Determine the possible harmonics in the longitudinal vibrations of a rod clampred in the middle.

Explanation

The rod of length L is clamped in the middle as shown in the adjoining figure. It has one node in the middle and two antinodes at its free ends in the fundamental mode.
`therefore" "L = 2.(lambda_(1))/(4) or lambda_(1) = 2L`
Fundamental frequency or first harmonic is
`v = (upsilon)/(lambda_(1)) = (upsilon)/(2L)`

Now, in the second mode, as shown in the adjoining figure, an additional node and antinode appear on the two sides of the clamp
`therefore" "L = 6.(lambda_(2))/(4) or lambda_(2) = (2L)/(3)`
Also, `v_(2) = (upsilon)/(lambda_(2)) = (3 upsilon)/(2L) = 3v`
This is the third harmonic or first overtone.
Similarly for third mode, two additional nodes and antinoes appear on the two sides of the clamp.
`therefore" "v_(3) = (5 upsilon)/(2L) = 5v`
This is the fifth harmonic or second overtone.
So, odd no. of harmonic are possible.