Question

A transverse wave is incident on a boundary separating two media of acoustic impdances Z1 and Z2.
(a) Write down the equations of incident, reflected and transmitted waves. .
(b) Write down the boundary conditions.
(c) Obtain expressions for reflection and
transmission coefficients

Answer 1

1) For example, consider the following incident wave with incident angle $\alpha$, refracted with angle $\beta$ and transmitted with angle $\gamma$:

$$p_i=p_0e^{ik_1\vec{r}}=p_0\text{exp}[ik_1(x\text{cos}(90^\circ-\alpha)+y\text{cos}\alpha],$$

reflected wave:

$$p_r=p_0R\text{exp}[ik_1(x\text{sin}\beta+y\text{sin}\beta]$$

and transmitted wave

$$p_t=p_0D\text{exp}[ik_2(x\text{sin}\gamma+y\text{cos}\gamma],$$

where $k_1,\space k_2$ - wavenumbers for the mediums.

2) The boundary conditions (y=0):

$$p_0\text{exp}(ik_1x\text{sin}\alpha)+Rp_0\text{exp}(ik_1x\text{sin}\beta)=Dp_0\text{exp}(ik_2x\text{sin}\gamma).$$

According to Snellius law:

$$k_1\text{sin}\alpha=k_1\text{sin}\beta=k_2\text{sin}\gamma,$$

then

$$D=R+1.$$

The boundary conditions at x=0:

$$\frac{\text{cos}\alpha}{c_1\rho_1}-R\frac{\text{cos}\beta}{c_1\rho_1}=D\frac{\text{cos}\gamma}{c_2\rho_2}.$$

3) The reflection and transmission coefficients are

$$R=\frac{Z_1-Z_2}{Z_1+Z_2},$$

$$D=\frac{2Z_1}{Z_1+Z_2}.$$