Lecture 27: 12 November 2010 Estimated effect of group velocity dispersion: showed that pulse widths grow asymptoticall proportional to the distance travelled, to the second derivative d2k/dω2, and to the bandwidth of the pulse (inversely with the initial pulse width). Discussed where dispersion comes from. Essentially, the lack of dispersion in the simple wave equation with constant coefficients comes from scale invariance: if u(x,t) is a solution, then u(sx,st) is a solution for any scale factor s, and so the wave speed cannot depend on the length or timescale (k or ω). To get dispersion, we need something that sets a length or timescale, and this comes in three forms: • Numerical dispersion: the discretization Δx and Δt set length/time scales, and the wave speed depends on how the wavelength/period compares to these. • Material dispersion: there is some underlying spatial or time scale set by the response of the medium that causes c to depend on ω. For example, deviations from ideal gases due to finite molecule sizes, resonant absorption that has a timescale, or a material that doesn't respond instantly to stimuli (e.g. an electrical material doesn't polarize instantly in response to an applied field, hence the index of refraction depends on ω). • Geometric/waveguide dispersion: the coefficients of the wave equation vary in space to set a lengthscale, e.g. for waves propagating in a hollow pipe the speed depends on how the wavelength compares to the diameter. Began discussing general topic of waveguides. Defined waveguides: a wave-equation system that is invariant (or periodic) in at least one direction (say y), and has some structure to confine waves in one or more of the other "transverse" directions. A simple example of a waveguide (although not the only example) consists of waves confined in a hollow pipe (either sound waves or electromagnetic waves, where the latter are confined in metal pipe). Began with a simple 2d example: a waveguide for a scalar wave equation that is invariant in y and confines waves with "hard walls" (Dirichlet boundaries at x=0 and x=L) in the x direction. In such a wave equation, or any wave equation that is invariant in y, the solutions are separable in the invariant direction, and the eigenfunctions u(x,y)e-iωt can be written in the form uk(x)ei(ky-ωt) for some function uk and some eigenvalues ω(k). In this case, plugged the separable form into the scalar wave equation and immediately obtained a 1d equation for uk: uk''-k2uk=-ω2uk, which we solved to find uk=sin(nπx/L) for ω2=k2+(nπ/L)2. Plotted the dispersion relation ω(k) for a few guided modes (different integers n). Commented on the k goes to 0 and infinity limits where the group velocity goes to 0 and 1 (c), respectively.MIT OpenCourseWare http://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and Numerics Fall 2010 For information about citing these materials or our Terms of Use, visit:
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