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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Chapter 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 2 π/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 2 π radians of oscillation. Although 2 π is dimensionless, it is not the ideal dimensionless number, which is unity. In 1 radian of oscillation, the wave travels a distance λ λ ¯ ≡ 2 π . The bar notation, meaning ‘divide by 2 π ’, is chosen by analogy with h and ~ . The one- radian forms such as ~ are more useful for approximations than the 2 π-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 and λ ¯ , 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 relation 6.055 / 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 ....
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may12 - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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