Boyce/DiPrima 9th ed, Ch 10.8 Appendix B: Derivation of the Wave EquationElementary Differential Equations and Boundary Value Problems, 9th edition, by William E. Boyce and Richard C. DiPrima, ©2009 by John Wiley & Sons, Inc.

In this appendix we derive the wave equation in one space dimension as it applies to the transverse vibrations of an elastic string or cable.

The elastic string can be thought of as a violin string, a guy wire, or possibly an electric power line.

The same equation, however, with the variables properly interpreted, occurs in many other wave phenomena having only one significant space variable.

Assumptions (1 of 2)

Consider a perfectly flexible elastic string stretched tightly between two fixed supports at the same horizontal level.

Let the x-axis lie along the axis of the string with the endpoints located at x = 0 and x = L.

If the string is set in motion at some initial time t = 0 and is thereafter left undisturbed, it will vibrate freely in a vertical plane, provided that the damping effects, such as air resistance, are neglected.

Assumptions (2 of 2)

To determine the differential equation governing this motion, we will consider forces acting on a small element of the string of length x lying between the points x = x0 and x = x0 + x.

Assume the motion of string is small and as a consequence, that each point on the string moves solely in a vertical line.

Let u(x,t) be the vertical displacement of the point x at time t.

Assume the tension T(x,t) in string acts in tangential direction, and let denote the mass per unit length of the string.

Horizontal Components

Newton’s law, as it applies to the element x of the string, states that the net external force, due to the tension at the ends of the element, must be equal to the product of the mass of the element and the acceleration of its mass center.

Since there is no horizontal acceleration, the horizontal components must satisfy

If we denote the horizontal component of the tension by H, then the above equation states that H is independent of x.

0)cos(),()cos(),( txTtxxT

Vertical Components (1 of 3)

The vertical components satisfy

where x is the coordinate of the center of mass of the element of string under consideration.

The weight of the string, which acts vertically downward, is assumed to be negligible.

),()sin(),()sin(),( txuxtxTtxxT tt

Vertical Components (2 of 3)

The vertical components satisfy

If the vertical component of T is denoted by V, then the above equation can be written as

Passing to the limit as x 0 gives

),()sin(),()sin(),( txuxtxTtxxT tt

),(),(),(

txux

txVtxxVtt

),(),( txutxV ttx

Vertical Components (3 of 3)

Note that

Then our equation from the previous slide,

becomes

Since H is independent of x, it follows that

If the motion of the string is small, then we can replace H = Tcos by T. Then our equation reduces to

),()(tan)(),( txutHtHtxV x

ttxx uHu )(

),,(),( txutxV ttx

/, 22 Tauua ttxx

ttxx uHu

Wave Equation

Our equation is

We will assume further that a2 is constant, although this is not required in our derivation, even for small motions.

This equation is the wave equation for one space dimension.

Since T has the dimension of force, and that of mass/length, it follows that a has the dimension of velocity.

It is possible to identify a as the velocity with which a small disturbance (wave) moves along the string.

The wave velocity a varies directly with the tension in the string and inversely with the density of the string material.

/, 22 Tauua ttxx

Telegraph Equation

There are various generalizations of the wave equation

One important equation is known as the telegraph equation

where c and k are nonnegative constants.

The terms cut, ku, and F(x,t) arise from a viscous damping force, elastic restoring force, and external force, respectively.

Note the similarity between this equation and the spring-mass equation of Section 3.8; the additional a2uxx term arises from a consideration of internal elastic forces.

The telegraph equation also governs the flow of voltage or current in a transmission line (hence its name).

ttxx uua 2

),(2 txFuakucuu xxttt

Multidimensional Wave Equations

For a vibrating system with more than one significant space coordinate, it may be necessary to consider the wave equation in two dimensions,

or in three dimensions,

ttyyxx uuua 2

ttzzyyxx uuuua 2