Question

Show that (i) the frequency of open organ pipes. is two times the frequency
of the fundamental note of a closed pipe of same length (ii) to produce a
fundamental note of same frequency, the length of the open pipe must be
two times the length of the closed pipe.

Answer 1

Answer:

The wavelength for a closed pipe

has a node at one end and an anti-node at the other end

the distance between  A----N  is 1/4 wavelength

since for a compressional wave a full wavelength consists of:

N------A-------N--------A----------N

and the fundamental frequency for the closed pipe is A--------N

L = wavelength / 4

Since the open pipe has anti-nodes at each end the fundamental

wavelength for the fundamental frequency is

A--------N---------A           (L = wavelength / 2)     where L = length of pipe

or twice  that for a closed pipe

and since V (speed of sound) = F * wavelength

F =  V / wavelength

F (open) = V / (2 L)

F (closed) = V / (4 L)

so for pipes of the same length the fundamental frequency for the

open pipe is twice of that for the closed pipe