Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo
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FUNDAMENTALS OF WAVE PROPAGATION |
v is also known as the phase velocity and can be larger than the vacuum speed of light in the case of light waves.
More generally, a 1-D wave given by Eq. (3.2-8) may satisfy another partial differential equation with respect to x and t, in addition to Eq. (3.2-5). A general wave equation that has a simple harmonic solution can be written as
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uðx; tÞ ¼ 0; |
ð3:2-11Þ |
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f |
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@x |
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where f is a polynomial function of |
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and |
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@x |
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@3u |
þ a |
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@3u |
B |
@u |
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¼ 0 |
ð3:2-12Þ |
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@x3 |
@x2@t |
@t |
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Using Eq. (3.2-8) for the simple |
harmonic |
solution allows |
the substitution of |
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! jo, and Eq. (3.2-12) becomes |
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@t |
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k3 ak2o Bo ¼ 0 |
ð3:2-13Þ |
Such an equation relating o and k is known as the dispersion equation. The phase velocity v in this case is given by
v ¼ |
o |
¼ |
k2 |
ð3:2-14Þ |
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k |
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ak2 þ B |
A wave with a phase velocity, which is a function of wave number or wavelength other than Eq. (3.2-10), is known as a dispersive wave. Hence, Eq. (3.2-14) is a dispersive wave equation.
Some examples of waves are the following:
A.Mechanical waves, such as longitudinal sound waves in a compressible fluid, are governed by the wave equation
@2uðx; tÞ |
¼ |
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@2uðx; tÞ |
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ð |
3:2-15 |
Þ |
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@x2 |
Kc @t2 |
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where is the fluid density, and Kc is the compressibility. Hence, the phase velocity is given by
s
Kc
v ¼ ð3:2-16Þ
WAVES |
29 |
B. Heat diffuses under steady-state conditions according to the wave equation
Kh |
@2uðx; tÞ |
¼ |
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@uðx; tÞ |
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ð |
3:2-17 |
Þ |
s @x2 |
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@t |
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where Kh is the thermal conductivity, s is the specific heat per unit volume, and uðx; tÞ is the local temperature.
C. The Schrodinger’s wave equation in quantum mechanics is given by
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h2 |
@2uðx; tÞ |
þ |
V x |
u |
x; t |
Þ ¼ |
j |
h |
@uðx; tÞ |
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ð |
3:2-18 |
Þ |
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4p2m @x2 |
2p @t |
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ð Þ |
ð |
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where m is mass, V(x) is potential energy of the particle, h is Planck’s constant, and uðx; tÞ is the state of the particle. The dispersion relation can be written as
ho ¼ |
h2k2 |
ð3:2-19Þ |
2m þ VðxÞ; |
where h equals h=2p.
D.Time-varying imagery can be considered to be a 3-D function gðx; y; tÞ. Its 3-D Fourier representation can be written as
1 |
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gðx; y; tÞ ¼ ð ð ð Gðfx; fy; f Þej2pðfxxþfyyþftÞdfxdfydf ; |
ð3:2-20Þ |
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1 |
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where |
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Gðfx; fy; f Þ ¼ ð ð ð gðx; y; tÞe j2pðfxxþfyyþftÞdxdydt |
ð |
3:2-21 |
Þ |
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¼ Aðfx; fy; f Þej ðfx;fy;f Þ
For real gðx; y; tÞ, Gðfx; fy; f Þ ¼ G ð fx; fy; f Þ so that Eq. (3.2-21) can be written as
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ð |
ð |
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gðx; y; tÞ ¼ 2 |
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Aðfx; fy; f Þ cosð2pðfxx þ fyy þ ftÞ þ ðfx; fy; f ÞÞdfxdfydf |
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4 1 1 |
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ð3:2-22Þ
30 |
FUNDAMENTALS OF WAVE PROPAGATION |
Aðfx; fy; f Þ cosð2pðfxx þ fyy þ ftÞ þ ðfx; fy; f ÞÞ is a 2-D plane wave with amplitude Aðfx; fy; f Þ, phase ðfx; fy; f Þ, spatial frequencies fx and fy, and time frequency f. In this case, the wave vector k is given by
k ¼ 2pðfxex þ fyeyÞ ¼ kxex þ kyey; |
ð3:2-23Þ |
where ex and ey are the unit vectors along the x- and y-directions, respectively. The components kx and ky are the radian spatial frequencies along the x-direction and y-direction, respectively. fx and fy are the corresponding spatial frequencies.
