The Wave EquationThe wave equation for a plane wave traveling in the x direction is where v is the phase velocity of the wave and y represents the variable which is changing as the wave passes. This is the form of the wave equation which applies to a stretched string or a plane electromagnetic wave. The mathematical description of a wave makes use of partial derivatives. In two dimensions, the wave equation takes the form which could describe a wave on a stretched membrane.

Index Wave concepts  

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Waves in Ideal StringThe wave equation for a wave in an ideal string can be obtained by applying Newton's 2nd Law to an infinitesmal segment of a string.

Index Wave concepts Reference: Kreyzig Ch. 9  

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String Wave Equation Development

Index Wave concepts Reference: Kreyzig Ch. 9  

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Ideal String ConstraintsIn order for the wave equation to apply to the waves in a string, it must meet certain constraints. For an ideal string, it is assumed that 1. The string is perfectly uniform with a constant mass per unit length, and is perfectly elastic with no resistance to bending. 2. The string tension is presumed to be large enough so that gravity can be neglected. 3. Small segments of the string are presumed to move transversely in a plane perpendicular to the string, and that the displacements and slopes of segments of the string are small. Though stringent, these idealizations permit the development of a wave equation which describes well the vibrations of thin real strings. 
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