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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
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 ):
φ(x, t) =
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)
2 λ 3λ
2 x 3T
2 t λ
(a) y(x, t) versus x at t = 0
y(0, t) A 0
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
P A -A λ
2 λ x 3λ
2 (b) t = T/4
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 . . .
λ (17) • For places other than peaks, this can be generalized for any
point on the wave
λ (18) • Take time derivative of above eq. to get velocity
2π 2π dx
= 0 ⇒ up =
T (19) • Phase velocity = propagation velocity (up ), is velocity of the
wave pattern. Consider water waves, if you follow one part of
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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.
- Fall '12