05 The Continuum Limit and the Wave Equation
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1 Utah State University Founations of Wave Phenomena Physics, Department of The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University, Recommene Citation Torre, Charles G., "05 The Continuum Limit an the Wave Equation" (2004). Founations of Wave Phenomena. Book This Book is brought to you for free an open access by the Physics, Department of at It has been accepte for inclusion in Founations of Wave Phenomena by an authorize aministrator of For more information, please contact
2 5. The Continuum Limit an the Wave Equation. Our example of a chain of oscillators is nice because it is easy to visualize such a system, namely, a chain of masses connecte by springs. But the ieas of our example are far more useful than might appear from this one simple mechanical moel. Inee, many materials (incluing solis, liquis an gases) have some aspects of their physical response to (usually small) perturbations behaving just as if they were a bunch of couple oscillators at least to a first approximation. In a sense we will explore later, even the electromagnetic fiel behaves this way! This harmonic oscillator response to perturbations leas in a continuum moel to the appearance of wave phenomena in the traitional sense. We caught a glimpse of this when we examine the normal moes for a chain of oscillators with various bounary conitions. Because the harmonic approximation is often a goo first approximation to the behavior of systems near equilibrium, you can see why wave phenomena are so ubiquitous. The key i erence between a wave in some meium an the examples of 4 is that wave phenomena are typically associate with propagation meia (stone, water, air, etc.) which are moele as continuous rather than iscrete. As mentione earlier, our chain of oscillators in 4 can be viewe as a iscrete moel of a continuous (one-imensional) material. We now want to introuce a phenomenological escription of the material in which we ignore the atomic iscreteness of matter. The basic physical iea is reasonably simple. Often times we are intereste in certain macroscopic properties of the material (e.g., the behavior of a plucke guitar string as a function of time an space) an we want to ignore most of the etails of the microscopic make-up of the material since they shoul be irrelevant for the most part. So long as the length scales associate with the macroscopic behavior of the material (e.g., wavelengths) are much larger than the length scales associate with the microscopic structure (e.g., the interparticle spacing) we can approximate the behavior of the material by taking a limit in which we set the interparticle spacing to zero while letting the number of oscillators become arbitrarily large ( approach infinity ). We will have to exercise a little care in this limiting process. Here care means that we keep fixe some macroscopic quantities characterizing the material in which the waves are propagating. Some goo examples of the materials to keep in min are: soun waves in an elastic soli, e.g., in a metal ro; a vibrating string or rope uner tension; soun waves in a gas. Each of these materials will have certain physical parameters which are relevant to the propagation of the wave an which are macroscopic reflections of the e ective oscillator parameters which moel the microscopic behavior of the material. For example each of the three illustrations just mentione will be characterize (in part) by their mass ensity. Let us emphasize that a continuum approximation, by its very nature, will not have universal valiity. For example, if we consier wave phenomena in which the wavelengths are comparable to (or smaller than) the interparticle spacing, then we on t expect our 41
3 moel will accurately moel what is actually happening physically. 5.1 Derivation of the Wave Equation As in 4, we suppose that the equilibrium separation of the oscillators is an we label the equilibrium position of the oscillators by x = j. We can then enote by q(x, t) the isplacement of the j th oscillator from its equilibrium position at time t. Our use of the symbol x, usually reserve for a continuous variable, anticipates our implementation of the strategy wherein the interparticle spacing is so small (compare to the typical sizes of macroscopic phenomena) that we can moel the particles as forming a continuous mass istribution. We rearrange the equations of motion (4.1) into the form (exercise) 2 apple apple q(x, t) t 2 =! {q(x, t) q(x, t)} +!2 {q(x +, t) q(x, t)}. (5.1) We now stuy the right-han sie of this equation in the limit where is very small. In this case we can view q(x, t) as a continuous function of x to a better an better approximation, an we have that (exercises) an 1 {q(x, t) q(x, t)} 1 {q(x +, t) q(x, x=j x=j+/2 In the same manner, the i erence of these terms efines, in the limit as! 0, the secon erivative of q @ 2 q(x, x=j 2 We can therefore write the equation of motion in this approximation as:. (5.4) 2 q(x, 2 =! 2 q(x, 2. (5.5) This is alreay a wave equation, but to get our final form of it we nee to consier the limit as! 0. We o this as follows. First recall that! = r k m, (5.6) Yes, those efinitions for erivatives as limits of i ferences that you learne in calculus class really o come in hany after all! 42
