Unless indicated otherwise, assume the speed of sound in air to be v = 344 m/s.

Question

Unless indicated otherwise, assume the speed of sound in air to be v = 344 m/s. Singing in the Shower. A pipe closed at both ends can have standing waves inside of it, but you normally don't hear them because little of the sound can get out. But you can hear them if you are inside the pipe, such as someone singing in the shower, (a) Show that the wavelengths of standing waves in a pipe of length L that is closed at both ends are An = 2L/n and the frequencies are given by fn = nv/2L = nf1, where n = 1, 2, 3 (b) Modeling it as a pipe, find the frequency of the fundamental and the first two overtones for a shower 2.50 m tall. Are these frequencies audible?

Explanation / Answer

a) pipe closed at both ends,

then nodes will be formed at closed end,

distance between two consecutive nodes = lambda/2

distance between nodes = lambad/2 + mlambda/2

so L = lambad/2 + mlambda/2

lambda(n + 1)/2 = L

lambda = 2L/(n +1)
for n = 0, 1, 2, 3 ...
n + 1 can be replace n then n = 1,2,3,...

lambda = 2L/n

b) for fundamental,

n =1 and frequency = v / lambda = 344/(2L/n)

f = 68.8n

fundamental, f0 = 68.8 Hz

1st: 2 * 688 = 137.6 Hz

2nd: 3 * 68.8 = 206.4 hz

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