Separation of Variables in Cartesian Coordinates

Transcription

1 Lecture 9 Separation of Variables in Cartesian Coordinates Phs 3750 Overview and Motivation: Toda we begin a more in-depth loo at the 3D wave euation. We introduce a techniue for finding solutions to partial differential euations that is nown as separation of variables. We first do this for the wave euation written in Cartesian coordinates. In subseuent lectures we will do the same using clindrical and spherical-polar coordinates. The techniue is applicable not onl to the wave euation, but to a wide variet of partial differential euations that are important in phsics. Ke Mathematics: The techniue of separation of variables! I. Separable Solutions Last time we introduced the 3D wave euation, which can be written in Cartesian coordinates as c + +, () t and we spent some time looing at the plane-wave solutions + i ( + + ω,,,, t A e, (a) + i ( + + +ω,,,, t A e, (b) where ω c is the familiar dispersion relation. Euation (a) describes a wave that travels in the ˆ ˆ ˆ + + direction with wavelength λ π, while E. (b) describes a similar wave that travels in the direction. Toda we introduce separation of variables, a techniue that leads to separable (also nown as produc solutions. A separable solution is of the form (,, t) X ( ) Y Z T ( t). (3) That is, the function (,, t) is a product of the functions X ( ), T () t. Y, Z (), and So how do we find these solutions? Well, let's just substitute the rhs of E. (3) into E. () and see what happens. With this substitution E. () becomes D M Riffe -- /6/03

2 Lecture 9 Phs 3750 c X Y Z T X YZT + XY ZT + XYZ T, (4) where we have suppressed the independent variables and the double prime indicates the second derivative of the function with respect to its argument. Note that because each of these functions is a function of onl one variable, all derivatives are now ordinar derivatives. Euation (4) loos ind of ugl, but notice what happens if we divide E. (4) b (,, t) X ( ) Y Z T ( t). We then get c T T X Y Z + +. (5) X Y Z Now here is where the magic happens. Notice that each term is onl a function of its associated independent variable. So for eample, if we var t onl the term on the lhs of E. (5) can var. But, because none of the other terms depends upon t, the rhs cannot var, which means that the lhs cannot var, which means that T T is independent of t! Following the same logic for each of the other terms means that each term is constant. So we can write c T X Y Z α, β, γ, δ T X Y Z (6a) (6d) where the constants α, β, γ, and δ are nown as separation constants. However, onl three of them are independent because Euation (5) tells us that α β + γ + δ. (7) Notice, b demanding a product solution the partial differential wave euation has transformed into four ordinar differential euations. We can easil solve all of these euations. Let's start with the euation for T ( t), T () t α c T () t 0. (8) This is essentiall the harmonic oscillator euation (if α is real and <0), and so it has the two, linearl independent solutions T ± c α t () t T 0 e, (9a) where 0 T is some undetermined constant. Similarl, the X, Y, and Z euations have solutions D M Riffe -- /6/03

3 Lecture 9 Phs 3750 X Y Z ± β () t X 0 e, (9b) ± γ () t Y 0 e, (9c) ± δ () t Z 0 e, (9d) Putting this all together gives us a solution of the form ± β ± γ δ ± c α t (,, t) X Y Z T e e e ± e. (0) II. Some Phsics Added In Notice that the solution in E. (0) can have all sorts of behavior. For eample, if the constant β is real and positive, then the dependent part of the solution can either eponentiall increase or decrease with increasing (depending upon the sign in the eponent). This points out the fact that separation-of-variables solutions often give ou more than ou reall need in an given situation. For eample, if we are looing for solutions to the wave euation that are phsicall meaningful as, we are (probabl) not going to be interested in a solution that eponentiall increases with increasing. With that in mind, let's see what the constraints on the separation constants must be if we are onl interested in purel oscillator solutions (in all independent variables). Let's consider the dependent part of the solution. If e ± β is to onl oscillate then i it must be of the form e ±, where is real. Then, if we write a generall comple β as β β + iβ we have β + i β i which implies β + iβ. Thus β 0 and β. That is, β is real and negative. Similarl, γ and δ must be real and negative. E. (7) then implies that α is also real and negative. So if we rename the constants as β, γ, δ, α ω c, (a) (d) (where,,, and ω are all real) then E. (0) can be rewritten as ± i ± i ± i iω t (,, t) A e e e e ± () where A X 0 Y0 Z0 T0 is an arbitrar constant. E. (7) can now be written as ω c + +, which is the well nown dispersion relation! Now,, and D M Riffe -3- /6/03

