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Question: (I) An organ pipe is 116 cm long. Determine the fundamental and first three audible overtones if the pipe is (a) closed at one end, and (b) open at both ends.

Short Answer

Expert verified
  1. The fundamental frequency for the closed pipe is 74.56Hz. The first three audible overtones are 223.6Hz, 372.8Hz and 521.92Hz.
  2. The fundamental frequency for the open pipe is149.12Hz. The first three audible overtones are 298.24Hz, 447.36Hz and 596.48Hz.

Step by step solution

01

Step-1:-Fundamental frequency of an organ pipe

In this problem, the fundamental frequency is calculated by dividing the wave speed (v) by the length of the tube (l).

02

Step-2:-Given data

The length of the pipe is l=116cm.

The speed of sound at room temperature is ฮฝ=346m/s.

03

Step-3:-Calculation of fundamental frequency and first three overtones of a closed pipe

The fundamental frequency for the closed pipe is calculated as:

f1=ฮฝ4l=346m/s4ร—(116cm(1m100cm))=74.56Hz

The first overtone will be,

f3=3f1=3ร—74.56Hz=223.6Hz

The second overtone is,

f5=5f1=5ร—74.56Hz=372.8Hz

The third overtone is,

f7=7f1=7ร—74.56Hz=521.92Hz

The fundamental frequency is 74.56Hz and the three overtones are 223.6Hz, 372.8Hz and 521.92Hz.

04

Step-4:-Calculation of fundamental frequency and first three overtones of an open pipe

The fundamental frequency for an open pipe is calculated as:

f1=ฮฝ2l=346m/s2ร—(116cm(1m100cm))=149.12Hz

The first overtone is,

f2=2f1=2ร—149.12Hz=298.24Hz

The second overtone is,

f3=3f1=3ร—149.12Hz=447.36Hz

The third overtone is,

f4=4f1=4ร—149.12Hz=596.48Hz

The fundamental frequency is 149.12Hz and the three overtones are 298.24Hz, 447.36Hz and 596.48Hz.

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