MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and EngineeringSpring 2008For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.97 9797 972008-01-14 22:31:34 / rev 55add9943bf1Chapter 10 Springs Everything is a spring! The main example in this chapter is waves, which illustrate springs, discretization, and special cases – a fitting, unified way to end the book. 10.1 Waves Ocean covers most of the earth, and waves roam most of the ocean. Waves also cross pud-dles and ponds. What makes them move, and what determines their speed? By applying and extending the techniques of approximation, we analyze waves. For concreteness, this section refers mostly to water waves but the results apply to any fluid. The themes of section are: Springs are everywhere and Consider limiting cases. 10.1.1 Dispersion relations The most organized way to study waves is to use dispersion relations. A dispersion re-lation states what values of frequency and wavelength a wave can have. Instead of the wavelength λ, dispersion relations usually connect frequency ω, and wavenumber k, which is defined as 2π/λ. This preference has an basis in order-of-magnitude reasoning. Wave-length is the the distance the wave travels in a full period, which is 2π radians of oscillation. Although 2π is dimensionless, it is not the ideal dimensionless number, which is unity. In 1 radian of oscillation, the wave travels a distance λ λ¯ ≡ 2π. The bar notation, meaning ‘divide by 2π’, is chosen by analogy with h and ~. The one-radian forms such as ~ are more useful for approximations than the 2π-radian forms. The Bohr radius, in a form where the dimensionless constant is unity, contains ~ rather than h. Most results with waves are similarly simpler using λ¯ rather than λ. A further refinement is to use its inverse, the wavenumber k = 1/λ¯. This choice, which has dimensions of inverse length, parallels the definition of angular frequency ω, which has dimensions of inverse time. A relation that connects ω and k is likely to be simpler than one connecting ω andλ¯, although either is simpler than one connecting ω and λ. The simplest dispersion relation describes electromagnetic waves in a vacuum. Their fre-quency and wavenumber are related by the dispersion relation98 9898 982008-01-14 22:31:34 / rev 55add9943bf16.055 / Art of approximation 98 ω = ck, which states that waves travel at velocity ω/k = c, independent of frequency. Dispersion relations contain a vast amount of information about waves. They contain, for example, how fast crests and troughs travel: the phase velocity. They contain how fast wave packets travel: the group velocity. They contain how these velocities depend on frequency: the dispersion. And they contain the rate of energy loss: the attenuation. 10.1.2 Phase and group velocities The usual question with a wave is how fast it travels. This question has two answers, the phase velocity and the group velocity, and both depend on the dispersion relation. To get a feel for how to use dispersion relations (most of the chapter is about how to calculate them), we discuss the simplest examples that illustrate these two velocities. These analyses start with the general form of a traveling wave: f (x, t) = cos(kx − ωt), where f is its amplitude. Phase velocity is an easier idea than group 0011t = t1:t = t2:cos(kx − ωt1)cos(kx − ωt2)velocity so, as an example of divide-and-conquer reasoning and of maximal lazi-ness, study it first. The phase, which is the argument of the cosine, is kx− ωt. A crest occurs when the phase is zero. In the top wave, a crest occurs when x = ωt1/k. In the bottom wave, a crest occurs when x = ωt2/k. The difference ω(t2 − t1)kis the distance that the crest moved in time t2 −t1. So the phase velocity, which is the velocity of the crests, is distance crest shifted ω = = .vph time taken kTo check this result, check its dimensions: ω provides inverse time and 1/k provides length, so ω/k is a speed. Group velocity is trickier. The word ‘group’ suggests that the concept involves more than one wave. Because two is the first whole number larger than one, the simplest illustration uses two waves. Instead of being a function relating ω and k, the dispersion relation here is a list of allowed (k, ω) pairs. But that form is just a discrete approximation to an official dis-persion relation, complicated enough to illustrate group velocity and simple enough to not create a forest of mathematics. So here are two waves with almost the same wavenumber and frequency:99 9999 992008-01-14 22:31:34 / rev 55add9943bf1      ! 99 10 Springs f1 = cos(kx − ωt), f2 = cos((k + ∆k)x − (ω + ∆ω)t), where ∆k and ∆ω are small changes in wavenumber and frequency, respectively. Each produce an envelope (thick line). The envelope itself looks like a cosine. After waiting a wave has phase velocity vph = ω/k, as long as ∆k and ∆ω are tiny. The figure shows their sum. xxxA+B=The point of the figure is that two cosines with almost the same spatial frequency add to velocity, which motivates the following definition: Group velocity is the phase velocity of the envelope. With this pictorial definition, you can compute group velocity for the wave f1 + f2. As suggested in the figures, adding two cosines produces a a slowly varying envelope times a rapidly oscillating inner function. This trigonometric identity gives the details: cos(A + B) = 2 cos B − A × cos A + B .2 2 | {z } | {z } envelope inner If A ≈ B, then A − B ≈ 0, which makes the envelope vary slowly. Apply the identity to the sum: f1 + f2 = cos(kx − ωt) + cos((k + ∆k)x− (ω + ∆ω)t).|{z}| {z } AB while, each wave changes because of the ωt or (ω + ∆ω)t terms in their phases. So the sum and its envelope change to this: xxxA+B=The envelope moves, like the crests and troughs of any wave. It is a wave, so it has a phase Then the envelope contains B − A x∆k− t∆ω cos = cos .2 2 The envelope represents a wave with phase100 100100 1002008-01-14 22:31:34 / rev 55add9943bf1100 6.055 / Art of approximation ∆k ∆ω 2 x − 2 t. So it is a wave with wavenumber ∆k/2 and frequency ∆ω/2. The envelope’s phase velocity is the group velocity of f1 + f2: frequency ∆ω/2 ∆ω vg = = = .wavenumber ∆k/2