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Unformatted text preview: 21 Monochromatic waves and phasor notation Recall that we reached the traveling-wave dAlembert solutions E , H f ( t z v ) via the superposition of time-shifted and amplitude-scaled versions of f ( t ) = cos( t ) , namely the monochromatic waves A cos[ ( t z v )] = A cos( t z ) , with amplitudes A where v = can be called wave-number in analogy with wave-frequency . T = 2 cos( t ) t 1-1 Period = 2 cos( z ) z 1-1 Wavelength As depicted in the margin, monochromatic solutions A cos( t z ) are periodic in position and time, with the wave-number being essentially a spatial-frequency , the spatial counterpart of . This is an important point that you should try to understand well it has implications for signal processing courses related to images and vision. 1 In general , monochromatic solutions of 1D wave-equations ob- tained in various branches of science and engineering can all be rep- resented in the same format as above in terms of wave-frequency / wave-wavenumber pairs and having a ratio v recognized as the wave-speed and specific dispersion relations such as: T = 2 cos( t ) t 1-1 Period = 2 cos( z ) z 1-1 Wavelength Dispersion relations between wavefrequency and wavenumber determine the propagation veloc- ity v = = f for all types of wave motions.wave motions....
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This note was uploaded on 06/20/2011 for the course ECE 329 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08