show that in a closed end organ pipe only odd harmonies can be produced - Physics - Waves

Question

show that in a closed end organ pipe only odd harmonies can be
produced

Shaheen . · Accepted Answer

Consider a closed pipe of length L, which encloses certain
specific amount of air called as air column This air column is set
into vibrations by holding a vibrating tuning fork near its mouth
These longitudinal waves of frequency u wavelength l travel with
a
velocity of v through the pipe and gets reflected at the other
end, because the other end acts like a boundary and this reflected
wave traveling in opposite direction superpose with incident wave
forming a stationary wave such that a node is formed at the closed
end and an antinodes at the open end, which is characterized by a
set of natural
frequencies called as normal modes of oscillation The first 3
nodes of oscillations are:-
The stationary waves formed is given by
y = 2a sin kx cos w t where 2a sin Kx gives its amplitude,
therefore the positions of nodes
are given by sin Kx = 0
and hence Kx = np where n = 0, 1, 2, 3,
since K =2p/ l; we get , x =nl/2
where n = 0, 1, 2, 3,
In the same way, the positions of the antinodes are given by
Sin Kx = 1
hence Kx =(n+1/2) p where n = 0, 1, 2, 3,
Since K= 2p/l ; we get, x = (n+1/2) l/2 where n = 0, 1, 2,
3,
Taking the closed end of the pipe to be x = 0, the condition for
node is satisfied and the other end of the pipe to be x = L where
an antinodes is formed, requires that the length L to be related to
wavelength l given by L = (n+1/2) l /2 for n = 0, 1, 2, 3
Thus, the possible wavelengths of stationary wave formed in
different modes of oscillation are given by
l =2L/(n+1/2)
where n = 0, 1, 2, 3, and corresponding frequencies are
given by u = (n+1/2)v/2L
where n = 0, 1, 2, 3,
Thus, for fundamental mode of oscillation (or) I harmonic n = 0,
and
corresponding frequency is given by
u1 =V/4L
For second mode of oscillation n =1, corresponding frequency is
given by u2 =3V/4L ; this is called as I overtone (or) III
harmonic
Similarly, for third mode of oscillation n =2, corresponding
frequency is given by u3 =5V/4L ; this is called as II overtone
(or) V harmonic and so on
Therefore, u1 : u2 : u3 : = 1 : 3 : 5:
Thus, a closed pipe produces odd harmonics i e , the ratio of
frequencies of overtone to that of fundamental frequency are odd
natural numbers

show that in a closed end organ pipe only odd harmonies can be produced.