Propagation of waves

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Propagation of waves

• Friday October 18, 2002

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Propagation of waves in 3D

• Imagine a disturbane that results in waves
  propagating equally in all directions
• E.g. sound wave source in air or water, light
  source in a dielectric medium etc..
• The generalization of the wave equation to
  3-dimensions is straight forward if the medium is
  homogeneous
• Let ? amplitude of disturbance (could be
  amplitude of E-field also)

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Propagation of waves in 3D
? depends on x, y and z such that it satisfies
the wave equation
or,
where in cartesian co-ordinates,
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1. Special Case Plane Waves along x

• Suppose ?(x, y, z, t)?(x, t) (depends only on x)
• Then ? f(kx-?t) g(kx?t)
• Then for a given position xo, ? has the same
  value for all y, z at any time to.
• i.e. the disturbance has the same value in the
  y-z plane that intersects the x-axis at xo.
• This is a surface of constant phase

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Plane waves along x
Planes perpendicular to the x-axis are wave
fronts by definition
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2. Plane waves along an arbitrary direction (n)
of propagation

• Now ? will be constant in plane perpendicular to
  n if wave is plane
• For all points P in plane

P
P
d
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2. Plane waves along an arbitrary direction (n)
of propagation
For all points P in plane
or, for the disturbance at P
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2. Plane waves along an arbitrary direction (n)
of propagation
If wave is plane, ? must be the same everywhere
in plane ? to n
This plane is defined by
P
P
d
is equation of a plane ? to n, a distance d from
the origin
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2. Plane waves along an arbitrary direction (n)
of propagation
is the equation of a plane wave propagating in
k-direction
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3. Spherical Waves

• Assume has spherical symmetry
  about origin (where source is located)
• In spherical polar co-ordinates

z
?
r
y
f
x
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3. Spherical Waves

• Given spherical symmetry, ? depends only on r,
  not f or ?
• Consequently, the wave equation can be written,

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3. Spherical Waves
Now note that,
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3. Spherical Waves
But,
is just the wave equation, whose solution is,
i.e. amplitude decreases as 1/ r !! Wave fronts
are spheres
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4. Cylindrical Waves (e.g. line source)
The corresponding expression is,
for a cylindrical wave traveling along positive ?
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Electromagnetic waves

• Consider propagation in a homogeneous medium (no
  absorption) characterized by a dielectric
  constant

?o permittivity of free space
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Electromagnetic waves
Maxwells equations are, in a region of no free
charges,
Gauss law electric field from a charge
distribution
No magnetic monopoles
Electromagnetic induction (time varying magnetic
field producing an electric field)
Magnetic fields being induced By currents and a
time-varying electric fields
µo permeability of free space (medium is
diamagnetic)
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Electromagnetic waves
For the electric field E,
or,
i.e. wave equation with v2 1/µo?
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Electromagnetic waves
Similarly for the magnetic field
i.e. wave equation with v2 1/µo?
In free space, ? ? ?o ?o
(? 1)
c 3.0 X 108 m/s
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Electromagnetic waves
In a dielectric medium, ? n2 and
? ? ?o n2 ?o
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Electromagnetic waves Phase relations
The solutions to the wave equations,
can be plane waves,