4 We express the spring constant k which represents microscopic information as k = apple, (5.7) where the physical interpretation of the macroscopic constant apple epens upon what we are moeling. For transverse vibrations of a string, apple will represent the tension on the string. For the longituinal vibrations of an elastic meium (e.g., soun waves in a soli), apple will represent the Young s moulus, which etermines the sti ness of the material making up the meium. For compression (soun) waves in air, apple will be the elastic moulus. The quantity apple is one of two macroscopic quantities that are hel fixe when taking the continuum limit. We now have (exercise)! 2 2 = apple m. (5.8) Next we express the mass of the oscillators another microscopic quantity as m = µ, (5.9) where µ, which is a macroscopic quantity, represents the mass per unit length of the continuum meium. The mass per unit length is the other macroscopic quantity that is hel fixe in the continuum limit. We now have! 2 2 = apple µ. (5.10) The continuum limit has the microscopic parameters behaving as! 0,!! 1(also m! 0, k! 1), with the macroscopic parameters apple an µ characterizing the continuous material in question hel fixe. Setting v 2 := µ apple,* (5.5) 2 q(x, 2 = v q(x, 2. (5.11) This is the one-imensional wave equation, which is a funamental example of a partial i erential equation. The equation (5.11), an its generalizations, will be the subject of all of our attention from now on. Note that we are now using partial time erivatives, i.e., we hol x fixe when we vary t to take the time erivative an we hol t fixe when we compute erivatives with Recall that transverse waves have a isplacement which is orthogonal to the irection of propagation of the isplacement, while longitunal waves have a isplacement which is parallel to the irection of propagation. * The notation a := b inicates that we are making a efinition, namely, a is efine to be the quantity b. Thus we can istinguish between equalities that we shoul be able to euce from some other facts (which use = ), an equalities true merely by efinition (which use := ). The notation a := b is close to, but not quite the same as, a b, which means a is ientically equal to b, as in
5 respect to x. Don t forget: x = j formerly labele equilibrium positions of the j th mass an q j (t) enote the isplacement of that mass from its equilibrium position at time t. In the continuum approximation, the chain of oscillators is represente by a line of fixe mass ensity. Points on the line are labele x an the isplacement from equilibrium of a point at x on the line at time t is enote by q(x, t). The secon time erivative of q(x, t) in the wave equation is just the acceleration that features in Newton s secon law. The secon spatial erivative of q(x, t) is the continuum limit of the harmonic nearest neighbor interaction. If the continuum is meant to escribe an elastic meium unergoing longituinal vibrations (soun waves) then the isplacement q(x, t) represents a compression or rarefication of the elastic meium the soun is traveling in at the point x an time t, that is, q represents a longituinal ensity wave. If the continuum is meant to represent a vibrating string uner tension, then q(x, t) represents the eflection of the string at (x, t) from its equilibrium position, that is, q represents a transverse isplacement wave. The parameter v that appears in (5.11) is easily seen to have units of spee (exercise). We shall see that v characterizes the spee of the waves that satisfy (5.11). By the way, an easy way to remember how the velocity factor enters the wave equation is to use imensional analysis: the v 1 is neee to balance the units (exercise). 2 To summarize: the 1-imensional wave equation escribes a continuum of matter in which isplacements of infinitesimal elements of mass experience a Hooke s law restoring force Bounary Conitions In our iscussion of the chain of oscillators we consiere various bounary conitions. Since the wave equation can be viewe as a limiting case of the chain of oscillators, there are corresponing bounary conitions here as well. Let us briefly escribe them here. Of course, the case of no bounary conitions has for the most part been ispense with in the previous paragraphs no bounary conitions were impose there. Typically, one will ignore bounary conitions if one is not near a bounary an one is consiering features of the wave which are much smaller than the spatial omain of the problem. One will then speak of, e.g., waves on a long string. The usual mathematical moel for such a situation is to suppose that the spatial omain x is all of the real numbers, 1 <x<1. Let us consier our original fixe-wall bounary conitions. These are sometimes calle Dirichlet conitions in this context. Here the spatial omain of interest is finite, say, 0 apple x apple L, an the isplacement q = q(x, t) vanishes at the bounaries for all time: q(0,t)=0=q(l, t). (5.12) 44
6 This bounary conition moels a string uner tension with fixe ens (e.g., a guitar string). It also moels soun waves in air in a finite, close region (in a one-imensional approximation e.g., a long close pipe.) A vibrating metal ro (whose spatial cross section is small compare to its length L) which is clampe at one en or a vibrating column of air in a pipe which is open at one en coul be moele with just the bounary conitions q(0,t)=0. Perioic bounary conitions can be hanle similarly, on a finite interval 0 apple x apple L we insist that q(0,t)=q(l, t). (5.13) This conition coul be use to moel a vibrating loop of string, or soun in a close circular pipe. 45
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