The 2-D plane wave can be written as
uðr; tÞ ¼ Aðk; oÞ cosðk r þ ot þ ðk; oÞÞ; |
ð3:2-24Þ |
where
r ¼ xex þ yey |
ð3:2-25Þ |
If we assume uðr; tÞ moves in the direction k with velocity v, uðr; tÞ can be shown to be a solution to the 2-D nondispersive wave equation given by
r |
2u r; t |
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@2uðr; tÞ |
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Þ ¼ v2 @t2 |
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ð |
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where
r2 |
¼ |
@2 |
þ |
@2 |
ð3:2-27Þ |
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@x2 |
@y2 |
In conclusion, time-varying imagery can be represented as the sum of a number of 2-D plane waves. This conclusion can be extended to any multidimensional signal in which one of the dimensions is treated as time.
Equation (3.2-26) is the generalization of Eq. (3.2-5) to two spatial coordinates x and y. If three spatial coordinates x, y, and z are considered with the same wave property, Eqs. (3.2-23) and (3.2-25) remain the same, with the following definitions of r and r2:
r ¼ xex þ yey þ zez; |
ð3:2-28Þ |
where ez is the additional unit vector along the z-direction, and
r2 |
¼ |
@2 |
þ |
@2 |
þ |
@2 |
ð3:2-29Þ |
@x2 |
@y2 |
@z2 |
ELECTROMAGNETIC WAVES |
31 |
The r2 operator is known as the Laplacian. The 3-D wave vector used in Eq. (3.2-24) is given by
k ¼ 2pðfxex þ fyey þ fzezÞ ¼ kxex þ kyey þ kzez |
ð3:2-30Þ |
Thus, kz is the additional radian frequency along the z-direction, with fz being the corresponding spatial frequency. The plane for which uðr; tÞ is constant is called the wavefront. It moves in the direction of k with velocity v. This is discussed further in Section 3.5.
In what follows, we will assume linear, homogeneous, and isotropic media unless otherwise specified. We will also assume that metric units are used.
EXAMPLE 3.1 Consider uðx1; t1Þ ¼ uðx2; t2Þ; x2x1; t2t1 for a right-traveling wave. Show how x1 and x2 are related.
Solution: uðx1; t1Þ can be considered to be moving to the right with velocity v. Letting t ¼ t2 t1, uðx1; t1Þ and uðx2; t2Þ will be the same when
x2 x1 ¼ v t
3.3ELECTROMAGNETIC WAVES
Electromagnetic waves are 3-D waves, with three space dimensions and one time dimension. There are four quantities D, B, H, and E, which are vectors in space coordinates and whose components are functions of x, y, z, and t. In other words, D, B, H, and E are directional in space. D is the electric displacement (flux) (vector)
field (C=m2), E is the electric (vector) field (V=m), B is the magnetic density (vector) field (Wb/m2), H is the magnetic (vector) field ðA=mÞ, is the charge density (C/ m3), and J is the current density (vector) field (A/m2).
Electromagnetic waves are governed by four Maxwell’s equations. In metric units, they are given by
r D ¼ |
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ð3:3-1Þ |
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r B ¼ 0 |
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ð3:3-2Þ |
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@D |
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r H ¼ |
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þ J |
ð3:3-3Þ |
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@t |
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r E ¼ |
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ð3:3-4Þ |
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@t |
where the divergence and curl of a vector A with components Ax, Ay, and Az are given by
r A ¼ |
@Ax |
þ |
@Ay |
þ |
@Az |
ð3:3-5Þ |
@x |
@y |
@z |
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FUNDAMENTALS OF WAVE PROPAGATION |
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@z |
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r ¼ |
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Ax |
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@Az |
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ex þ |
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ey þ |
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ez |
ð3:3-6Þ |
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@y |
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Two more relations are |
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D ¼ eE |
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ð3:3-7Þ |
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B ¼ mH; |
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ð3:3-8Þ |
where e is the permittivity, and m is the permeability. Their values in free space or vacuum are given by
10 9
e0 ¼ 36p F=m
m0 ¼ 4p 10 7 H=m
In dielectrics, e and m are greater than e0 and m0, respectively. In such media, we also consider dipole moment density P (C/m2), which is related to the electric field E by
P ¼ we0E; |
ð3:3-9Þ |
where w is called the electric susceptibility. w can be considered as a measure for electric dipoles in the medium to align themselves with the electric field.
