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Sturm-Liouville Differential Equation

One of the original purposes of the Sturm-Liouville differential equation is the mathematical formulation of the vibration frequency and the amplitude profile of a vibrating string. Such a string has generally a space dependent tension and mass density:

A cable of variable mass density, say , suspended vertically from a fixed support is a good example. Because of its weight, this cable is under variable tension, say , along its length. Let be the instantaneous transverse displacement of the string.

Application of Newton's law of motion, mass acceleration=force, to the mass of each segment leads to the wave equation for the transverse amplitude ,

The force (per unit length) on the right hand side is due to the bending of the cable. Suppose the cable is imbedded in an elastic medium. The presence of such a medium is taken into account by augmenting the force density on the right-hand side. Being linear in the amplitude , the additional restoring force density is

Here is the position dependent Hooke's constant experienced by the cable segment . Consequently, the augmented wave equation is

 (31)

This is the equation of motion for a string imbedded in an elastic medium. Being linear and time-independent, the system has normal modes. They have vibrational frequencies and amplitudes

Thus the spatial amplitude profile of such a mode satisfies

 (32)

For the purpose of mathematical analysis one writes this order linear o.d.e. in terms of the standard mathematical notation

and thereby obtains what is known as the Sturm-Liouville (S-L) equation,

However, it is appropriate to point out that actually any order linear o.d.e. can be brought into this Sturm-Liouville'' form. Indeed, consider the typical order homogeneous differential equation

We wish to make the first two terms into a total derivative of something. In that case, the d.e. will have its S-L form. To achieve this, divide by and then multiply by

The result is

or

in terms of newly defined coefficients. Combining the first two terms one has
 (33)

This is known as the Sturm-Liouville equation. In considering this equation, we shall make two assumptions about its coefficients.

The first one is

 0 0

in the domain of definition, . We make this assumption because nature demands it in the problems that arise in engineering and physics.

The second assumption we shall make is that the coefficients , , and are continuous on . We make this assumption because it entails less work. It does happen, though, that , , or are discontinuous. This usually expresses an abrupt change in the propagation medium of a wave, for example, the tension or the mass density of string, or the refractive index in a wave propagation medium. This discontinuity can be handled by applying junction conditions'' for across the discontinuity.

Next: Homogeneous Boundary Conditions Up: Sturm-Liouville Systems Previous: Sturm-Liouville Systems   Contents   Index
Ulrich Gerlach 2010-12-09