Harmonics and Overtones in a Standing Wave

Physics

definition

Hormonics and Overtone

An overtone is any frequency greater than the fundamental frequency of a sound. Using the model of Fourier analysis, the fundamental and the overtones together are called partials. Harmonics, or more precisely, harmonic partials, are partials whose frequencies are integer multiples of the fundamental (including the fundamental which is 1 time itself).

example

Solve problems on standing waves with given information about its possible harmonics

Example: A rope, under a tension of 200 N and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by : . Where x = 0 at one end of the rope, x is in meters and t is in seconds. If the rope oscillates in a third-harmonic standing wave pattern, the period of oscillation is sec. Find .

Solution:
In the second harmonic, .Hence the time period Frequency=For a standing wave pattern,

definition

Define and find higher harmonics and overtones for standing wave in a string fixed at one end

 .

These are the normal frequencies of vibration. The fundamental frequency is obtained when n=0, ie,

.

The overtone frequencies are

,

,

, etc.

definition

Understand and use the relation between fundamental frequency and higher overtones for string fixed at one end

Vibration of a string fixed at one end standing waves can be produced on a string which is fixed at one end and whose other end is free to move in a transverse direction. Such a free end can be nearly achieved by connecting the string to a very light thread. If the vibrations are produced by a source of correct frequency, standing waves are produced. If the end is fixed and is free.

with the boundary condition that is an antinode. The boundary condition that is a node is automatically satisfied by the above equation. For to be an antinode,



Or,
d
Or,

Or,

Or, .

These are the normal frequencies of vibration. The fundamental frequency is obtained when , ie,

.

The overtone frequencies are

,
,

, etc.

We see that all the harmonics of the fundamental are not the allowed frequencies for the standing waves. Only the odd harmonics are the overtones. Figure shows shapes of the string for some of the normal modes.