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:

$$T(x)= \hbox{tension ~[force]}$$

$$\rho (x)=\hbox{density }~\left[ \frac{\hbox{mass}}{\hbox{length}} \right]$$

A cable of variable mass density, say $\rho (x)$ , suspended vertically from a fixed support is a good example. Because of its weight, this cable is under variable tension, say $T(x)$ , along its length. Let $v(x,t)$ be the instantaneous transverse displacement of the string.

Figure 3.2: Instantaneous amplitude profile of a suspended cable with variable tension and variable mass density.

Application of Newton's law of motion, mass$\times$ acceleration=force, to the mass $\rho (x) \Delta x$ of each segment $\Delta x$ leads to the wave equation for the transverse amplitude $v(x,t)$ ,

$$\rho (x)\frac{\partial^2 v(x,t)}{\partial t^2} = \frac{\partial}{\partial x} T(x)\frac{\partial v(x,t)}{\partial x}\,.$$

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 $v(x,t)$ , the additional restoring force density $[\textrm{force/length}]$ is

$$-k(x)v(x,t) ~~.$$

Here $k(x)\Delta x$ is the position dependent Hooke's constant experienced by the cable segment $\Delta x$ . Consequently, the augmented wave equation is

$$\boxed{ \rho (x)\frac{\partial^2 v(x,t)}{\partial t^2} = \frac{\partial}{\partial x} T(x)\frac{\partial v(x,t)}{\partial x} -k(x)v(x,t) \,. }$$ (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 $\omega$ and amplitudes

$$v(x,t)= u(x)\cos\omega (t-t_0)\,.$$

Thus the spatial amplitude profile $u(x)$ of such a mode satisfies

$$\left[ \frac{d}{dx} T(x) \frac{d}{dx} +\lambda\rho (x)-k(x)\right] u(x)=0\,, ~~\qquad~~\lambda =\omega^2\,.$$ (32)

For the purpose of mathematical analysis one writes this $2^{\textrm{nd}}$ order linear o.d.e. in terms of the standard mathematical notation

$$p(x) = T(x)$$

$$\rho (x) = \rho (x)$$

$$q(x) = -k(x)~~,$$

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

$$\frac{d}{dx} \left(p(x)\frac{du}{dx}\right) +[q(x)+\lambda \rho (x)] u=0\,.$$

However, it is appropriate to point out that actually any $2^{\textrm{nd}}$ order linear o.d.e. can be brought into this ``Sturm-Liouville'' form. Indeed, consider the typical $2^{\textrm{nd}}$ order homogeneous differential equation

$$P(x)u'' +Q(x)u' +(R(x)+\lambda )u=0\,.$$

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 $P(x)$ and then multiply by

$$e^{ \int^x\frac{Q}{P} dt} = p(x)\,.$$

The result is

$$e^{ \int^x \frac{Q}{P} dt} u'' +\frac{Q}{P}e^{ \int^x\frac{Q}{P}... ...\int^x\frac{Q}{P}dt} +\frac{\lambda}{P} e^{ \int^x \frac{Q}{P}dt}\right) u=0$$

or

$$p(x) u'' +p'(x) u' +(q(x)+\lambda \rho (x))u=0$$

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

$$\boxed{ \frac{d}{dx} \left( p(x) \frac{du}{dx}\right) +[q(x)+\lambda \rho (x)]u=0\,. }$$ (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

$$\rho (x) >$$ 0

$$p(x) >$$ 0

in the domain of definition, $a<x<b$ . 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 $q(x)$ , $\rho (x)$ , $p(x)$ and $p'(x)$ are continuous on $a<x<b$ . We make this assumption because it entails less work. It does happen, though, that $p'(x)$ , $q(x)$ , or $\rho (x)$ 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 $u(x)$ across the discontinuity.