GRE Physics Companion: Fourier Analysis, Wave Packets, and Dispersion

GRE Physics Companion: Fourier Analysis, Wave Packets, and Dispersion

A wave packet can be represented as

(1)

The central velocities are

(2)

1 High-value GRE facts

1. A localized disturbance requires a range of wave numbers.
2. v_p = ω∕k.
3. v_g = dω∕dk.
4. A linear dispersion relation gives v_p = v_g.
5. Nonzero d²ω∕dk² produces leading-order narrowband spreading.
6. For a Gaussian amplitude, narrow spatial width means broad spectral width.
7. A real field has conjugate-symmetric complex Fourier coefficients.
8. Linear propagation changes spectral phase even when spectral magnitude is unchanged.
9. Parseval’s relation equates quadratic content in real and spectral representations.
10. A discrete normal-mode sum becomes a Fourier integral in the infinite-domain limit.

Part I: Original GRE-style problems

Problem 1: phase velocity

For a Fourier component with frequency

(3)

and wave number

(4)

the phase velocity is

1. dω∕dk
2. ω∕k
3. k∕ω
4. d²ω∕dk²
5. always zero

Problem 2: group velocity

For a narrowband packet, the envelope velocity is

1. ω∕k
2. k∕ω
3. dω∕dk
4. d²ω∕dk²
5. ωk

Problem 3: nondispersive relation

Which dispersion relation is nondispersive for positive

(5)

1. ω = ck
2. ω = αk²
3. ω =
4. ω = ω₀ + αk²
5. ω = 1∕k

Problem 4: Gaussian localization

If a Gaussian amplitude has characteristic spectral width

(6)

its characteristic spatial amplitude width is

1. σ_k
2. 1∕σ_k
3. σ_k²
4. 2πσ_k
5. independent of σ_k

Problem 5: two-wave envelope

For two nearby components, the finite-difference envelope velocity is

1. Δk∕Δω
2. Δω∕Δk
3. ω₁∕k₂
4. ω₂∕k₁
5. zero

Problem 6: spectral evolution

In an ideal linear time-invariant dispersive system, the spectrum evolves as

1. A(k,t) = A(k, 0)e^−iω(k)t
2. A(k,t) = A(k, 0)e^−ikt
3. A(k,t) = A(k, 0) + ωt
4. A(k,t) = 0
5. A(k,t) = A(k, 0)e^+k2t necessarily

Problem 7: packet spreading

Leading-order narrowband spreading is controlled by

1. ω
2. k
3. dω∕dk
4. d²ω∕dk²
5. only the wave amplitude

Problem 8: lattice phase velocity

For the monatomic chain example with

(7)

and

(8)

the phase velocity is closest to

1. 1.00 m/s
2. 1.41 m/s
3. 1.80 m/s
4. 2.00 m/s
5. 2.83 m/s

Problem 9: lattice group velocity

For the same lattice state, the group velocity is closest to

1. 1.00 m/s
2. 1.41 m/s
3. 1.80 m/s
4. 2.00 m/s
5. 2.83 m/s

Problem 10: real field symmetry

For a real periodic field with complex Fourier coefficients,

1. c_−n = c_n
2. c_−n = −c_n
3. c_−n = c_n^∗
4. c_−n = 0
5. c_n must be real for every n

Problem 11: Parseval meaning

Parseval’s relation states that

1. only the zero-frequency component matters
2. a quadratic norm can be computed equivalently in real or spectral space
3. every waveform has one Fourier component
4. phase and group velocities are equal
5. dispersion is impossible

Problem 12: Gaussian width after dispersion

A Gaussian amplitude has

(9)

After

(10)

its quadratic-dispersion width is closest to

1. 0.250 m
2. 0.500 m
3. 0.707 m
4. 0.943 m
5. 1.60 m

Part II: Complete worked solutions

Solution 1

A constant phase satisfies

(11)

Therefore

(12)

Answer: (B).

Solution 2

The narrowband envelope velocity is

(13)

Answer: (C).

Solution 3

If

(14)

then

(15)

and

(16)

Answer: (A).

Solution 4

For the amplitude convention used here,

(17)

Thus

(18)

Answer: (B).

Solution 5

The slowly varying envelope has phase

(19)

Therefore

(20)

Answer: (B).

Solution 6

Each Fourier component acquires its own propagation phase:

(21)

Answer: (A).

Solution 7

The first derivative moves the envelope. The second derivative produces leading-order narrowband spreading. Therefore the controlling quantity is

(22)

Answer: (D).

Solution 8

At

(23)

(24)

and

(25)

Hence

(26)

Answer: (C).

Solution 9

(27)

Answer: (B).

Solution 10

A real field requires conjugate symmetry:

(28)

Answer: (C).

Solution 11

Parseval’s relation equates a quadratic norm in the two representations, up to the transform-convention factor. Answer: (B).

Solution 12

σx(t)	= 0.500	(29)
	≈ 0.943 mm.	(30)

Answer: (D).

2 GRE checklist

1. Decide whether the problem concerns one Fourier component or a packet.
2. Use v_p = ω∕k for a phase point.
3. Use v_g = dω∕dk for a narrowband envelope.
4. Check whether ω(k) is linear before assuming nondispersive propagation.
5. Use ω′′(k₀) to diagnose leading-order packet spreading.
6. Remember that narrow spatial localization requires broad spectral support.
7. Keep the Fourier-transform convention consistent.
8. On a dispersion graph, distinguish the secant slope from the tangent slope.

References

References