (B).

Solution 3

At the string center,

(9)

which vanishes for even n. Answer: (B).

Solution 4

Substitution of

(10)

gives

(11)

Answer: (B).

Solution 5

At resonance,

(12)

Answer: (B).

Solution 6

For the damping convention 2γq,

(13)

Answer: (C).

Solution 7

(14)

Answer: (C).

Solution 8

The causal impulse response is

(15)

Answer: (B).

Solution 9

Causality requires no response before the source acts:

(16)

Answer: (C).

Solution 10

The denominator is

(17)

which vanishes at

(18)

Answer: (B).

Solution 11

The source coupling contributes Φ_n(r′) and the observation point contributes Φ_n(r). Answer: (B).

Solution 12

Using sin x ≈ x for small x,

(19)

Answer: (C).

2 GRE checklist

1. Project both source and field onto the homogeneous eigenmodes.
2. Check source-mode overlap before declaring a resonance.
3. At exact undamped resonance, replace the divergent algebraic amplitude by the linearly growing solution.
4. Keep the damping convention consistent when using Q or bandwidth.
5. Include the Heaviside factor in a retarded Green function.
6. Check the derivative jump of the impulse response.
7. Treat zero modes by a limit rather than division by zero.

References

References