GRE Physics Companion: Acoustic Waves, Sound Intensity, and Acoustic Impedance

GRE Physics Companion: Acoustic Waves, Sound Intensity, and Acoustic Impedance

For a plane acoustic wave in a fluid,

(1)

The pressure and particle-velocity amplitudes satisfy

(2)

for positive-direction propagation.

The average intensity is

(3)

1 High-value GRE facts

1. c² = (∂p∕∂ρ) _s.
2. For an ideal gas, c = .
3. z₀ = ρ₀c.
4. For a right-moving plane wave, p′ = z₀u.
5. ⟨I⟩ = p_rms²∕z ₀ = z₀u_rms².
6. L_I = 10 log ₁₀(I∕I_ref).
7. L_p = 20 log ₁₀(p_rms∕p_ref).
8. Doubling ideal free-field distance lowers level by about 6 dB.
9. Pressure reflection at a fluid interface is (z₂ − z₁)∕(z₁ + z₂).
10. A rigid wall is a pressure antinode and particle-velocity node.

Part I: Original GRE-style problems

Problem 1: characteristic impedance

For a progressive plane wave in a fluid, the characteristic specific acoustic impedance is

1. ρ₀∕c
2. ρ₀c
3. c∕ρ₀
4. ρ₀c²
5. 1∕(ρ₀c)

Problem 2: intensity

The average intensity of a harmonic plane wave is

1. p_rmsρ₀c
2. p_rms²∕(ρ ₀c)
3. p_rms∕(ρ₀c)
4. ρ₀c∕p_rms²
5. independent of pressure

Problem 3: right-moving pressure and velocity

For a right-moving progressive plane wave,

1. pressure and particle velocity differ by 90^∘
2. pressure and particle velocity are in phase
3. particle velocity is always zero
4. pressure is always zero
5. pressure and particle velocity have unrelated frequencies

Problem 4: intensity level

Increasing intensity by a factor of

(4)

changes the sound intensity level by

1. 2 dB
2. 10 dB
3. 20 dB
4. 40 dB
5. 100 dB

Problem 5: pressure level

Increasing RMS pressure by a factor of

(5)

at fixed impedance changes sound pressure level by

1. 3 dB
2. 6 dB
3. 10 dB
4. 20 dB
5. 100 dB

Problem 6: distance doubling

For an ideal isotropic point source in a lossless free field, doubling distance changes the intensity level by approximately

1. +6 dB
2. −6 dB
3. +3 dB
4. −3 dB
5. 0 dB

Problem 7: pressure reflection

At Normal incidence from medium 1 to medium 2, the pressure reflection coefficient is

1. (z₁ − z₂)∕(z₁ + z₂)
2. (z₂ − z₁)∕(z₁ + z₂)
3. 2z₁∕(z₁ + z₂)
4. 2z₂∕(z₁ + z₂)
5. z₁z₂

Problem 8: matched interface

If

(6)

then the pressure reflection coefficient is

1. −1
2. +1
3. 0
4. 1∕2
5. infinite

Problem 9: rigid wall

At an ideal rigid wall,

1. pressure is a node and particle velocity an antinode
2. pressure is an antinode and particle velocity a node
3. both pressure and particle velocity are nodes
4. both are antinodes
5. no reflection occurs

Problem 10: air intensity

Take

(7)

and

(8)

The intensity is closest to

1. 9.7 × 10⁻⁷ W∕m²
2. 9.7 × 10⁻⁵ W∕m²
3. 4.1 × 10⁻² W∕m²
4. 0.200 W∕m²
5. 413 W∕m²

Problem 11: pressure level

For

(9)

and

(10)

the sound pressure level is

1. 20 dB
2. 40 dB
3. 60 dB
4. 80 dB
5. 100 dB

Problem 12: two equal incoherent sources

Two independent incoherent sources each produce intensity

(11)

Together they increase the level by approximately

1. 1 dB
2. 3 dB
3. 6 dB
4. 10 dB
5. 20 dB

Part II: Complete worked solutions

Solution 1

For a progressive plane wave,

(12)

Answer: (B).

Solution 2

Using RMS pressure,

(13)

Answer: (B).

Solution 3

For positive-direction propagation,

(14)

Thus pressure and particle velocity are in phase. Answer: (B).

Solution 4

(15)

Answer: (C).

Solution 5

(16)

Answer: (D).

Solution 6

Ideal spherical spreading gives

(17)

Doubling distance gives

(18)

Answer: (B).

Solution 7

Pressure and normal particle-velocity continuity give

(19)

Answer: (B).

Solution 8

If

(20)

then

(21)

Answer: (C).

Solution 9

A rigid wall requires

(22)

Thus particle velocity is a node and pressure is an antinode. Answer: (B).

Solution 10

First,

(23)

Then

⟨I⟩	=	(24)
	≈ 9.7 × 10−5 W∕m2.	(25)

Answer: (B).

Solution 11

Lp	= 20 log 10	(26)
	= 20 log 10(104)	(27)
	= 80 dB.	(28)

Answer: (D).

Solution 12

The total intensity is

(29)

Thus

(30)

Answer: (B).

2 GRE checklist

1. Distinguish wave speed from particle velocity.
2. Use ρ₀c as the plane-wave specific acoustic impedance.
3. Use RMS pressure in the standard intensity formula.
4. Remember the different logarithmic factors: 10 for intensity, 20 for pressure amplitude.
5. Apply pressure and normal particle-velocity continuity at a fluid interface.
6. Use the acoustic pressure reflection sign, not the string displacement sign.
7. A rigid wall is a pressure antinode and velocity node.
8. Add physical intensities before converting independent-source results back to decibels.

References

References