## 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

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.*,

and

### 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).

**Next Section:**

Properties of Gases

**Previous Section:**

Properties of Elastic Solids