329lect20 - 20 Monochromatic wave solutions and phasor...

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20 Monochromatic wave solutions 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( ωt ) , namely the monochromatic waves A cos[ ω ( t z v )] = A cos( ωt βz ) , 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( ωt ) t 1 -1 Period λ = 2 π β cos( βz ) z 1 -1 Wavelength As depicted in the margin, monochromatic solutions A cos( ωt βz ) 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 speciFc dispersion relations such as: T = 2 π ω cos( ωt ) t 1 -1 Period λ = 2 π β cos( βz ) z 1 -1 Wavelength Dispersion relations between wavefrequency ω and wavenumber β determine the propagation veloc- ity v = ω β = λf for all types of wave motions. 1. TEM waves in perfect dielectrics: β = ω μ±, 2. Acoustic waves in monoatomic gases with temperature T (K) and atomic mass m (kg): β = ω ± m 5 3 KT , 3. TEM waves in collisionless plasmas (ionized gases) with plasma frequency ω p = ² Ne 2 o : β = 1 c ² ω 2 - ω 2 p .
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329lect20 - 20 Monochromatic wave solutions and phasor...

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