Mechanical waves

We study the reflection and transmission of a wave at normal incidence.

Medium \((1)\) and \((2)\) have different physical characteristics a priori but the pressure at equilibrium is identical, denoted \(P_0\).

A surface which delimits two different media is called "acoustic diopter".

We study the reflection and transmission of plane sound waves at the connection of two pipe of cross sections \(S_1\) and \(S_2\), separated between two materials of acoustic impedances \(z_1\) for \(x<0\) and \(z_2\) for \(x>0\).

Recall that the acoustic impedance is defined from :

\(z = \frac {p}{v}\)

We note :

\(z_1=\mu_1c_1\;\;\;\;\;and\;\;\;\;\;z_2=\mu_2c_2\)

Continuity of the volume flow rate at the interface :

It is considered that the two fluids do not mix, so that the interface is an impermeable membrane.

\(V\) is a volume in the connection area between the two cross sections \(\Sigma_1\) and \(\Sigma_2\), located in \(x=-\varepsilon\) and \(x=+\varepsilon\).

One can imagine the volume (\(V\)) which oscillates to both sides of the separation membrane and directly write the conservation of volume, resulting in the conservation of volume flow rate :

\({v_1}{S_1} = {v_2}{S_2}\)

In the particular case where the two cross sections are equal, then there is continuity of the speed of both sides of the membrane.

Continuity of pressure :

In the acoustic approximation, the displacement of the interface is small compared to the wavelength and can be neglected, so that the interface is confused with the plane \(x=0\).

It is assumed that the pressure at rest \(P_0\) has the same value at any point in space.

Considering a surface element \(dS\) of the interface between the two fluids as a fictitious massless membrane, is shown with the Newton's second law in projection on the \((Ox)\) axis that pressure is continuous at the interface.

Indeed :

\(({P_0} + {p_1}(x=0^-,t))dS - ({P_0} + {p_2}(x=0^+,t))dS = 0\;\;\;\;\;so\;\;\;\;\;{p_1}(x=0^-,t) = {p_2}(x=0^+,t)\)

Remark :

In case the interface is treated as a flat plate of mass surface density \(\sigma\), relationship between the pressures on either side of the interface can now be written :

\(\sigma \frac {\partial v}{\partial t} (0,t)=p_1(x=0^-,t)-p_2(x=0^+,t)\)

It is assumed that the two cross sections are identical :

\(S_1=S_2\)

It is considered that the incident wave is a HPPW :

\(\underline v_i(x,t)=\underline A_i \;exp(j(\omega t - k_1x))\)

The reflected and transmitted waves are noted :

\(\underline v_r(x,t)=\underline A_r \;exp(j(\omega t + k_1x))\)

And :

\(\underline v_t(x,t)=\underline A_t \;exp(j(\omega t - k_2x))\)

The wave vectors are given by :

\(k_1=\frac{\omega}{c_1}\)

And :

\(k_2=\frac{\omega}{c_2}\)

Where \(c_1\) and \(c_2\) are the sound velocities in the two medium.

Overpressure are deduced using the relationship \(\underline p=\pm\mu c \;\underline v=\pm z\;\underline v\), so :

\(\underline p_i(x,t)=z_1\underline A_i \;exp(j(\omega t - k_1x))\)

\(\underline p_r(x,t)=-z_1\underline A_r \;exp(j(\omega t + k_1x))\)

\(\underline p_t(x,t)=z_2\underline A_t \;exp(j(\omega t - k_2x))\)

Using the boundary condition :

• Continuity of the volume flow rate at the interface \(x=0\) (continuity of speed since the cross sections are the same) :

  \(\underline A_i+\underline A_r= \underline A_t\)

• Continuity of overpressure at the interface \(x=0\) :

  \(z_1(\underline A_i-\underline A_r)=z_2 \underline A_t\)

Expression of the coefficients of reflection and transmission in amplitude :

The previous system is solved as :

\(\underline r_v=\frac{\underline A_r}{\underline A_i}=\frac{z_1-z_2}{z_1+z_2}\)

And :

\(\underline t_v=\frac{\underline A_t}{\underline A_i}=\frac{2z_1}{z_1+z_2}\)

\(\underline r_v\) is the reflection coefficient in amplitude for speed and \(\underline t_v\) is that for the transmission for speed also.

The coefficients of reflection and transmission to the pressure are deduced :

\(\underline r_p=-\frac{\underline A_r}{\underline A_i}=\frac{z_2-z_1}{z_1+z_2}=-\underline r_v\)

And :

\(\underline t_p=\frac{\mu_2c_2}{\mu_1c_1}\;\frac{\underline A_t}{\underline A_i}=\frac{2z_2}{z_1+z_2}\)

Some notes :

• The transmission coefficients (\(\underline t_v\) and \(\underline t_p\)) are real positives : the transmitted wave is always in phase with the incident wave.

• The reflection coefficients may be positive or negative : the reflected wave can be in phase or in phase opposition with the incident wave.

• When \(z_1=z_2\), there is no reflected wave (called adaptation of impedance).

• The sound wave in the medium \((1)\) results from the superposition of two HPPW propagating in opposite direction and whose amplitudes generally differ.

  It is not either of a traveling wave or a standing wave but a wave having a SWR (standing wave ratio).

• In the special case \(z_2>>z_1\) :

  \(r_v\approx-1\;\;\;\;\; ;\;\;\;\;\;r_p\approx1\;\;\;\;\; ;\;\;\;\;\;t_v\approx0\;\;\;\;\; ;\;\;\;\;\;t_p\approx2\)

  There is no more transmitted wave. Standing waves is shown that is obtained in the medium \((1)\) with speed node and pressure anti-node in \(x=0\).

  This is the case of an air interface to solid.

• If \(z_2<<z_1\), so :

  \(r_v\approx1\;\;\;\;\; ;\;\;\;\;\;r_p\approx -1\;\;\;\;\; ;\;\;\;\;\;t_v\approx 2\;\;\;\;\; ;\;\;\;\;\;t_p\approx 0\)

  There is no more transmitted wave. Standing wave is always obtained in the medium \((1)\), but with a speed anti-node and pressure node in \(x=0\).

  This is the case of a solid-to-air interface.

On average, the "surface density of sound power" vectors of the three incident, reflected and transmitted waves are :

\(\left\langle {{\Pi _i}} \right\rangle = \frac{1}{2}{z_1}A_i^2\;\;\;\;\;;\;\;\;\;\;\left\langle {{\Pi _i}} \right\rangle = \frac{1}{2}{z_1}A_r^2\,\,\,\,\,;\,\,\,\,\,\left\langle {{\Pi _i}} \right\rangle = \frac{1}{2}{z_2}A_t^2\)

We deduce the coefficients of reflection and transmission in energy :

\(R = \frac{{\left\langle {{\Pi _r}} \right\rangle }}{{\left\langle {{\Pi _i}} \right\rangle }} = r_v^2 = {\left( {\frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}} \right)^2}\;\;\;\;\;and\;\;\;\;\;T = \frac{{\left\langle {{\Pi _t}} \right\rangle }}{{\left\langle {{\Pi _i}} \right\rangle }} = \frac{{4{z_1}{z_2}}}{{{{({z_1} + {z_2})}^2}}}\)

Remarks :

• Energy conservation :

  \(R+T=1\)

• When \(z_1>>z_2\) or \(z_2>>z_1\), \(R\approx 1\) and \(T\approx 0\) : there is no transmitted wave.

• For \(z_1\) data, the coefficient \(T\) is maximum when \(z_2=z_1\) : there is the adaptation of impedance.

  An interface correctly transmits sound waves only if the impedances of both sides are almost equal.