We also have
D ¼ e0E þ P ¼ e0ð1 þ wÞE ¼ eE; |
ð3:3-10Þ |
where |
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e ¼ e0ð1 þ wÞ |
ð3:3-11Þ |
In a uniform isotropic dielectric medium in which space charge and current density J are zero, Maxwell’s equations become
r D ¼ er E ¼ 0 |
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ð3:3-12Þ |
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r B ¼ mr H ¼ 0 |
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ð3:3-13Þ |
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@D |
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r H ¼ |
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¼ e |
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ð3:3-14Þ |
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r E ¼ |
@B |
¼ m |
@H |
ð3:3-15Þ |
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@t |
@t |
PHASOR REPRESENTATION |
33 |
3.4PHASOR REPRESENTATION
The electric and magnetic fields we consider are usually sinusoidal with a timevarying dependence in the form
uðr; tÞ ¼ AðrÞ cosðk r wtÞ |
ð3:4-1Þ |
We can express uðr; tÞ as |
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uðr; tÞ ¼ Real½A0ðrÞejwt&; |
ð3:4-2Þ |
where |
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A0ðrÞ ¼ AðrÞejk r |
ð3:4-3Þ |
A0ðrÞ is called the phasor (representation) of uðr; tÞ. It is time independent. We note the following:
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A0ðrÞejwt ¼ jwA0 |
ðrÞejwt |
ð3:4-4Þ |
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dt |
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ð A0 |
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A0 |
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ðrÞejwtdt ¼ |
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ð3:4-5Þ |
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jw |
Hence, differentiation and integration are equivalent to multiplying A0ðrÞ by jw and 1=jw, respectively.
The electric and magnetic fields can be written in the phasor representation
as
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Eðr; tÞ ¼ |
~ |
jwt |
& |
ð3:4-6Þ |
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Real½EðrÞe |
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Hðr; tÞ ¼ |
~ |
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jwt |
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ð3:4-7Þ |
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Real½HðrÞe |
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~ |
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where EðrÞ and HðrÞ are the phasors. The corresponding phasors for D, B, and J are |
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defined similarly. |
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Maxwell’s equations in terms of EðrÞ and HðrÞ can be written as |
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ð3:4-8Þ |
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r D |
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ð3:4-9Þ |
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r B |
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r H ¼ jweE |
þ J |
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ð3:4-11Þ |
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r E ¼ jwB |
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34 |
FUNDAMENTALS OF WAVE PROPAGATION |
3.5WAVE EQUATIONS IN A CHARGE-FREE MEDIUM
Taking the curl of both sides of Eq. (3.3-3) gives |
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r r H ¼ e |
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r E ¼ em |
@2H |
@t |
@t2 |
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r r H can be expanded as |
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r r H ¼ rðr HÞ r rH;
where rH is the gradient of H. This means rHi; i ¼ x, y, z is given by
rHi ¼ |
@Hi |
ex þ |
@Hi |
ey þ |
@Hi |
ez |
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r rHi ¼ r2Hi is given by |
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@2Hi |
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@2Hi |
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r2Hi ¼ |
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ex þ |
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ey þ |
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ez |
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@x2 |
@y2 |
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@z2 |
ð3:5-1Þ
ð3:5-2Þ
ð3:5-3Þ
ð3:5-4Þ
Thus, |
r rH |
is a vector whose |
components along the |
three directions are |
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r |
Hi; i ¼ x, y, z, respectively. |
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It can be shown that rðr HÞ, which is the gradient vector of r H, equals zero. |
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Hence, |
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r r H ¼ r rH ¼ r2H |
ð3:5-5Þ |
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Substituting this result in Eq. (3.5-1) gives |
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r2H ¼ em |
@2H |
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ð3:5-6Þ |
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Similarly, it can be shown that |
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r2E ¼ em |
@2E |
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ð3:5-7Þ |
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@t2 |
Equations (3.5-6) and (3.5-7) are called the homogeneous wave equations for E and H, respectively.
