$4\mathrm{.}46$ & $4\mathrm{.}47$ Consider a long pipe filled with air at standard sea$-$level conditions. Let $x$ be the

longitudinal coordinate measured along the pipe. The air is stationary inside the pipe, that

is$,$ the flow velocity is zero everywhere inside the pipe. A small firecracker is mounted

inside the tube at an axial location $x=0.$ When the firecracker is detonated, two weak

pressure disturbances $($pressure waves$)$ are created at $x=0$ that propagate along the pipe,

one to the right and the other to the left. Assume that these weak pressure distributions

travel at the local speed of sound. Using SI units, calculate:

$($a$)$ the speed of the waves relative to the pipe

$($b$)$ the $x-$location of each wave $0\mathrm{.}2$ seconds after detonation of the firecrackers

$($c$)$ the x$-$location of each wave if the air inside the pipe is flowing from left to right in the

direction of the positive $x-$axis with a flow velocity of $40\frac{m}{s}$

$($d$)$ the x$-$location of each wave if the air inside the pipe is flowing from left to right with a

flow velocity of $420\frac{m}{s}$