329lect19 - 19 dAlembert wave solutions, radiation from...

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19 d’Alembert wave solutions, radiation from current sheets, Poynting theorem d’Alembert wave solutions of Maxwell’s equations for homogeneous and source-free regions obtained in the last lecture having the forms E , H f ( t z v ) are classiFed as uniform plane-TEM waves . TEM stands for T ransverse E lectro M agnetic, and the reason for this designation is: x y z H E xf ( t - z v ) E × H x y z H E xf ( t + z v ) E × H viable solutions satisfying ∇ · E = H =0 conditions have their E and H vectors transverse to the direction of propagation which always coincides with the direction of vector S E × H known as Poynting vector — more on this later on. Poynting vector E × H d’Alembert wave solutions such as E xf ( t - z v ) and H y f ( t - z v ) η are also designated as uniform plane waves because: these wave-Felds are constant (have the same vector value) at planes of constant phase , e.g., on planes deFned by t - z v = const., 1
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which are planes transverse to the propagation direction (direction of vector E × H ). Not all waves solutions of Maxwell’s equations are uniform plane — for in- stance non-uniform TEM waves with spherical surfaces of constant phase are ubiquitous, but they will be examined later on (in ECE 450, mainly). After the next set of examples we will examine how uniform plane waves can be radiated by inFnite planes of surface currents. By contrast, spherical waves are produced by compact antennas having Fnite dimensions. Example 1: Let E x ± ( t - y/c τ ) be a wave solution in free space where ± ( t τ ) is a triangular waveform of duration τ peaking at t =0 (deFned in ECE 210). We will next provide two di±erent solutions demonstrating how the wave Feld B accompanying E can be found. Solution 1: We recognize the given wave Feld E as a TEM uniform plane wave travel- ing in y -direction given the t - y/c dependence of phase. Consequently, we obtain H by dividing E with η = η o and rotating it by 90 from ˆ x -direction to co-align it with E × H vector. As a result,
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329lect19 - 19 dAlembert wave solutions, radiation from...

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