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Unformatted text preview: it moves at the phase velocity, however, the water itself
is moving up and down only.
Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors 29 • What about direction of propagation? If the signs of the terms
in the phase,
2πt 2πx
φ(x, t) =
−
(20)
T
λ
are diﬀerent ⇒ wave travels in +x direction, otherwise in −x
direction.
• Frequency is the reciprocal of time period T: f = 1/T (Hz)
• ⇒ up = λ/T = f λ.
• Things are “simpliﬁed” further by deﬁning:
1. Angular velocity: ω = 2πf (rad/s)
2. Phase constant (also called the wavenumber): β = 2π/λ
so substituting and taking the propagation direction in the
positive x direction: Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors y (x, t) = A cos 2πf t − 2π
x
λ = A cos (ωt − βx) 30 (21) • For the −x direction:
y (x, t) = A cos (ωt + βx) (22) What about the phase reference φ0 ? If it is not zero, then we have
y (x, t) = A cos (ωt − βx + φ0 ) (23) Fig. 12 shows what happens for diﬀerent phase references at a ﬁxed
position x = 0. Note that,
• Negative φ0 results in a lag behind the reference wave,
• While positive φ0 leads the reference wave. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors y Leads ahead of
reference wave 31 Reference wave (φ0 = 0)
Lags behind reference wave A
φ0 = π/4 φ0 = π/4 T
2 T A Figure 12: Eﬀect of diﬀerent phase reference on y (0, t).
Figure 113
Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. 3T
2 t Electromagnetics I: Introduction: Waves and Phasors 32 • Sinusoidal wave in a lossy medium
So far, wave’s amplitude did not change with distance ⇒ lossless case.
If it changes (decreases) ⇒ lossy case (lossy medium).
• Attenuation constant α characterizes how lossy the medium
is
• α measured in Np/m
• Falloﬀ given by an exponential function exp(−αx) so that full
wave is given by
y (x, t) = Ae−αx cos (ωt − βx + φ0 ) (24) • Example of such function given in Fig. 13
To get the feel for the numbers and their meaning, do examples. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors 33 y(x)
y(x) 10 m 10e0.2x 5m
0 1 2 3 4 5 6 7 8 x (m) 5 m
10 m Figure 114 Figure 13: Exponentially attenuated wave. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors 34 1.4. EM spectrum
What are electromagnetic waves?
• EM waves consist of electric and magnetic ﬁeld components of
the same frequency (to be discussed further later)
• EM wave phase velocity in vacuum is constant and is the socalled velocity of light in vacuum (or free space)...
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 Fall '12
 MartinSiderious
 Electromagnet

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