## 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
is the
vector

having magnitude

### Non-Stiff String

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

*Young's modulus*(stress/strain--the ``spring constant'' for solids--see §B.5.1), then

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.

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

(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 , it is found that (B.31) separates into the three wave equations

(B.37) | |||

(B.38) | |||

(B.39) |

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.*,

### 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]

(B.40) | |||

(B.41) | |||

(B.42) |

where and denote longitudinal and transverse displacement, respectively, and the commonly used ``dot'' and ``prime'' notation for partial derivatives has been introduced,

*e.g.*,

(B.43) | |||

(B.44) |

(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 [391] for further details (including computer simulations).

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