Cut-off Frequencies and Pressure Distribution in a Pipe with Square Cross Section

What are the cutoff frequencies and how can we sketch the pressure distribution across a pipe with square cross section of dimension L?

Cut-off Frequencies Calculation

To find the cutoff frequencies in a pipe with square cross section of dimension L, we first need to consider the modes of vibration in the pipe. The possible modes of vibration for such a pipe can be determined using the equation λ = 2L/n, where λ is the wavelength and n is the mode number. For the lowest five sets of modes (n = 1, 2, 3, 4, 5), we can substitute these values into the equation to find the cutoff wavelengths for each mode. The cutoff frequency can then be obtained by dividing the speed of sound in the fluid by the cutoff wavelength.

Pressure Distribution Sketch

Sketching the pressure distribution across the pipe involves understanding the nature of the modes of vibration. For the first mode (n = 1), a half-wavelength standing wave is formed along the length of the pipe, resulting in sinusoidal pressure variation with high pressure at the middle and low pressure at the ends. For the second mode (n = 2), two half-wavelength standing waves are formed, leading to two pressure nodes and one pressure antinode. The pressure distribution for the third, fourth, and fifth modes can be similarly determined based on the number of standing waves formed. Unfortunately, the exact sketch of the pressure distribution cannot be provided in this text-based format, but you can refer to detailed calculations and graphical representations for better visualization.

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