Thomas Beecham

- Consider a traveling wave on a string moving from left to
right. If the far right end of the string is fixed the wave will be
reflected (moving right to left). The incident and reflected waves will
undergo "superposition", interfering constructively as indicated below.
- Fixing the far left of the string also will result in a standing
wave pattern. The whole string will look as if it is vibrating in
Simple Harmonic Motion. Points on the string which remain
stationary are called
, maximum displacement positions are called*nodes**antinodes.* - For a string of fixed length L, the frequency of the standing wave is given by . The wavelength () is given by 2L/n, where n can take on any positive integer value. In the above diagram the upper standing wave has n=1 and the lower wave has n=2.
- The general expression for the frequency is given by
(n = 1,2,3,4....)

f

_{1}is called the fundamental frequency or first harmonic.f

_{2}f_{3}f_{4}... are called the 2nd, 3rd, 4th ... harmonics.The standing wave patterns for n=1-6 are shown in the diagram at right.

- These standing wave patterns are exactly the modes of vibration
of the strings of stringed musical instruments, e.g.
guitar,
violin ,
cello . When such a string is set vibrating the sound produced is a
combination of the fundamental and all harmonics.
- The velocity v, of a wave on a string is given by

where T is the tension in the string and (m/L) is the mass per unit length of the string (kg/m). Tuning of a stringed instrument is achieved by adjusting the tension, which in turn changes the wave velocity and therefore the frequency.

- A vibrating column of air (constrained in a tube) will produce
similar standing wave patterns. By adjusting the length of the
column different fundamental frequencies are obtained. This is
the operating principle of woodwind
and brass
instruments.

- Resonance:

Normally the vibrations in the stretched string or column of air die away naturally due to various energy losses, e.g. sound, heat, air resistance. However, if it is possible to apply a periodic force which inputs enough energy to overcome these losses the vibrations will continue. If the energy input exceeds the losses the amplitude of vibrations will increase and it is possible the string could break. This phenomenon is known as resonance.

Resonance occurs when the "driving frequency" is equal to the "natural frequency" of the oscillating system.

"Following the
dispute with the domestic servants' union at

*Dr. C. L. Davis*

*Physics Department*

*University of Louisville*

*email*: c.l.davis@louisville.edu