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tutorial15Solution - The University of Western Australia...

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The University of Western Australia School of Electrical, Electronic and Computer Engineering Solutions - Tutorial Sheet 15, 2006 Electromagnetic Theory ENGT3303 Question 1 The resulting plane wave is obtained by superposition and is given by ( ) ( ) ( ) z t A z t A t z 2 2 1 1 cos cos , β ω β ω + = Ψ If we substitute in the four equations for ω 1 , ω 2 , β 1 , and β 2 , as given in the question and use the identity ( ) β α β α β α sin sin cos cos cos m = ± , the resultant wave becomes ( ) ( ) ( ) z t z t A t z β ω β ω = Ψ cos cos 2 , This resultant wave is a carrier wave at frequency ω = ω 1 + ω 2 ( ) /2 and phase velocity v = ω / β . The amplitude of the carrier wave is a wave of frequency ω , called the beat frequency, and propagation constant β traveling at velocity v g = ∆ ω / β . time group carrier This velocity is called the group velocity, and represents the rate of flow of energy. This simple argument can be extended to n waves, whereupon, the “group” nature becomes more apparent.
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