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329lect21 - 21 Monochromatic waves and phasor notation...

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21 Monochromatic waves and phasor notation Recall that we reached the traveling-wave d’Alembert solutions E , H f ( t z v ) via the superposition of time-shifted and amplitude-scaled versions of f ( t ) = cos( ω t ) , namely the monochromatic waves A cos[ ω ( t z v )] = A cos( ω t β z ) , with amplitudes A where β ω v = ω μ can be called wave-number in analogy with wave-frequency ω . T = 2 π ω cos( ω t ) t 1 -1 Period λ = 2 π β cos( β z ) z 1 -1 Wavelength As depicted in the margin, monochromatic solutions A cos( ω t β z ) are periodic in position and time, with the wave-number β being essentially a spatial-frequency , the spatial counterpart of ω . 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
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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 ω and β having a ratio v ω β recognized as the wave-speed and specific dispersion relations such as: T = 2 π ω cos( ω t ) t 1 -1 Period λ = 2 π β cos( β z ) z 1 -1 Wavelength Dispersion relations between wavefrequency ω and wavenumber β determine the propagation veloc- ity v = ω β = λ f for all types of wave motions.
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