Engineering Acoustics/Reflection and transmission of plane waves

Before discussing the reflection and transmission of plane 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 plane waves is defined.

$$   \mathbf{Z}=\frac{\mathbf{p}}{\mathbf{u}}=\rho_o C_o $$

Consider an incident plane wave in a 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.

$$   \mathbf{p_2}=\mathbf{p_t}=\mathbf{P_t}e^{j(\omega t -K_2x)} $$ 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

Applying the first boundary condition,

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

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

Applying the second boundary condition and using the definition of specific impedance,

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

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

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

The pressure reflection and transmission coefficients are defined as the ratio of the reflected and transmitted acoustic pressure over the incident pressure, respectively.

$$   \mathbf{R}=\frac{\mathbf{P_r}}{\mathbf{P_i}} $$

$$   \mathbf{T}=\frac{\mathbf{P_t}}{\mathbf{P_i}} $$

The specific acoustic impedance ratio is also defined.

$$   \zeta=\frac{r_2}{r_1} $$

Applying the above definitions on the boundary conditions,

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

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

Solving for the pressure reflection coefficient,

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

Solving for the pressure transmission coefficient,

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

Solving for the specific acoustic impedance ratio,

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

Case 1: Consider an incident plane 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: Consider an incident plane 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 $$180^\circ$$ 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: Consider two media with the same specific acoustic impedance so that the specific acoustic impedance ratio is 1, the reflection coefficient is zero and the transmission coefficient is 1. Therefore, the wave is not reflected, only transmitted. It behaves as if there was no boundary.