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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.

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  • $\begingroup$ If the pipe moves with constant velocity then it makes no difference. Are you considering attenuation as well? $\endgroup$ – Deep Sep 5 '17 at 5:11
  • $\begingroup$ @Deep Thank you for you interest. Pipe is not moving. The fluid inside the pipe is moving perpendicularly (in small portion of the pipe). You may think the system is equilibrium under resonance, meaning dissipation is there to stop the amplitude increase infinitely. $\endgroup$ – Creator Sep 5 '17 at 5:31
  • $\begingroup$ @Deep this question is when moving portion is perpendicular to the wave propagation so Doppler is not straight forward and the structure is in the range of wavelength so far field equations are not relevant. $\endgroup$ – Creator Sep 5 '17 at 5:34
  • $\begingroup$ It is not clear how in a closed pipe only a small portion of fluid moves in perpendicular direction. What about the rest of the fluid? Some clarification regarding flow configuration would be desirable, and if possible including a sketch would be even better. $\endgroup$ – Deep Sep 5 '17 at 5:37
  • $\begingroup$ @Deep So I said as hypothetical pipe. It resembles a physical situation where the breadth of the pipe can be considered as infinite or very long compare to L. $\endgroup$ – Creator Sep 5 '17 at 5:42
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The pressure variation at the close end will be sinusoidal, depending on the frequency of excitation. Even if the fluid flows perpendicular to the wave direction, the resultant speed would be combination of the wave speed and the speed of the fluid. The additional distance that needs to be covered by the wave would depends on the speed of the fluid. Higher the speed fluid higher the distance. This means the time required by the wave to reach the sensor would be higher in case of moving fluid compare to stationary fluid.

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