The wave equation. The basic equation and it's fundamental solutions on bounded domains. u t t = c 2 u x x. An amalgam of sections 1.5 and 2.2.

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Beverley Perkins
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1 The wave equation An amalgam of sections.5 and.. The basic equation and it's fundamental solutions on bounded domains Suppose that uhx, tl describes the displacement from equlibrium of a vibrating string whose ends are fixed. The fundamental equation governing this motion is u t t = c u x x. This is called the wave equation. It's pretty easy to see that any function of the form u n Hx, tl = b n sinhn xl coshc n tl is a solution. Possible values for n are dictated by the length of the string. If the length is p, then n may be any integer. Here's an animation of perhaps the simplest such solution. Do@Plot@Sin@xD Cos@tD, 8x,, Pi<, AspectRatio Ø Automatic, PlotRange Ø 8-, <, Ticks Ø 88, p<, 8-, <<D, 8t,, Pi, Pi ê <D; p - Here's a higher frequency solution. DoAPlotA ÅÅÅÅ Sin@3 xd Cos@3 td, 8x,, Pi<, 3 AspectRatio Ø Automatic, PlotRange Ø 8-, <, Ticks Ø 88, p<, 8-, <<E, 8t,, Pi, Pi ê <E;

2 p - A fundamental property of linear PDEs is that solutions may be super-imposed. DoAPlotASin@xD Cos@tD + ÅÅÅÅ Sin@3 xd Cos@3 td, 8x,, Pi<, 3 AspectRatio Ø Automatic, PlotRange Ø 8-, <, Ticks Ø 88, p<, 8-, <<E, 8t,, Pi, Pi ê <E; p - Just as with the heat equation, a major question is how to deal with more or less arbitrary conditions. We will look quite closely at this question later. Derivation of the wave equation The wave equation is essentially a consequence of Newton's second law. We assume that the string has a uniform linear density r and constant tesion T. We then consider the motion of a small portion of the string over the interval from x to x + D x. The vertical accelleration of this portion is due entirely to the vertical components of the tension in the string: VHx, tl and VHx + D x, tl.

3 3 x x+dx The total force on this portion of the string is the difference between these two components and, by Newton's second law, this total force must equal the mass of this portion times its acceleration. In symbols, or, dividing by D x, V Hx + D x, tl - V Hx, tl = r Dx u tt V Hx + D x, tl - V Hx, tl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ D x = r u tt. After taking a limit as D x Ø, we get V x = r u t t. It turns out that the horizontal H and vertical V components of the tension at a point are related to one another via the slope of the string, u x : V = H u x. T V H

4 4 Assuming the displacement of the string is small, H º T. Thus plugging T u x for V into V x = r u t t, we get the wave equation in the form T u x x = r u t t. Solution on It's important to realize that this equation, or a close relative, describes a vast array of vibrational phenomena. The book, for example, uses the fundamental conservation law to derive the wave equation to describe the propogation of waves through elastic media. In this context, it makes sense to study the wave equation on. ü d'alembert's formula The general Cauchy problem for the wave equation on is u tt = c u xx, u Hx, L = f HxL, and u t Hx, L = g HxL for x œ. The term Cauchy problem is just a fancy word for IVP. Thus our mission is to solve the wave equation subject to a given intial postion and velocity. Note that FHx ± c tl provides two independent solutions of the wave equation for any twice differentiable function F. The most general solution of the wave equation may be written in the form u Hx, tl = F Hx + c tl + G Hx - c tl, where F and G are arbitrary twice differentiable functions. Thus any solution consists of a left travelling wave and a right travelling wave. We would now like to choose F and G to satisfy the initial conditions. This is accomplished by the following formula, called d'alembert's formula: u Hx, tl = ÅÅÅÅ x+c t Hf Hx + c tl + f Hx - c tll + ÅÅÅÅÅÅÅÅ c g HsL s. x-c t It may be verified by direct substitution that u is a solution to the problem. Note that d'alembert's formula is of the form FHx + c tl + GHx - c tl, where F HxL = G HxL = ÅÅÅÅ f HxL + ÅÅÅÅÅÅÅÅ c x g HsL s. ü Notes on FTC It's easy to verify that d'alembert's formula satifies the first initial condition, uhx, L = f HxL. To verify the second, it is neccessary to compute u t. To do so, you must use the fundamental theorem of calculus in the form t d ÅÅÅÅÅÅÅ dt g HsL s = g HtL. Used in conjuction with the chain rule, this yields x+c t d ÅÅÅÅÅÅÅ dt g HsL s = c g Hx + c tl.

