Traveling Sound Waves — Question 7

Question 7

A pressure reflection coefficient differs from a velocity coefficient

A normally incident plane sound wave crosses a stationary fluid interface. The impedances are $Z_1=400$ and $Z_2=1200\,\mathrm{Pa\,s/m}$. Incident, reflected and transmitted signed pressure amplitudes are $P_i,P_r,P_t$. Pressure and normal velocity are continuous.

1. Derive $P_r/P_i$ and $P_t/P_i$.

2. For $P_i=1.0\,\mathrm{Pa}$, find the interface velocity on both sides and explain how $P_t>P_i$ is possible.

Question 7 - Solution

A pressure reflection coefficient differs from a velocity coefficient

(a) Method and calculation

At the interface,$$P_i+P_r=P_t,\qquad\frac{P_i-P_r}{Z_1}=\frac{P_t}{Z_2}.$$Solving,$$r_p=\frac{Z_2-Z_1}{Z_2+Z_1}=\frac12,\qquad t_p=\frac{2Z_2}{Z_1+Z_2}=\frac32.$$The minus sign for reflected velocity is essential.

(b) Interpretation and further calculation

The total normal velocities agree:$$u_1=\frac{1-0.5}{400}=0.00125\,\mathrm{m/s},\quad u_2=\frac{1.5}{1200}=0.00125\,\mathrm{m/s}.$$Pressure amplitude alone is not an energy fraction. The higher-impedance transmitted medium has a smaller velocity for a given pressure; continuity allows a transmitted pressure larger than the incident pressure.

Physical check

When $Z_1=Z_2$, $r_p=0$ and $t_p=1$. The interface then has no reflected plane wave in this ideal model.