The amplitude of a standing sound wave in a long pipe closed at the left end is

Question

The amplitude of a standing sound wave in a long pipe closed at the left end is sketched below.

Useful Hints:

A.) The vertical axis is the maximum displacement of the air, and the horizontal axis is along the length of the pipe. What is the harmonic number for the mode of oscillation illustrated?

B.)The length of the pipe is 0.320 m. What is the pitch (frequency) of the sound? Use 340 m/s for the speed of sound in air.

C.) The pipe is now held vertically, and a small amount of water is poured inside the pipe. The first harmonic frequency is then measured to be 292.2 Hz. Measuring from the bottom of the pipe, what is the level of the water?

The harmonic number is usually associated with the ratio of the mode frequency to the frequency of the fundamental, in which case for a pipe closed at one end, the harmonic # for the lowest (fundamental) mode is 1, for the next higher mode (mode2) the harmonic # is 3, for mode 3 it is 5, for mode 5 is 7, etc... .

Explanation / Answer

A) 7th harmonic

B) from the figure, 7*lamda/4 = 0.32

lamda = 0.32*4/7

= 0.183 m

use the relation, v = lamda*f

==> f = v/lamda

= 340/0.183

= 1859 hz

C) use, v = lamda*f

lamda = v/f

= 340/292.2

= 1.163

in the first harmonic, lamda = L/4 (here L is effective length of air column)

==> L = lamda/4

= 1.163/4

= 0.29 m

so, water level = 0.32 - 0.29

= 0.03 m or 3 cm

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