Non-Stiff String

Let's now assume the string is perfectly flexible (zero stiffness) so that the direction of the force vector $\mathbf{K}$ is given by the unit vector $d{\bf s}/ds$ tangent to the string. (To accommodate stiffness, it would be necessary to include a force component at right angles to the string which depends on the curvature and stiffness of the string.) The magnitude of $\mathbf{K}$ at any position is the rest tension $K$ plus the incremental tension needed to stretch it the fractional amount

$$\frac{ds - dx}{dx} = \frac{ds}{dx} - 1.$$

If $S$ is the constant cross-sectional area of the string, and $Y$ is the Young's modulus (stress/strain--the ``spring constant'' for solids--see §B.5.1), then

$$\left\vert\mathbf{K}\right\vert = K+ SY\left(\frac{ds}{dx} - 1\right),$$

so that

$$\mathbf{K}= \left[K+ SY\left(\frac{ds}{dx} - 1\right)\right]\frac{d{\bf s}}{ds}$$ (B.35)

where no geometrical limitations have yet been placed on the magnitude of $\mathbf{p}$ and $\partial\mathbf{p}/\partial x$ , other than to prevent the string from being stretched beyond its elastic limit.

The four equations (B.31) through (B.35) can be combined into a single vector wave equation that expresses the propagation of waves on the string having three displacement components. This differential equation is nonlinear, so that superposition no longer holds. Furthermore, the three displacement components of the wave are coupled together at all points along the string, so that the wave equation is no longer separable into three independent 1D wave equations.

To obtain a linear, separable wave equation, it is necessary to assume that the strains $\partial\xi/\partial x$ , $\partial\eta/\partial x$ , and $\partial\zeta/\partial x$ be small compared with unity. This is the same assumption ( $\vert\partial\eta/\partial x\vert\ll 1$ ) necessary to derive the usual wave equation for transverse vibrations only in the $y$ -$x$ plane.

When (B.35) is expanded into a Taylor series in the strains (see (B.33)), and when only the first-order terms are retained, we obtain

$$\mathbf{K}= \mathbf{i}\left(K+ SY \frac{\partial \xi}{\partial x}\right) + \mathbf{j}K\frac{\partial \eta}{\partial x} + \mathbf{k}K\frac{\partial \zeta}{\partial x}.$$ (B.36)

This is the linearized wave equation for the string, based only on the assumptions of elasticity of the string, and strain magnitudes much less than unity. Using this linearized equation for the force $\mathbf{K}$ , it is found that (B.31) separates into the three wave equations

$$\frac{\partial^2 \xi}{\partial x^2} = \frac{1}{c_b^2} \frac{\partial^2 \xi}{\partial t^2}$$ (B.37)

$$\frac{\partial^2 \eta}{\partial x^2} = \frac{1}{c_t^2} \frac{\partial^2 \eta}{\partial t^2}$$ (B.38)

$$\frac{\partial^2 \zeta}{\partial x^2} = \frac{1}{c_t^2} \frac{\partial^2 \zeta}{\partial t^2}$$ (B.39)

where $c_b=\sqrt{SY/\epsilon }$ is the longitudinal wave velocity, and $c_t=\sqrt{K/\epsilon }$ is the transverse wave velocity.

In summary, the two transverse wave components and the longitudinal component may be considered independent (i.e., ``superposition'' holds with respect to vibrations in these three dimensions of vibration) provided powers higher than 1 of the strains (relative displacement) can be neglected, i.e.,

$$\left\vert\frac{\partial \xi}{\partial x}\right\vert \ll 1, \quad \left\vert\frac{\partial \eta}{\partial x}\right\vert \ll 1,$$   and$$\left\vert\frac{\partial \zeta}{\partial x}\right\vert \ll 1.$$