GRE Physics Companion: Beats, Interference, and Coherence

GRE Physics Companion: Beats, Interference, and Coherence

For two coherent waves,

(1)

For nearby frequencies,

(2)

For path difference,

(3)

1 High-value GRE facts

1. Superpose amplitudes before squaring for coherent waves.
2. Equal amplitudes and zero phase difference give maximum reinforcement.
3. Equal amplitudes and phase difference π give complete cancellation.
4. Δϕ = (2π∕λ)Δr + ϕ₀.
5. In-phase sources constructively interfere at Δr = mλ.
6. In-phase sources destructively interfere at Δr = (m + 1∕2)λ.
7. f_beat = |f₁ − f₂|.
8. T_beat = 1∕f_beat.
9. Unequal beat amplitudes give nonzero minima.
10. Narrow bandwidth implies long coherence time.

Part I: Original GRE-style problems

Problem 1: equal-amplitude constructive interference

Two coherent waves each have amplitude

(4)

and zero phase difference. The resultant amplitude is

1. 0
2. A∕2
3. A
4. 2A
5. 4A

Problem 2: destructive interference

Two equal-amplitude coherent waves cancel completely when their phase difference is

1. 0
2. π∕2
3. π
4. 2π
5. 4π

Problem 3: equal-source intensity maximum

Each of two equal coherent sources individually produces intensity

(5)

At complete constructive interference, the total intensity is

1. I₀
2. 2I₀
3. 3I₀
4. 4I₀
5. 8I₀

Problem 4: path difference

Two in-phase sources have wavelength

(6)

Complete destructive interference occurs at path difference

1. 0
2. λ∕4
3. λ∕2
4. λ
5. 2λ

Problem 5: beat frequency

Two tones have frequencies

(7)

and

(8)

The beat frequency is

1. 2 Hz
2. 4 Hz
3. 8 Hz
4. 442 Hz
5. 884 Hz

Problem 6: beat period

If the beat frequency is

(9)

the beat period is

1. 0.05 s
2. 0.10 s
3. 0.20 s
4. 1.0 s
5. 5.0 s

Problem 7: unequal beat amplitudes

Two nearby-frequency waves have amplitudes

(10)

and

(11)

The minimum beat-envelope amplitude is

1. 0
2. A
3. 2A
4. 5A
5. 8A

Problem 8: coherent visibility

Perfectly coherent sources have intensities

(12)

The fringe visibility is

1. 0.20
2. 0.40
3. 0.60
4. 0.80
5. 1.00

Problem 9: incoherent averaging

If the relative phase of two sources varies randomly and rapidly compared with the averaging time, the average interference term is approximately

1. maximum
2. minimum
3. zero
4. infinite
5. equal to I₁I₂

Problem 10: coherence-time estimate

A source has bandwidth

(13)

Using

(14)

the coherence time is approximately

1. 0.001 s
2. 0.010 s
3. 0.10 s
4. 1.0 s
5. 100 s

Problem 11: coherence length

For the source in Problem 10, take wave speed

(15)

The coherence length is approximately

1. 0.03 m
2. 0.30 m
3. 3.0 m
4. 30 m
5. 300 m

Problem 12: partial coherence

Two equal-intensity fields have

(16)

Their maximum possible fringe visibility is

1. 0.30
2. 0.49
3. 0.70
4. 1.00
5. 1.40

Part II: Complete worked solutions

Solution 1

In phase, the amplitudes add:

(17)

Answer: (D).

Solution 2

For equal amplitudes, complete destructive interference occurs at

(18)

The simplest choice is

(19)

Answer: (C).

Solution 3

Constructive interference doubles the amplitude. Since intensity is proportional to amplitude squared,

(20)

Answer: (D).

Solution 4

For in-phase sources, destructive interference occurs at

(21)

The smallest positive value is

(22)

Answer: (C).

Solution 5

(23)

Answer: (B).

Solution 6

(24)

Answer: (C).

Solution 7

For unequal amplitudes,

(25)

Answer: (C).

Solution 8

𝒱	=	(26)
	=	(27)
	= 0.60.	(28)

Answer: (C).

Solution 9

Rapid random phase variation gives

(29)

Thus the stable interference term averages away. Answer: (C).

Solution 10

(30)

Answer: (B).

Solution 11

(31)

Answer: (C).

Solution 12

For equal intensities,

(32)

Therefore

(33)

Answer: (C).

2 GRE checklist

1. Add coherent amplitudes first, then determine intensity.
2. Use phase difference, not source separation alone.
3. Convert path difference to phase using 2π∕λ.
4. Beat frequency is the absolute frequency difference.
5. The average of the two frequencies is not the beat frequency.
6. Unequal beat amplitudes give incomplete minima.
7. Random phase variation removes the stable interference term.
8. Use coherence estimates only at the level justified by the bandwidth convention.

References

References