Consider a hypothetical pipe of length $L$ with both end closed. The pipe is filled with a some fluid. Let there be a ultrasound source of frequency $f=v/2L$ at one end of the pipe and a sensor to measure pressure at the other end of the pipe, where $v$ is the speed of ultrasound wave. Clearly the energy will be trapped in the cylinder and resonance phenomenon will be observed. The question is what pressure measurement will be observed at the sensor? Specifically we know the pressure at the ultrasound source at the other end. What is the relation between pressure at the source and sensor?
More importantly, I want to know if a "portion" of the fluid in the if the pipe moves across the diameter (not in the direction of length) , this means fluid is moving perpendicular to the wave propagation: in this case how the pressure at the sensor would differ compare to earlier case where there the fluid is stationary?
Any hint how to think of this problem would be highly appreciated.