When the wave motion is constrained in one or more directions, the travelling wave can become a “standing wave”. And example of a standing wave is the vibration produced by any string instrument and the sound vibration produced in air columns.

\[ y(x,t) = A \sin(\omega_n t) \sin\left( \frac{n\pix}{\lambda} \right) \]

. In order to understand music and how guitars produce it, it is helpful to understand the physics of sound. Sound is created when the vibration of material bodies causes energy to propagate in pressure waves through a medium, usually air.

All forms of musical instruments create vibrations in order to produce the sound waves that make music.

Guitars produce sound by amplifying the sound of the vibrations of their strings. The strings on a guitar are fixed at both ends and are elastic. The vibrations of the string travel back and forth along the string. The energy is reflected when it reaches one end of the string. Vibrations of certain natural frequencies for the string will get reinforced, while all other frequencies will disappear.

The natural frequency of a string of length $L$ which has mass density $\mu$[kg/m] and is under tension $T$[N] is given by the equation: \[ f = \frac{ \sqrt{ T / \mu } }{ 2L}. \] The first harmonic refers to a standing wave with this frequency A standing wave has crests and troughs certain positions along its length. The displacement of the envelope of the wave as a whole increases and decreases over time..

The above equation comes from the combination of the following two facts:

• The wavelength of the first harmonic is $\lambda = 2L$.
• The velocity of the energy propagation in a string with density $\mu$[kg/m] under tension $T$[N] is:

\[ v = \sqrt{ \frac{ T }{ \mu } }. \]

  Waves will travel faster when the tension of the string is higher.

We obtain the frequency equation by solving for $f$ in the general equation $v = \lambda f$.

Observe that the equation $f = \frac{ \sqrt{ T / \mu } }{ 2L}$ confirms your experience with guitars. The frequency of a string will be higher if the tension is increased or if a lighter string is used.

[ Nice illustrations of standing waves. ]
http://en.wikipedia.org/wiki/Standing_wave

NOINDENT [ Examples of standing waves on a string and many other waves. ]
http://www.youtube.com/watch?v=6dzEfhjQYRY