4 Lecture 9 Phs 3750 can be positive or negative, so we don't need the ± on the spatial-function eponents, and so we are left with two linearl independent solutions + i ( + + ω,,,, t A e, (3a) + i ( + + +ω,,,, t A e, (3b) which are the plane-wave solutions of E. ()! So, b demanding that the separable solutions to the wave euation oscillate, we have ended up with the plane-wave solutions that we discussed in the last lecture. III. Utilit of the Separable Solutions Because the are product solutions, the separable solutions are prett specialied. That is, there are man solutions to the wave euation that cannot be written as a product solution. And so ou ma as, what is the usefulness of these solutions? There are two parts to the answer. The first is that sometimes these solutions have intrinsic interest. For eample, in the case of the time-dependent Schrödinger euation the separable solutions (with some appropriate phsics thrown in) are the energ eigenstates of the sstem. The second, and perhaps more important, part of the answer is that a basis can be constructed from the set of separable solutions that can be used to represent an solution. Let's thin about this statement with regards to the wave euation. Let's first consider the D wave euation. Bac in the Lecture 7 notes we solved the initial-value problem on the interval < <. The solution to that problem can be written as bˆ ( ) ( + ) ( ) i t bˆ ω i ωt, t d aˆ + e + aˆ e, (4) π i c i c where â ( ) and bˆ ( ) are the Fourier transforms of the initial conditions, and ω c is the dispersion relation. Notice that in E. (4) we have written the general solution (, t) as a linear combination of the functions i( + ω e i and ( ωt e ). Although we did not go through the separation of variables procedure for the D wave euation to i t produce these solutions, b writing these functions as e e i ω i iωt and e e, we readil see that the are indeed product solutions. Similarl, for the 3D wave euation we can write the solution to the initial value problem as a linear combination of the separable (plane-wave) solutions [E. (3)] D M Riffe -4- /6/03

5 Lecture 9 where ( r t) Phs 3750 bˆ ( ) i + t bˆ r ω i( r ωt d ) aˆ + e + aˆ e, (5) 3 ( π ) i c i c 3, r + +, d 3 d d d, c + + bˆ ( ) are the 3D Fourier transforms of the initial conditions. ω, and â ( ) and In the coming lectures we will loo at separable solutions in clindrical and sphericalpolar coordinates. While we will not do a lot with the general solution written as a linear combination of these solutions, we should eep in mind that, in principle, it can be done. Eercises **9. Heat Euation. A partial differential euation nown as the heat euation, which is used to describe heat or temperature flow in an object, is given b, where λ >0. λ t (a) If there is no or dependence to the problem, write down a simplified version of this euation. (b) Use separation of variables to find two ordinar differential euations, one in and one in t. What are the orders of these two euations? Are the linear or nonlinear? (c) Find the general solutions to the two ordinar differential euations. Thus write down the general separable solutions to the heat euation. [Note: there should be two linearl independent solutions.] (d) In solving the heat euation, ou should have found that X ( ) X ( ) C, where C is some constant. Assume that this constant is real. If C > 0, describe the behavior of the solutions vs and t. In what was are these solutions lie solutions to the wave euation? In what was are the different? (e) If C < 0, describe the behavior of the solutions vs and t. In what was are these solutions lie solutions to the wave euation? In what was are the different? D M Riffe -5- /6/03

6 Lecture 9 Phs 3750 **9. Normal Modes Inside a Rectangular Room: Here we consider sound waves inside a rectangular room with one corner located at the origin and occuping space as indicated: 0 < < L, 0 < < L, 0 < < L. While we could deal with displacement (a vector field), it is easier to thin about the pressure which is also governed b the wave euation, but is a scalar field. The boundar conditions on the p p pressure p (,, t) are 0 at 0 and L, 0 at 0 and L, and p 0 at 0 and L. (a) As discussed in the notes, the separable traveling-wave solutions to the wave ± ± i euation can be written (with replaced b p ) as ( + + mω t p ),,,, t A e where ω c + +. However, we could have epressed the separable solutions as the standing-wave solutions p ± [ ][ sin( ) + B cos( ) ] (,, t) A sin( ) + B cos( ) A ± iωt [ A sin( ) + B cos( ) ] e Starting with this standing-wave form, find the specific solutions that satisf the above boundar conditions. Note, these solutions should be labeled with three integers; call these integers n, n, and n. Epress the allowed freuencies ω as a function of n, n, and n, the wave velocit c, and the dimensions of the room. (b) The solutions that ou found in (a) describe the normal modes for sound waves in a tpical room where L and L are the widths and L is the height of the room. Consider a room of dimensions 0 ft ft 8 ft. Calculate the freuencies ( f ( n, n, n ) ω( n, n, n ) π ) of the 0 lowest-freuenc room modes, and order them from lowest to highest freuenc. For the speed of sound ou ma use 330 m/s. (c) These lowest-freuenc modes often wrea havoc with sound reproduction because the serve to amplif, through resonance, reproduced freuencies near their (resonance) freuencies. This is especiall troublesome if two (or more) modes have freuencies that are close together (degenerate). For eample a 0' 0' 0' room would have its three lowest modes at eactl the same freuenc. Are an of the modes ou calculated in (b) degenerate or nearl so? D M Riffe -6- /6/03