Wave Equation for the Vibrating String

Consider an elastic string under tension which is at rest along the dimension. Let , , and denote the unit vectors in the , , and directions, respectively. When a wave is present, a point originally at along the string is displaced to some point specified by the displacement vector

Note that typical derivations of the wave equation consider only the displacement in the direction. This more general treatment is adapted from [122]. An alternative clear development is given in [394].

The displacement of a neighboring point originally at along the string can be specified as

Let denote string tension along when the string is at rest, and denote the vector tension at the point in the present displaced scenario under analysis. The net vector force acting on the infinitesimal string element between points and is given by the vector sum of the force at and the force at , that is, . If the string has stiffness, the two forces will in general not be tangent to the string at these points. The mass of the infinitesimal string element is , where denotes the mass per unit length of the string at rest. Applying Newton's second law gives

where has been canceled on both sides of the equation. Note that no approximations have been made so far.

The next step is to express the force
in terms of the tension
of the string at rest, the elastic constant of the string, and
geometrical factors. The displaced string element
is the
vector

having magnitude

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