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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

$\displaystyle 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 [391].

The displacement of a neighboring point originally at $ \mathbf{q}=(x+d
x,0,0)$ along the string can be specified as

$\displaystyle 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

$\displaystyle \frac{\partial \mathbf{K}}{\partial x} = \epsilon \frac{\partial^2 \mathbf{p}}{\partial t^2}$ (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

$\displaystyle d{\bf s}$ $\displaystyle =$ $\displaystyle \mathbf{i}(dx + d\xi) + \mathbf{j}d\eta + \mathbf{k}d\zeta$ (B.32)
  $\displaystyle =$ $\displaystyle \left[\mathbf{i}\left(1+\frac{\partial \xi}{\partial x}\right)+
\...
...artial \eta}{\partial x} +
\mathbf{k}\frac{\partial \zeta}{\partial x}\right]dx$ (B.33)

having magnitude

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

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

$\displaystyle \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

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

so that

$\displaystyle \mathbf{K}= \left[K+ SY\left(\frac{ds}{dx} - 1\right)\right]\frac{d{\bf s}}{ds}$ (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, and when only the first-order terms are retained, we obtain

$\displaystyle \mathbf{K}= \mathbf{i}\left(K+ SY \frac{\partial \xi}{\partial x}...
...frac{\partial \eta}{\partial x} + \mathbf{k}K\frac{\partial \zeta}{\partial x}.$ (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
$\displaystyle \frac{\partial^2 \xi}{\partial x^2}$ $\displaystyle =$ $\displaystyle \frac{1}{c_b^2} \frac{\partial^2 \xi}{\partial t^2}$ (B.37)
$\displaystyle \frac{\partial^2 \eta}{\partial x^2}$ $\displaystyle =$ $\displaystyle \frac{1}{c_t^2} \frac{\partial^2 \eta}{\partial t^2}$ (B.38)
$\displaystyle \frac{\partial^2 \zeta}{\partial x^2}$ $\displaystyle =$ $\displaystyle \frac{1}{c_t^2} \frac{\partial^2 \zeta}{\partial t^2}$ (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.,

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


Wave Momentum

The physical forward momentum carried by a transverse wave along a string is conveyed by a secondary longitudinal wave [391].

A less simplified wave equation which supports longitudinal wave momentum is given by [391, Eqns. 38ab]

$\displaystyle \epsilon {\ddot \xi}$ $\displaystyle =$ $\displaystyle \left(SY+K\right) \xi^{\prime\prime} +
SY\eta^\prime\eta^{\prime\prime}$ (B.40)
$\displaystyle \epsilon {\ddot \eta}$ $\displaystyle =$ $\displaystyle K \eta^{\prime\prime} +
SY\left(\frac{3}{2}(\eta^\prime)^2\eta^{\...
...rime}
+\xi^{\prime}\eta^{\prime\prime} + \eta^{\prime}\xi^{\prime\prime}\right)$ (B.41)
  $\displaystyle \approx$ $\displaystyle K\eta^{\prime\prime},$ (B.42)

where $ \xi$ and $ \eta$ denote longitudinal and transverse displacement, respectively, and the commonly used ``dot'' and ``prime'' notation for partial derivatives has been introduced, e.g.,
$\displaystyle {\dot \xi}$ $\displaystyle \isdef$ $\displaystyle \frac{\partial \xi}{\partial t}$ (B.43)
$\displaystyle {\xi^{\prime}}$ $\displaystyle \isdef$ $\displaystyle \frac{\partial \xi}{\partial x}.$ (B.44)

(See also Eq.$ \,$(C.1).) We see that the term $ SY\eta^\prime\eta^{\prime\prime}$ in the first equation above provides a mechanism for transverse waves to ``drive'' the generation of longitudinal waves. This coupling cannot be neglected if momentum effects are desired.

Physically, the rising edge of a transverse wave generates a longitudinal displacement in the direction of wave travel that propagates ahead at a much higher speed (typically an order of magnitude faster). The falling edge of the transverse wave then cancels this forward displacement as it passes by. See [391] for further details (including computer simulations).


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