Acoustic impedance and specific acoustic impedance are measures of the opposition that a system presents to the acoustic flow resulting from an acoustic pressure applied to the system. The SI unit of acoustic impedance is the pascal-second per cubic metre (symbol Pa·s/m<sup>3</sup>), or in the MKS system the rayl per square metre (Rayl/m<sup>2</sup>), while that of specific acoustic impedance is the pascal-second per metre (Pa·s/m), or in the MKS system the rayl (Rayl). There is a close analogy with electrical impedance, which measures the opposition that a system presents to the electric current resulting from a voltage applied to the system.

Mathematical definitions

Acoustic impedance

For a linear time-invariant system, the relationship between the acoustic pressure applied to the system and the resulting acoustic volume flow rate through a surface perpendicular to the direction of that pressure at its point of application is given by:

: <math>p(t) = [R * Q](t),</math>

or equivalently by

: <math>Q(t) = [G * p](t),</math>

where

  • p is the acoustic pressure;
  • Q is the acoustic volume flow rate;
  • <math>*</math> is the convolution operator;
  • R is the acoustic resistance in the time domain;
  • G = R is the acoustic conductance in the time domain (R is the convolution inverse of R).

Acoustic impedance, denoted Z, is the Laplace transform, or the Fourier transform, or the analytic representation of time domain acoustic resistance:

: <math>\rho \frac{\partial^2 \delta}{\partial t^2} = -\frac{\partial p}{\partial x}.</math>

Combining this equation with the previous one yields the one-dimensional wave equation:

: <math>\frac{\partial^2 \delta}{\partial t^2} = c^2 \frac{\partial^2 \delta}{\partial x^2}.</math>

The plane waves

: <math>\delta(\mathbf{r},\, t) = \delta(x,\, t)</math>

that are solutions of this wave equation are composed of the sum of two progressive plane waves traveling along x with the same speed and in opposite ways:

: <math>\delta(\mathbf{r},\, t) = f(x - ct) + g(x + ct)</math>

from which can be derived

: <math>v(\mathbf{r},\, t) = \frac{\partial \delta}{\partial t}(\mathbf{r},\, t) = -c\big[f'(x - ct) - g'(x + ct)\big],</math>

: <math>p(\mathbf{r},\, t) = -\rho c^2 \frac{\partial \delta}{\partial x}(\mathbf{r},\, t) = -\rho c^2 \big[f'(x - ct) + g'(x + ct)\big].</math>

For progressive plane waves:

: <math>

\begin{cases}

p(\mathbf{r},\, t) = -\rho c^2\, f'(x - ct)\\

v(\mathbf{r},\, t) = -c\, f'(x - ct)

\end{cases}

</math>

or

: <math>

\begin{cases}

p(\mathbf{r},\, t) = -\rho c^2\, g'(x + ct)\\

v(\mathbf{r},\, t) = c\, g'(x + ct).

\end{cases}

</math>

Finally, the specific acoustic impedance z is

: <math>z(\mathbf{r},\, s) = \frac{\mathcal{L}[p](\mathbf{r},\, s)}{\mathcal{L}[v](\mathbf{r},\, s)} = \pm \rho c,</math>

: <math>z(\mathbf{r},\, \omega) = \frac{\mathcal{F}[p](\mathbf{r},\, \omega)}{\mathcal{F}[v](\mathbf{r},\, \omega)} = \pm \rho c,</math>

: <math>z(\mathbf{r},\, t) = \frac{1}{2}\!\left[p_\mathrm{a} * \left(v^{-1}\right)_\mathrm{a}\right]\!(\mathbf{r},\, t) = \pm \rho c.</math>

The absolute value of this specific acoustic impedance is often called characteristic specific acoustic impedance and denoted z<sub>0</sub>: (It is possible to have no reflections when the pipe is very long, because of the long time taken for the reflected waves to return, and their attenuation through losses at the pipe wall.

See also

  • Acoustic attenuation
  • Earthquake bomb
  • Impedance analogy
  • Mechanical impedance

References

  • The Wave Equation for Sound
  • What Is Acoustic Impedance and Why Is It Important?