A pipe is open at both ends. We can play pure notes into the pipe using a speake

Question

A pipe is open at both ends. We can play pure notes into the pipe using a speaker and a signal generator. But, like most speakers, our speakers will not produce very low frequencies well. So we are restricted to testing frequencies above about 50 Hz. Playing sound of different frequencies into the pipe we find that the pipe “rings” loudly when we play sounds of frequencies 54 Hz, 72 Hz, 90 Hz and 108 Hz. It does not ring at frequencies between these.

(a) What is the frequency of the fundamental mode of this pipe?

(b) Which harmonics (i.e. what numbers) are the ones that we found using the speaker?

(c) How long is the pipe?

(d) Suppose we close one end of the pipe. List the 4 lowest frequency modes that the pipe will now have and indicate which harmonics they are.

Explanation / Answer

Part a

fundamental harmonic

F1 = 72-54 = 18Hz

Part B

given harmonics

54Hz = 3*18 = third harmonic

72 = 4*18 = fourth harmonic

90Hz = 5*18 = fifth harmonic

108hz = 6th harmonic

Part C

let length of pipe is L then

f1 = V/2L

V = 340m/s

L = 340/2*18 = 9.5m

Part D

If pipe is closed at one end then we observe only odd harmonics

fundamental = V/4L = 340/4*9.5 = 9Hz

so four lowest frequencies observed are

63Hz , 81Hz , 99Hz, 117Hz

7th, 9th, 11th and 13th harmonic

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