5 5 Thus, d ÅÅÅÅÅÅÅ dt x+c t i j ÅÅÅÅÅÅÅÅ k c x-c t = ÅÅÅÅÅÅÅÅ c g HsL s y z = { ÅÅÅÅÅÅÅÅ c This allows us to plug in t = to obtain u t Hx, L = ghxl. d ÅÅÅÅÅÅÅ dt i j k x+c t x-c t g HsL s - g HsL s y z { Hg Hx + c tl c - g Hx - c tl H-cLL = ÅÅÅÅ Hg Hx + c tl + g Hx - c tll. ü Applying d'almebert's formula Suppose we wish to solve the wave equation u t t = 4 u x x applied to a very long string (essentially infinite in length) whose initial position is given by a bell shape curve uhx, L = -x and whose initial velocity is zero. d'alembert's formula implies that the solution is u Hx, tl = ÅÅÅÅ I -Hx+ tl + -Hx- tl M. Here's an animation of the motion of the string. You should be able to see this directly from the graph of the initial condition. DoA PlotAI -Hx+ tl + -Hx- tl M ë, 8x, -5, 5<, PlotRange Ø 88-5, 5<, 8, <<, AspectRatio Ø ê 3E, 8t,,,.5<E; An alternative way to visualize the motion is via the solution surface.

6 6 Plot3DAI -Hx+ tl + -Hx- tl M ë, 8x, -5, 5<, 8t,, <, ViewPoint -> 8, -,.<, AxesLabel Ø 8"x", "t", ""<E; t -.5 x 4 Application of d'alembert's formula is more difficult when the initial velocity is non-zero, due to the integration required. In some cases, this may be done analytically, but it may require a numerical integration. ü Example Solve the wave equation u t t = u x x subject to uhx, L = and u t Hx, L = x ê H + x 4 L. Sketch the solution for several choices of t and interpret the result physically. ü Solution In this case, we may use d'alembert's formula to find the solution analytically. u Hx, tl = ÅÅÅÅ x+t s ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ x-t + Hs L s = ÅÅÅÅ arctan Hs x+t L À x-t = ÅÅÅÅ Harctan HHx + tl L - arctan HHx - tl LL Here's the graph of arctanhx L.

7 7 D, 8x, -, <, AspectRatio Ø AutomaticD; The solution is the superpostion of two waves half this size; one is shifted to the left and the other is shifted to the right and reflected. At time t =, the two waves cancel each other out and we get zero - the initial condition. A short time later, a downward hump appears to the left and an upward hump appears to the right. These two waves travel in their respective direction as time progresses. Here's an animation of this process. g@x_d := x ê H + x 4 L; u@x_, t_d = ÅÅÅÅ Integrate@g@sD, 8s, x - t, x + t<d; Do@ Plot@u@x, td, 8x, -5, 5<, PlotRange Ø 88-5, 5<, 8-, <<, AspectRatio Ø ê 3D, 8t,, 4,.<D; We can interpret this as follows. We impart an inital velocity to a very long string at rest. The precise initial velocity is given by ghxl = x ê H + x 4 L. Here is the graph of this function.

8 8 x ê H + x 4 L, 8x, -, <, AspectRatio Ø AutomaticD; We might do this by striking the string in two locations with equal, but opposite vertical forces. The animation shows the evolution of the resulting wave. ü Example Solve the wave equation u t t = 9 u x x subject to uhx, L = -x and u t Hx, L = - -x. In this case, the integral may be expressed in terms of the error function. f@x_d := -x ; g@x_d := - -x ; u@x_, t_d = ÅÅÅÅ Hf@x + 3 td + f@x - 3 tdl + ÅÅÅÅ Integrate@g@sD, 8s, x - 3 * t, x + 3 * t<d 6 ÅÅÅÅ I -H-3 t+xl + -H3 t+xl M - ÅÅÅÅÅÅÅ è!!! p HErf@3 t - xd + Erf@3 t + xdl While Mathematica has built in techniques to evaluate the error function, a more generally applicable technique is to use the numerical integration command NIntegrate. u@x_, t_d := ÅÅÅÅ Hf@x + 3 td + f@x - 3 tdl + ÅÅÅÅ NIntegrate@g@sD, 8s, x - 3 * t, x + 3 * t<d; 6 Do@ Plot@u@x, td, 8x, -5, 5<, PlotRange Ø 88-5, 5<, 8-, <<, AspectRatio Ø ê 3D, 8t,,,.<D;

9 Such a wave might be produced by holding the string at a particular initial displacement, then throwing it in the other direction.

Applications of integrals

ApplicationsofIntegrals.nb Applications of integrals There are many applications of definite integrals and we cannot include all of them in a document intended to be a review. However there are some very