Acoustic impedances
Fondamental : Calculate the acoustic impedance
Consider a harmonic progressive plane sound wave (HPPSW) propagating in ascending order \(x>0\).
The speed and pressure may be written :
\(\underline v =A\;exp(j(\omega t-kx))\)
And :
\(\underline p=B\;exp(j(\omega t-kx))\)
With the dispersion relation :
\(k=\frac{\omega}{c}\)
Where \(c\) is the speed of sound waves.
The following equation (see course sheet on sound waves) :
\(\frac{{\partial p}}{{\partial t}} = - \frac{1}{{\chi _S }}div(\vec v)=- \frac{1}{{\chi _S }}\frac{\partial v}{\partial x}\)
allows to determine a relation between the speed and the acoustic pressure.
Indeed :
\(j\omega \underline p=- \frac{1}{{\chi _S }}(-jk)\underline v\)
Is :
\(\underline p = \frac{1}{{c\chi _S }}\underline v=\mu_0 c \;\underline v\)
Attention : Definition of the acoustic impedance
For HPPSW moving in the direction of \(x>0\) :
\(\underline p = z\underline v\)
With :
\(z=\mu_0 c\)
\(z\) is called acoustic impedance.
For HPPSW moving in the direction of \(x<0\) :
\(\underline p = -z\underline v\)
Remarque : Other definition of acoustic impedance
The acoustic impedance can be defined according to :
\(\underline p = Z (S\underline v)\)
Where \(S\) is the cross section of the sound conduct.
This definition is similar to that of the electric resistance : the flow rate \(Sv\) is comparable to the electrical current (charge flow rate).
The acoustic impedance is defined as :
\(Z=\frac{\mu_0c}{S}=\frac{z}{S}\)
See also the lesson about "Thermal resistance".