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
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
Let's now assume the string is perfectly flexible (zero stiffness) so that the direction of the force vector is given by the unit vector 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 at any position is the rest tension plus the incremental tension needed to stretch it the fractional amount
where no geometrical limitations have yet been placed on the magnitude of and , 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 , , and be small compared with unity. This is the same assumption ( ) necessary to derive the usual wave equation for transverse vibrations only in the - plane.
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 , it is found that (B.31) separates into the three wave equations
where is the longitudinal wave velocity, and 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.,
where and denote longitudinal and transverse displacement, respectively, and the commonly used ``dot'' and ``prime'' notation for partial derivatives has been introduced, e.g.,
(See also Eq.(C.1).) We see that the term 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  for further details (including computer simulations).
Properties of Gases
Properties of Elastic Solids