In conclusion, each component of the electric and magnetic field vectors satisfies p
the nondispersive wave equation with phase velocity v equal to 1= em. In free space, we get
1 |
3 108 m= sec |
ð3:5-8Þ |
c ¼ v ¼ pe0m0 |
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WAVE EQUATIONS IN A CHARGE-FREE MEDIUM |
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35 |
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Let us consider one such field component as uðr; tÞ. It satisfies |
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1 @2u r; t |
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r2uðr; tÞ ¼ |
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uðr; tÞ ¼ |
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ð |
Þ |
ð3:5-9Þ |
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@x2 |
@y2 |
@z2 |
c2 |
@t2 |
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A 3-D plane wave solution of this equation is given by |
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uðr; tÞ ¼ Aðk; oÞ cosðk r otÞ; |
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ð3:5-10Þ |
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where the wave vector k is given by |
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k ¼ kx þ kyey þ kzez |
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ð3:5-11Þ |
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and the phase velocity v is related to k and o by |
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v ¼ |
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o |
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ð3:5-12Þ |
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jkj |
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A phase front is defined by |
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k r ot ¼ constant |
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ð3:5-13Þ |
This is a plane whose normal is in the direction of k. When ot changes, the plane changes, and the wave propagates in the direction of k with velocity c. We reassert that cosðk r otÞ can also be chosen as cosðk r þ otÞ. Then, the wave travels in the direction of k.
The components of the wave vector k can be written as
ki ¼ |
2p |
ai ¼ kai |
i ¼ x; y; z; |
ð3:5-14Þ |
l |
where k ¼ jkj, and ai is the direction cosine in the ith direction. If the spatial frequency along the ith direction is denoted by fi equal to ki=2p, then the direction cosines can be written as
ax ¼ lfx |
ð3:5-15Þ |
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ay ¼ lfy |
ð3:5-16Þ |
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¼ q |
ð3:5-17Þ |
az |
1 l2fx2 l2fy2 |
as
ax2 þ ay2 þ az2 ¼ 1 |
ð3:5-18Þ |
36 |
FUNDAMENTALS OF WAVE PROPAGATION |
3.6 WAVE EQUATIONS IN PHASOR REPRESENTATION IN A CHARGE-FREE MEDIUM
This time let us start with the curl of Eq. (3.4-11):
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~ |
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ð3:6-1Þ |
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r r E |
¼ jwmH; |
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~ |
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¼ 0 in Eq. (3.6-1) yields |
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where B ¼ mH is used. Utilizing Eq. (3.4-10) with J |
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~ |
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ð3:6-2Þ |
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r r E ¼ w |
meE |
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As in Section 3.4, we have |
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~ |
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ð3:6-3Þ |
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r r E ¼ rðr EÞ r r~E |
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~ |
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As r E ¼ 0 by Eq. (3.4-8), we have |
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ð3:6-4Þ |
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r |
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meE ¼ 0 |
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We define the wave number k by |
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Then, Eq. (3.6-4) becomes |
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E. The homogeneous wave |
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equation for H can be similarly derived as |
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Let E be written as |
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E ¼ Exex þ Eyey þ Ezez |
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Equation (3.6-6) can now be written for each component Ei of E as |
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@z2 |
EXAMPLE 3.2 (a) Simplify Eq. (3.6-9) for a uniform plane wave moving in the
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z-direction, (b) show that the z-component of E and H of a uniform plane wave equals zero, using the phasor representation, (c ) repeat part (b) using Maxwell’s equations.
PLANE EM WAVES |
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Solution: (a) A uniform plane wave is characterized by |
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Hence, Eq. (3.6-9) simplifies to
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(c) We write
E ¼ ExejðkzþwtÞex þ EyejðkzþwtÞey þ EzejðkzþwtÞez
Substituting E in r E ¼ 0, we get
@E ejðkzþwtÞ ¼ 0
@z z
implying Ez ¼ 0. We can similarly consider the magnetic field H as
H ¼ HxejðkzþwtÞex þ HyejðkzþwtÞey þ HzejðkzþwtÞez
Substituting H in r H ¼ 0, we get
@H ejðkzþwtÞ ¼ 0
@z z
implying Hz ¼ 0.
3.7PLANE EM WAVES
Consider a plane wave propagating along the z-direction. The electric and magnetic fields can be written as
E ¼ Exejðkz wtÞ^ex þ Eyejðkz wtÞ^ey
H ¼ Hxejðkz wtÞ^ex þ Hyejðkz wtÞ^ey;
ð3:7-1Þ