Acoustic Impedance Reflection

Worked Example

Consider a plane sound wave incident on a boundary between two lossless media. The first medium has specific acoustic impedance Z₁ = 4, and the second has Z₂ = 1. The incident pressure amplitude is A = 1. Let us find the reflection and transmission coefficients and amplitudes.

1. Calculate the pressure reflection coefficient, r_p:
   r_p = (Z₂ - Z₁) / (Z₂ + Z₁) = (1 - 4) / (1 + 4) = -3 / 5 = -0.6
2. Calculate the pressure transmission coefficient, t_p:
   t_p = 2Z₂ / (Z₁ + Z₂) = 2(1) / (4 + 1) = 2 / 5 = 0.4
3. Calculate the energy reflection coefficient, R:
   R = r_p² = (-0.6)² = 0.36
4. Calculate the energy transmission coefficient, T:
   T = (Z₁ / Z₂) * t_p² = (4 / 1) * (0.4)² = 4 * 0.16 = 0.64
5. Confirm energy conservation:
   R + T = 0.36 + 0.64 = 1.0
6. Calculate the reflected and transmitted amplitudes:
   Reflected amplitude = r_p * A = -0.6 * 1 = -0.6
   Transmitted amplitude = t_p * A = 0.4 * 1 = 0.4

Experiment

Explore how varying impedances changes the reflected and transmitted waves in the Acoustic Impedance Reflection module.

1. Set Specific impedance 1 to 1 and Specific impedance 2 to 4.
2. Set Incident pressure to 1.
3. Observe that the pressure reflection is 0.6 and the pressure transmission is 1.6. Notice that the transmitted amplitude can exceed the incident amplitude without violating energy conservation, because the impedance ratio scales the energy.
4. Swap the impedances: set Specific impedance 1 to 4 and Specific impedance 2 to 1.
5. Observe that the pressure reflection becomes -0.6, indicating a phase inversion of the reflected wave. The energy reflection remains 0.36.

Limitations

This model assumes normally incident plane waves traveling through lossless media. It does not account for oblique incidence, which would introduce angular dependence and phenomena like total internal reflection. The model also assumes a continuous, perfectly sharp interface between the two media. For boundaries that are rough or comparable in size to the acoustic wavelength, scattering would occur, making this simple planar reflection model insufficient. Finally, the coefficients are valid for specific acoustic impedance (pressure / particle velocity) rather than volume-flow tube impedance.

References

The physics of acoustic impedance and energy transmission at an interface are standard in acoustics literature. For an interactive explanation, refer to:

• UNSW Physics Animations: Acoustic impedance, intensity and power: This resource supports the definitions of specific acoustic impedance, reflection/transmission coefficients, and the distinction between pressure amplitude and energy.