Acoustic Impedance and Reflection Quiz
Acoustic impedance is the quantity that decides how much of a sound wave crosses a boundary and how much turns back. It converts the qualitative idea that “sound reflects” into a calculable ratio: the pressure reflection coefficient depends only on the impedances of the two media, not on the wave’s frequency or amplitude.
This quiz tests the core relations for a plane wave in a medium of density ##\rho## and sound speed ##v##:
- Specific acoustic impedance: ##Z = \rho v## (unit: rayl, ##\text{Pa}\cdot\text{s}/\text{m}##)
- Pressure reflection coefficient: ##R = \dfrac{Z_2 – Z_1}{Z_2 + Z_1}##
- Pressure transmission coefficient: ##T = \dfrac{2Z_2}{Z_1 + Z_2}##
- Intensity coefficient: ##\alpha = \dfrac{4Z_1Z_2}{(Z_1+Z_2)^2}##
Typical values: air ##Z \approx 415## rayl, water ##Z \approx 1.48\times10^{6}## rayl, steel ##Z \approx 4.5\times10^{7}## rayl. These numbers explain why sound barely enters a pool from the air above it, why ultrasound coupling gel is used, and why a stethoscope’s chest piece is matched to tissue rather than to air.
Instructions: Attempt all 15 questions. Each has exactly one correct option. Use the explanations to consolidate the physics of impedance mismatch, impedance matching, and reflection at boundaries.
Q1. For a travelling plane wave in a medium of density ##\rho## and sound speed ##v##, the specific acoustic impedance is given by:
For a travelling plane wave the ratio of acoustic pressure to particle velocity is real and equals ##\rho v##, so ##Z = \rho v##.
Q2. The SI unit of specific acoustic impedance is the rayl. Its dimensions are:
Pressure has dimensions ##ML^{-1}T^{-2}## and velocity ##LT^{-1}##, so their ratio gives ##ML^{-2}T^{-1}##, i.e. mass per unit area per unit time.
Q3. At normal incidence between media 1 and 2, the pressure amplitude reflection coefficient is:
The pressure reflection coefficient is ##R = (Z_2 - Z_1)/(Z_2 + Z_1)##; the expression ##2Z_2/(Z_1+Z_2)## is the transmission coefficient.
Q4. The pressure transmission coefficient at normal incidence between media 1 and 2 is:
The pressure transmission coefficient is ##T = 2Z_2/(Z_1 + Z_2)##, which follows from continuity of pressure and particle velocity at the boundary.
Q5. The intensity transmission coefficient at normal incidence is:
The intensity coefficient is ##\alpha = 4Z_1Z_2/(Z_1+Z_2)^2##; the expression ##(Z_2-Z_1)^2/(Z_2+Z_1)^2## is the intensity reflection coefficient ##R^2##.
Q6. The pressure reflection coefficient ##R## at a boundary depends on:
The lesson stresses that ##R## depends only on the impedances of the two media, not on the wave's frequency or amplitude.
Q7. Air at 20 °C has ##Z \approx 415## rayl and water has ##Z \approx 1.48\times10^{6}## rayl. The fraction of incident intensity reflected at an air–water interface is approximately:
Because the impedance mismatch is enormous, the air–water interface reflects about 99.9% of incident intensity, which is why sound barely enters a pool from the air above it.
Q8. Ultrasound coupling gel is used between the transducer and the skin mainly because:
The gel acts as an impedance-matching medium, reducing the strong reflection that would otherwise occur at a transducer–air–skin interface.
Q9. A stethoscope's chest piece is matched to tissue rather than to air because:
Impedance matching to tissue minimizes reflection at the chest wall so that body sounds are transmitted efficiently into the stethoscope.
Q10. For a travelling plane wave, the specific acoustic impedance is real. This means that pressure and particle velocity:
For a travelling plane wave the ratio ##p/u## is real, so pressure and particle velocity oscillate in phase.
Q11. For standing waves or near-field sources, the acoustic impedance becomes complex. The reactive part of the impedance:
In standing waves or near-field situations the impedance is complex, and its reactive part stores energy rather than transporting it.
Q12. In air, ##v \propto \sqrt{T}## while ##\rho \propto 1/T##. Therefore the acoustic impedance of air varies with absolute temperature as:
Multiplying the two dependences gives ##Z = \rho v \propto (1/T)(\sqrt{T}) = 1/\sqrt{T}##, so impedance falls slowly as air warms.
Q13. The acoustic analogue of electrical resistance is acoustic impedance, with pressure playing the role of:
In the acoustic–electrical analogy, pressure plays the role of voltage and volume velocity the role of current, so impedance is analogous to resistance.
Q14. A high-impedance medium, compared with a low-impedance one, requires:
Impedance measures the pressure needed to drive a given particle velocity, so a high-impedance medium demands large pressure for small displacement.
Q15. If the impedances of the two media are equal (##Z_1 = Z_2##), then at the boundary:
With ##Z_1 = Z_2##, ##R = (Z_2-Z_1)/(Z_2+Z_1) = 0## and ##\alpha = 4Z_1Z_2/(Z_1+Z_2)^2 = 1##, so the wave passes without reflection.