Chapter1_Notes_v0

# Phasors cylindrical wavefront two dimensional wave a

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Unformatted text preview: illustrated with surface waves (e.g. water). 3-D ones may have different shapes, e.g. plane waves, cylindrical, spherical. Nice pictures, but to do anything useful we need mathematical description! Look at simplest case ﬁrst, i.e. sinusoidal waves in 1-D. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors 25 • Sinusoidal wave in lossless medium Lossless medium: It does not attenuate the amplitude of the wave traveling within it or on its surface. Take water surface waves, where y denotes the height of water relative to unperturbed state, then y (x, t) = A cos 2πt 2πx − + φ0 T λ (m) (14) A is amplitude of the wave, T is its time period , λ is spatial wavelength, and φ0 is reference phase. Even simpler form is obtained if the argument of the cosine term is called phase of the wave (not to be confused with the reference phase φ0 ): 2πt 2πx − + φ0 (15) φ(x, t) = T λ which is measured in radians or degrees (rad = ? degrees?). The quantity y (x, t) can be written, y (x, t) = cos φ(x, t) Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (16) Electromagnetics I: Introduction: Waves and Phasors 26 y(x, 0) A 0 -A λ 2 λ 3λ 2 x 3T 2 t λ (a) y(x, t) versus x at t = 0 y(0, t) A 0 -A T 2 T T (b) y(x, t) versus t at x = 0 Figure 1-11 Figure 10: 1-D wave “snapshots”. Let’s do some plotting of the wave y (x, y ). Math tells us that this is a periodic function. First look at y (x, t) by ﬁxing time to t = 0 and then by ﬁxing position x = 0. The wave repeats itself with a spatial period λ and time period T . This is shown in Fig. 10. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors 27 y(x, 0) P A -A λ 2 x 3λ 2 λ (a) t = 0 up y(x, T/4) P A -A λ 2 λ x 3λ 2 (b) t = T/4 y(x, T/2) P A λ 2 λ 3λ 2 x -A Figure 1-12 (c) t = T/2 Figure 11: Wave snapshots illustrating wave travel. What happens if we take a snapshot at diﬀerent times (within one time period T )? That is shown in Fig. 11. If we look at position of, e.g., peak value P, we notice that it moves in the +x direction. If we can ﬁnd what distance P travels in a given time then we can calculate phase velocity . Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors 28 • To have a peak, phase must be zero or multiples of 2π (setting relative phase φ0 = 0) φ(x, t) = 2πt 2πx − = 2nπ, n = 0, 1, 2 . . . T λ (17) • For places other than peaks, this can be generalized for any point on the wave 2πt 2πx − = const. T λ (18) • Take time derivative of above eq. to get velocity 2π 2π dx dx λ − = 0 ⇒ up = = (m/s) T λ dt dt T (19) • Phase velocity = propagation velocity (up ), is velocity of the wave pattern. Consider water waves, if you follow one part of the wave...
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## This note was uploaded on 09/25/2013 for the course ECE 331 taught by Professor Martinsiderious during the Fall '12 term at Portland State.

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