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In a stretched string the speed of the wave traveling through this depends upon the mass per unit length (µ) and the tension (T) in the string. Thus, the wave speed on a stretched string is give as:
V= √(T/µ)
A pulse of any shape travels along the string with speed v = √(T/µ) without any change in pulse shape.
The Wave Equation:
The wave equation for a plane wave traveling in the x-direction is:
where v is the phase velocity of the wave and y is the variable that changes as the wave travels in the forward direction. This is the one dimensional wave equation. It is applied to the stretched string or a plane electromagnetic wave. In two dimensions the wave equation is given by,
This is the wave equation, which describes a wave on a stretched membrane. There are varieties of wave equations. For example, there is wave equation for electromagnetic wave:
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Waves and Optics