Problem $2$

Consider a piston of diameter $1$ cm located at $x=0.$ The piston vibrates at a frequency $f=2kHz$ in semi$-$ infinite tube of diameter $1$ cm $.$ A plane wave propagates down the tube in the $+x$ direction, with a measured peak pressure amplitude of $1\mathrm{.}1$ Pa at $1$ m from the piston. Assume that the duct is filled with air with the following properties ${}_0=1\mathrm{.}184k\frac{g}{m^3}$ and $c=346\frac{m}{s}.$

a$.$ Draw a schematic of the system.

b$.$ Calculate the peak amplitude of the piston velocity $($m$/$s$)?$ Hint: the piston velocity should

match the value of the fluid particle velocity at $x=0.$

c$.$ Calculate the peak amplitude of the piston displacement $($m$)?$

d$.$ Compute the acoustic Mach number, which is the ratio of the peak amplitude of the particle velocity to the speed of sound in the fluid.