Wave Equation for the Vibrating String

Consider an elastic string under tension which is at rest along the $x$ dimension. Let $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ denote the unit vectors in the $x$ , $y$ , and $z$ directions, respectively. When a wave is present, a point $\mathbf{p}=(x,0,0)$ originally at $x$ along the string is displaced to some point $\mathbf{a}=\mathbf{p}+d\mathbf{p}$ specified by the displacement vector

$$d\mathbf{p}= \mathbf{i}\xi + \mathbf{j}\eta + \mathbf{k}\zeta.$$

Note that typical derivations of the wave equation consider only the displacement $\eta$ in the $y$ 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 $\mathbf{q}=(x+d x,0,0)$ along the string can be specified as

$$d\mathbf{q}= \mathbf{i}(\xi+d\xi) + \mathbf{j}(\eta+d\eta) + \mathbf{k}(\zeta+d\zeta).$$

Let $K$ denote string tension along $x$ when the string is at rest, and $\mathbf{K}$ denote the vector tension at the point $\mathbf{p}$ in the present displaced scenario under analysis. The net vector force acting on the infinitesimal string element between points $\mathbf{p}$ and $\mathbf{q}$ is given by the vector sum of the force $-\mathbf{K}$ at $\mathbf{p}$ and the force $\mathbf{K}+ (\partial \mathbf{K}/\partial x)d x$ at $\mathbf{q}$ , that is, $(\partial \mathbf{K}/\partial x)dx$ . 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 $\epsilon \,dx$ , where $\epsilon$ denotes the mass per unit length of the string at rest. Applying Newton's second law gives

$$\frac{\partial \mathbf{K}}{\partial x} = \epsilon \frac{\partial^2 \mathbf{p}}{\partial t^2}$$ (B.31)

where $d x$ 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 $\mathbf{K}$ in terms of the tension $K$ of the string at rest, the elastic constant of the string, and geometrical factors. The displaced string element $\mathbf{p}\mathbf{q}$ is the vector

$$d{\bf s} = \mathbf{i}(dx + d\xi) + \mathbf{j}d\eta + \mathbf{k}d\zeta$$ (B.32)

$$= \left[\mathbf{i}\left(1+\frac{\partial \xi}{\partial x}\right)+ \mathbf{j}\frac{\partial \eta}{\partial x} + \mathbf{k}\frac{\partial \zeta}{\partial x}\right]dx$$ (B.33)

having magnitude

$$ds = \sqrt{\left(1+\frac{\partial \xi}{\partial x}\right)^2 + \left(\frac{\partial \eta}{\partial x}\right)^2 + \left(\frac{\partial \zeta}{\partial x}\right)^2}.$$ (B.34)