# Engineering Acoustics/Reflection and transmission of planar waves

## Specific acoustic impedance

Before discussing the reflection and transmission of planar waves, the
relation between particle velocity and acoustic pressure is
investigated.

$\frac{\partial u}{\partial t}= \frac{-1}{\rho_o}\frac{\partial p}{\partial x}$

The acoustic pressure and particle velocity can be described in complex
form.

$\mathbf{p}=\mathbf{P}e^{j(\omega t -Kx)}$

$\mathbf{u}=\mathbf{u_o}e^{j(\omega t -Kx)}$

Differentiating and substituting,

$-j \omega \mathbf{u}=\frac{-j K \mathbf{p}}{\rho_o}=\frac{-j \omega \mathbf{p}}{\rho_o c_o}$

$\mathbf{u}=\frac{\mathbf{p}}{\rho_o c_o}$

The specific acoustic impedance for planar waves is defined.

$\mathbf{z}=\frac{\mathbf{p}}{\mathbf{u}}=\rho_o c_o=r_o$

## Planar wave: Normal incidence

Consider an incident planar wave traveling in an infinite medium with
specific impedance $r_1=\rho_1 c_1$ which encounters the boundary
between medium 1 and medium 2. Part of the wave is reflected back into
medium 1 and the remaining part is transmitted to medium 2 with specific
impedance $r_2=\rho_2 c_2$. The pressure field in medium 1 is described
by the sum of the incident and reflected components of the wave.

$\mathbf{p_1}=\mathbf{p_i}+\mathbf{p_r}=\mathbf{P_i}e^{j(\omega t -K_1x)}+\mathbf{P_r}e^{j(\omega t +K_1x)}$

The pressure field in medium 2 is composed only of the transmitted
component of the wave.

$\mathbf{p_2}=\mathbf{p_t}=\mathbf{P_t}e^{j(\omega t -K_2x)}$
!Reflection and transmission of normally incident planar
wave.\|440x440px
Notice that the frequency of the wave remains constant across the
boundary, however the specific acoustic impedance changes across the
boundary. The propagation speed in each medium is different, so the wave
number of each medium is also different. There are two boundary
conditions to be satisfied:

1.  The acoustic pressure must be continuous at the boundary
2.  The particle velocity must be continuous at the boundary

Imposition of the first boundary condition yields

$\mathbf{p_1}(x=0)=\mathbf{p_2}(x=0),$

$\mathbf{P_i} + \mathbf{P_r} = \mathbf{P_t}.$

Imposition of second boundary condition yields

$\mathbf{u_1}(x=0)=\mathbf{u_2}(x=0),$

$\mathbf{u_i}(x=0)+ \mathbf{u_r}(x=0)= \mathbf{u_t}(x=0),$

and using the definition of specific impedance, the equations are
expressed in terms of pressure

$\frac{\mathbf{P_i}}{r_1} - \frac{\mathbf{P_r}}{r_1} = \frac{\mathbf{P_t}}{r_2}.$

The pressure reflection coefficient is the ratio of the reflected
acoustic pressure over the incident acoustic pressure,
$\mathbf{R}=\frac{\mathbf{P_r}}{\mathbf{P_i}}$. The pressure
transmission coefficient is the ratio of the transmitted acoustic
pressure over the incident acoustic pressure,
$\mathbf{T}=\frac{\mathbf{P_t}}{\mathbf{P_i}}$. The specific acoustic
impedance ratio is also defined as: $\zeta=\frac{r_2}{r_1}$. Applying
the above definitions, the boundary conditions can be rewritten as:

$1+\mathbf{R}=\mathbf{T}$

$1-\mathbf{R}=\frac{\mathbf{T}}{\zeta}.$

Solving for the pressure reflection coefficient yields:

$\mathbf{R}=\mathbf{T}-1=\frac{\zeta-1}{\zeta+1}=\frac{r_2-r_1}{r_2+r_1}.$

Solving for the pressure transmission coefficient yields:

$\mathbf{T}=\mathbf{R}+1=\frac{2 \zeta}{\zeta +1}=\frac{2r_2}{r_2+r_1}.$

Solving for the specific acoustic impedance ratio yields:

$\zeta = \frac{1+\mathbf{R}}{1-\mathbf{R}} = \frac{\mathbf{T}}{2-\mathbf{T}}.$

## Case 1: Rigid boundary

Consider an incident planar wave that encounters a rigid boundary. This
is the case if the specific impedance of medium 2 is significantly
larger than the specific impedance of medium 1. Thus, the specific
acoustic impedance ratio is very large, the reflection coefficient
approaches 1 and the transmission coefficient approaches 2.

$\mathbf{R}=1=\frac{\mathbf{P_r}}{\mathbf{P_i}} \Rightarrow \mathbf{P_r} = \mathbf{P_i} \Rightarrow \mathbf{u}(x=0)=0$

$\mathbf{T}=2=\frac{\mathbf{P_t}}{\mathbf{P_i}} \Rightarrow \mathbf{P_t} = 2 \mathbf{P_i} \Rightarrow \mathbf{p}(x=0)= 2 \mathbf{P_i}$

The amplitudes of the incident and reflected waves are equal. The
reflected wave is in phase with the incident wave. The particle velocity
at the boundary is zero. The acoustic pressure amplitude at the boundary
is equal to twice the pressure amplitude of the incident wave and it is
maximum.

## Case 2: Resilient boundary

Consider an incident planar wave that encounters a resilient boundary.
This is the case if the specific impedance of medium 2 is significantly
smaller than the specific impedance of medium 1. Thus, the specific
acoustic impedance ratio approaches zero, the reflection coefficient
approaches -1 and the transmission coefficient approaches zero.

$\mathbf{R}=-1=\frac{\mathbf{P_r}}{\mathbf{P_i}} \Rightarrow \mathbf{P_r} = - \mathbf{P_i} \Rightarrow \mathbf{u}(x=0)= \frac{\mathbf{2P_i}}{r_1}$

$\mathbf{T}=0=\frac{\mathbf{P_t}}{\mathbf{P_i}} \Rightarrow \mathbf{P_t} = 0 \Rightarrow \mathbf{p}(x=0)= 0$

The amplitudes of the incident and reflected waves are equal. The
reflected wave is 180\\degree out of phase with the incident wave. The
particle velocity at the boundary is a maximum. The acoustic pressure at
the boundary is zero.

## Case 3: Equal impedance in both media

Consider two media with the same specific acoustic impedance so that the
specific acoustic impedance ratio is unity, the reflection coefficient
is zero and the transmission coefficient is unity. Therefore, the wave
is not reflected, only transmitted. It behaves as if there was no
boundary.

