Wave Impedance
Not much has been said so far about various factors that might determine the amplitude of a propagating wave. For a given displacement of a sound source, how much air compression is achieved in the wavefront? The answer involves both the elastic and inertial properties of the medium, combined in the concept of wave impedance.
Physical Basis of Wave Impedance
For a given displacement of a piston (or vocal fold surface), the amount of pressure generated depends on:
- Stiffness (elasticity) of the medium: The stiffer the medium, the greater the pressure peak will be for a given displacement
- Inertia (mass) of the medium: The medium resists rapid movement—air mass is being accelerated and decelerated
Momentum and Inertial Pressure
Air particles are continually being accelerated and decelerated by oscillating surfaces. Their momentum is being reversed periodically. This momentum creates inertial pressure as air particles move past their equilibrium position and crowd their neighbors.
The key insight is that the combined elastic and inertial pressure depends on the rate of change of displacement—that is, the speed of the particles. The greater the speed, the more momentum is imparted and the more pressure is exerted onto neighboring particles.
Definition of Wave Impedance
The wave impedance z quantifies the amount of pressure p that is produced for a given speed of the air particles v:
$$z = \frac{p}{v}$$
where:
- z = wave impedance (acoustic ohms, Pa·s/m, or rayls)
- p = acoustic pressure (Pa)
- v = particle velocity (m/s)
This relationship is analogous to electrical impedance (voltage/current) and has similar properties.
Relationship to Sound Velocity
In the development of sound velocity, we reasoned that propagation speed c should depend on both inertial and elastic properties of the medium. Now we have reasoned that wave impedance should depend on these same properties. It is not unreasonable to assume that wave impedance relates to sound velocity.
In standard acoustic theory, it is shown that:
$$z = \rho c$$
where:
- ρ (rho) = medium density (kg/m³)
- c = sound propagation velocity (m/s)
Typical Values
Air (at standard conditions):
- ρ ≈ 1.2 kg/m³
- c ≈ 343 m/s
- z ≈ 412 Pa·s/m (or rayls)
Water:
- ρ ≈ 1000 kg/m³
- c ≈ 1,480 m/s
- z ≈ 1.48 × 10⁶ rayls (about 3,600 times greater than air)
Soft tissue:
- ρ ≈ 1,050 kg/m³
- c ≈ 1,540 m/s
- z ≈ 1.62 × 10⁶ rayls
Implications for Reflection
The concept of wave impedance is crucial for understanding reflections at interfaces between media. When a wave encounters a boundary between two media with different impedances, the amount reflected depends on the impedance mismatch.
Impedance Mismatch
The reflection coefficient r at an interface is given by:
$$r = \frac{z_2 - z_1}{z_2 + z_1} = \frac{\rho_2 c_2 - \rho_1 c_1}{\rho_2 c_2 + \rho_1 c_1}$$
where:
- z₁ = impedance of medium 1
- z₂ = impedance of medium 2
This relationship shows that:
- Large impedance difference → Large reflection
- Small impedance difference → Small reflection
- Equal impedances → No reflection (perfect transmission)
Clinical Relevance: Air-Tissue Interface
The large impedance mismatch between air and tissue has important consequences:
At the mouth opening (air-to-air):
- Small impedance change
- Minimal reflection (for plane waves)
At vocal fold surfaces (air-to-tissue):
- z_tissue/z_air ≈ 3,600
- Nearly complete reflection
- Very little sound enters the tissue
At the glottis (subglottal to supraglottal):
- Impedance changes due to area change
- Creates reflection important for vocal fold oscillation
Particle Velocity versus Sound Velocity
It is critical to distinguish between two types of velocity:
Particle Velocity (v)
- Local oscillatory motion of air particles
- Typically very small: micrometers per second to millimeters per second
- Varies with position in a standing wave
- Related to pressure by impedance: p = zv
Sound Velocity (c)
- Propagation speed of the pressure disturbance
- Approximately 343 m/s in air (much faster than particle motion)
- Constant for a given medium (depends on temperature)
- Does not depend on amplitude or frequency of sound
Important Distinction: Sound propagates at ~343 m/s, but individual air particles oscillate with velocities typically less than 0.1 m/s. The wave pattern moves rapidly through the medium, but particles themselves move very little from their equilibrium positions.
Characteristic Impedance versus Input Impedance
Characteristic Impedance
The quantity z = ρc is the characteristic impedance of the medium—the impedance of a freely propagating plane wave in an infinite medium. It depends only on medium properties.
Input Impedance
In confined spaces like the vocal tract, the input impedance at a given location depends on:
- Characteristic impedance of the medium
- Geometry of the space (cross-sectional area)
- Boundary conditions (open, closed, or complex termination)
- Frequency of oscillation
Input impedance will be explored in detail in Chapter 6 when discussing vocal tract resonance.
Summary
Wave impedance quantifies the pressure-to-velocity ratio in propagating acoustic waves, defined as z = p/v. For plane waves in an infinite medium, wave impedance equals ρc (the product of density and sound velocity). This characteristic impedance determines how much pressure is generated for a given particle motion and plays a crucial role in reflection phenomena at media interfaces. The large impedance mismatch between air and tissue (approximately 3,600:1) means that sound reflects strongly at air-tissue boundaries, with minimal energy transmission into tissue. Understanding wave impedance is essential for analyzing vocal tract acoustics and sound radiation.
Key Takeaways
- ✅ Wave impedance (z) is the ratio of acoustic pressure to particle velocity: z = p/v
- ✅ For plane waves in infinite media, characteristic impedance is z = ρc
- ✅ Air has impedance ~412 rayls; tissue has impedance ~1.6 × 10⁶ rayls (about 3,600 times greater)
- ✅ Impedance mismatch at interfaces determines reflection: large mismatch → strong reflection
- ✅ Sound velocity (~343 m/s) is much greater than typical particle velocities (~0.01-0.1 m/s)
- ✅ Wave impedance depends on both elastic (stiffness) and inertial (mass) properties of the medium
- ✅ The air-tissue impedance mismatch causes nearly complete reflection at vocal fold surfaces
Related Topics
Further Reading
- Kinsler, L., & Frey, A. (1962). Fundamentals of Acoustics (2nd ed.). New York: Wiley.
- Morse, P. M. (1947). Vibration and Sound. New York: McGraw-Hill.
- Beranek, L. L. (1954). Acoustics. New York: McGraw-Hill.