Impedance matching

An industrious student decides to experiment with the effects of the damped boundary condition from Problem 6 of Prelim 1,

$$A B \frac{\partial s(x=0,t)}{\partial x} = b \frac{\partial s(x=0,t)}{\partial t},$$

where $B$ is the bulk modulus of the gas in the tube and $b$ the damping coefficient in the shock absorber. (See Figure 3.)

Figure 3: Boundary condition with drag from Prelim 1

(a)

Following the procedure from class and in the lecture notes, determine the form of the reflected pulse $f(u)$ in terms of the incoming pulse $g(u)$ where the general solution is $s(x,t)=f(x-ct)+g(x+ct)$.
Note: Because the shock absorber is on the left of the tube, the incoming pulse ``g'' is traveling from right to left and thus has a ``$+$'' sign in front of the factor of $c$. Your task is to solve for the reflected pulse ``f'', which travels from left to right.
Hint: You should check your answer by investigating the limits $b\rightarrow 0$ and $b \rightarrow \infty$.

(b)

By experimenting with shock absorbers with different damping coefficients $b$, the student finds--amazingly--that she can send a sound pulse down the tube without any reflection coming back! Find the value(s) of $b$ for which no reflection occurs. Express your answer in terms of the ``impedance'' $Z\equiv \rho_{\scriptscriptstyle 0}c = B/c$ of the gas in the tube.

Note: Proper termination of waveguides through impedance matching like this to avoid unwanted reflections is very important in many electronic communications applications.