An aluminimum rod `1.60 m` long is held at its centre. It is stroked with a rosin - coated cloth to set up a longitudinal vibration. The speed of sound in thin rod of aluminium is `5100 m//s`. (a) What is the fundamental frequency of the waves established in the rod ? (b) What harmonics are set up in the rod held in this manner ? (c ) What would be the fundamental frequency if the rod were copper , in which the speed of sound is `3650 m//s`?

Explanation

Standing waves are important because any wave confined to a restricted region of space will be reflected bak onto itself by the boundaries of the region . Then the travelling waves moving in opposite directions constitute a standing wave . The rod can sing a few hundred hertz and at integer - multiple higher harmonics . The frequency will be proportionately lower with copper .
We must identify where nodes and antinodes are used use the fact that antinodes are separated by half a wavelength with ` v = f lambda`.
a. Since the central clamp established a node at the centre, the fundamental node of vibration will be ANA. Thus , the rod length is `L = d_(AA) = lambda//2`.
Our first harmonic frequency is
`f_(1) = (v)/( lambda_(1)) = (v)/( 2L) = ( 5100 m//s)/( 3.20 m) = 1.59 Hz`
b. Since the rod is free at each end , the ends will be antinodes . The next vibration state will not have just one more node and one more antinode , reading ANANA, as shown in the diagram , with a wavelength and frequency of
`lambda = (2 L)/( 3) and f = (v)/( lambda) = ( 3v)/( 2L) = 3 f_(1)`
Since `f_(2) = 2f_(1)` was bypasses as having an antinode at the centre rather than the node required , we know that we get only odd harmonics .
` f = ( nv)/( 2L) = n ( 1.59 KHz)`
for ` n = 1 , 3 , 5 , ....`
c. For a copper rod , the density is higher , so the speed of sound is lower , and the fundamental frequency is lower .
` f_(1) = (v) /( 2L) = ( 3560 m//s)/( 3.20 m) = 1.11 kHz`
The second is not at just a few hertz , but is a sqeak or a squeal at over a thousand hertz . Only a few higher harmonics are in the audible range . Sound really moves fast in materials that are stiff against compression . For a thin rod , it is young's modulus that determines the speed of a longittudinal wave .