Prelim Practice: Generalized Boundary Conditions for a Sound Tube

A tube of length $L$ is filled with air of bulk modulus $B$ and mass density $\rho_o$ at atmospheric pressure $P_o$. One end of the tube $(x = L)$ is closed while the other end $(x = 0)$ is attached to a massless piston of area $A$ that can slide freely (without friction) along the tube. (We consider low amplitude waves so that you can take $L \approx \mathrm{const}$.) The piston is attached to an ideal spring of spring constant $k$ that exerts a horizontal force on the piston; the spring is relaxed when the piston is at $x=0$. (See Figure 3.) We denote the sound displacement inside the tube as $s(x,t)$ and the pressure inside the tube as $P(x,t)$.

(a)

Air Pressure and Displacement of the Piston

Draw the free body diagram for the piston, indicating the directions and the magnitudes of all the forces, using no quantities other than $P(x=0,t)$, $s(x=0,t)$, $k$, $A$, $\rho_o$, $L$, and $P_o$.
Note: You do not need all of these quantities.

(b)

Boundary Condition at the Piston (Challenge Problem!)

Write the Equation of Motion for the piston in terms of the degrees of freedom and known constants; i.e., using only

• the displacement function $s(x,t)$ and its derivatives evaluated at the $x=0$ end of the tube,
• $k$, $A$, $B$, $\rho_o$, $L$, and $P_o$.

Figure 3: Generalized boundary conditions for a sound tube.