6.1: The Continuum Limit

Consider a discrete space translation invariant system in which the separation between neighboring masses is \(a\). If \(a\) is very small, the discrete system looks continuous. To understand this statement, consider the action of the \(M^{- 1}K\) matrix, (5.8), in the notation of the last chapter in which the degrees of freedom are labeled by their equilibrium positions. The matrix \(M^{- 1}K\) acts on a vector to produce another vector. We have replaced our vectors by functions of \(x\), so \(M^{- 1}K\) is something that acts on a function \(A(x)\) to give another function. Let’s call it \(M^{- 1}K A(x)\). It is easiest to see what is happening for the beaded string, for which \(B = C = T / ma\). Then

\[M^{-1} K A(x)=\left(\frac{T}{m a}\right)(2 A(x)-A(x+a)-A(x-a)). \label{6.1}\]

So far, Equation \ref{6.1} is correct for any \(a\), large or small.

Whenever you say that a dimensional quantity, like the length \(a\), is large or small, you must specify a quantity for comparison. You must say large or small compared to what?¹ In this case, the other dimensional quantity in the problem with the dimensions of length is the wavelength of the mode that we are interested in. Now here is where small \(a\) enters. If we are interested only in modes with a wavelength \(\lambda=2 \pi / k\) that is very large compared to \(a\), then \(ka\) is a very small dimensionless number and \(A(x + a)\) is very close to \(A(x)\). We can expand it in a Taylor series that is rapidly convergent. Expanding Equation \ref{6.1} in a Taylor series gives

\[M^{-1} K A(x)=-\frac{T a}{m} \frac{\partial^{2} A(x)}{\partial x^{2}}+\cdots \label{6.2}\]

where the \(\cdots\) represent higher derivative terms that are smaller by powers of the small number \(ka\) than the first term in Equation \ref{6.2}. In the limit in which we take a to be really tiny (always compared to the wavelengths we want to study) we can replace \(m / a\) by the linear mass density \(\rho_{L}\), or mass per unit length of the now almost continuous string and ignore the higher order terms. In this limit, we can replace the \(M^{- 1}K\) matrix by the combination of derivatives that appear in the first surviving term of the Taylor series (Equation \ref{6.2}),

\[M^{-1} K \rightarrow-\frac{T}{\rho_{L}} \frac{\partial^{2}}{\partial x^{2}} . \label{6.3}\]

Then the equation of motion for \(\psi(x,t)\) becomes the wave equation:

\[\frac{\partial^{2}}{\partial t^{2}} \psi(x, t)=\frac{T}{\rho_{L}} \frac{\partial^{2}}{\partial x^{2}} \psi(x, t) . \label{6.4}\]

The dispersion relation is \[\omega^{2}=\frac{T}{\rho_{L}} k^{2} . \label{6.5}\]

This can be seen directly by plugging the normal mode \(e^{i k x}\) into Equation \ref{6.4}, or by taking the limit of (5.37)-(5.38) as \(a \rightarrow 0\). Equation (6.5) is the dispersion relation for the ideal continuous string. The quantity, \(\sqrt{T / \rho_{L}}\), has the dimensions of velocity. It is called the “phase velocity”, \(v_{\varphi}\). As we will discuss in much more detail in chapter 8 and following, this is the speed with which traveling waves move on the string.

We will call the approximation of replacing a discrete system with a continuous system that looks approximately the same for \(k_{\rightarrow} \gg 1 / a\) the continuum approximation. Really, all of the mechanical systems that we will consider are discrete, at least on the atomic level. However, if we are concerned only about waves with macroscopic wavelengths, the continuum approximation is a very good one.

Philosophy and Speculation

Our treatment of the wave equation in Equation \ref{6.4} is a little unusual. In many treatments of wave phenomena, the wave equation is given a place of honor. In fact, the wave equation is only a restatement of the dispersion relation, Equation \ref{6.5}, which is usually just an approximation to what is really going on. Almost all of the systems that we usually treat with the wave equation are actually discrete at very small distances. We cannot really get all the way to the continuum limit that gives Equation \ref{6.5}. Light waves, which we will study in the chapters to come, for all we know, may be an exception to this rule, and be completely continuous. However, we don’t really have the right to assume even that. It could be that at very short distances, far below anything we can look at today, the nature of light and even of space and time changes in some way so that space and time themselves have some tiny characteristic length scale \(a\). The analysis above shows that this doesn’t matter! As long as we can only look at space and time at distances much larger than \(a\), they look continuous to us. Then because we are scientists, concerned about how the world looks in our experiments, and not how it behaves in some ideal regime far beyond what we can probe experimentally, we might as well treat them as continuous.

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¹A dimensionless quantity does not require this step. A dimensionless number is large if it is much greater than one and small if it is much smaller than one.