Wave Motion
• Introduction
– At the heart of optics is the question, ”Is light a wave or particle
phenomenon?”
– In some situations, think of light as a stream of particles (particles
localized, interact via fields)
– In some situations, think of light as a wave (waves non-localized)
– Need to make use of the mathematical description of waves.
• One Dimensional Waves
– Classical travelling wave: self-sustaining disturbance of a medium
which moves through space transporting energy and momentum eg.
mechanical waves on strings, water waves, sound waves.
– Longitudenal (sound) and Transverse (water).
– The disturbance advances, not the material medium.
– Different to a stream of particles.
– Suppose a disturbance or pulse travels at speed v.
– Disturbance must be a function of both position and time,
ψ(x, t) = f (x, t),
– Then,
ψ(x, t)|t=0 = f (x, 0) = f (x).
– If we have a coordinate system that moves with the distrubance,
ψ = f (x0 ),
where
x0 = x − vt.
– Thus
ψ(x, t) = f (x − vt).
– This is the most general form of the one-dimensional wavefunction.
– Whats f (x + vt)?
• The Differentail Wave Equation
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– One - dimensional differential wave equation is
1 ∂2ψ
∂2ψ
=
.
∂x2
v 2 ∂t2
– Homogeneous wave equation - no source or sink terms - wave equation
for undamped sysems.
• Harmonic Waves
– Waves for which the profile is a sine or cosine wave.
ψ(x, t) = Asin(kx ± ωt),
where A = amplitude, k is a positive constant called the propogation
number, k = 2π/λ, where λ is he wavelength. Period T = λ/v, where
v is the speed of the wave, ν = 1/T is the frequency or number of waves
per unit time. Angular frequency is ω = 2πν, in units of radians per
second. The wavenumber is κ = 1/λ, measured in units of inverse
meters.
– ψ = Asin(kx ± vt) is at a single frequency - monochromatic or monoenergetic. Real waves contain a range of frequencies and therefore
a range of energies.
• Phase and Phase Velocity
– For ψ(x, t) = Asin(kx − ωt), the argument, φ = (kx − ωt) is called
the phase, φ of the wave.
– Can write ψ(x, t) = Asin(kx − ωt + ), where is the initial phase.
– So the phase is
φ(x, t) = (kx − ωt + ),
a function of x, t.
– The rate of change of phase with time is
|
(∂φ
|x = ω,
∂t)
the angular frequency of the wave.
– The rate of change of phase with distance is,
|
2
(∂φ
|t = k.
∂)
– What is
∂x
)φ ?
∂t
– Its the speed of propogation of the condition of constant phase, and is
equal to,
∂x
−(∂φ/∂t)x
( )φ =
.
∂t
(∂φ/∂x)t
∂x
ω
( )φ = ± = ±v.
∂t
k
– This is the speed at which the profile moves - known as the phase
velocity, positive/negative sign when the profile moves in the positive/negative x direction.
(
• The Superposition Principle
– If ψ1 and ψ2 are solutions of the wave equation, then ψ1 + ψ2 are also
solutions of the wave equation.
– OR: When two separate waves overlap in space or arrive at the same
spatial location, the resulting disurbance in the region is the algebraic
sum of the individual constituent waves at that location.
– Leads to constructive, destructive interference, in phase and out of
phase etc.
• The Complex Representation
– Complex representation leads to simpler mathematical processing.
√
– −1 = i.
– Complex number z = x + iy.
– Represented on an ”Argand diagram” as (x, y).
– Then x = rcosθ, y = rsinθ, where r 2 = x2 + y 2, tan(θ) = y/x.
– Then
z = x + iy = r(cosθ + isinθ).
– Note
eiθ = cosθ + isinθ.
z = reiθ = rcosθ + irsinθ.
– r is the magnitude of z and θ is its phase angle.
– |z| is the magnitude of z.
– The complex conjugate of z = x + iy is z∗ = x − iy.
z∗ = r(cosθ − isinθ) = re−iθ .
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– Re(z) = rcosθ, and Im(z) = rsinθ.
– So a wave
ψ(x, t) = re[Aei(ωt−kx+) ].
• Plane Waves
– Simplest example of a 3-D wave.
– Surfaces on which a given disturbance has constant phase form a set
of planes, each of which is generally perpendicular to the direction of
propogation.
– Surfaces joining all points of equal phase known as wavefronts.
– So
ψ(r, t) = Aei(k.r±ωt) ,
with A, ω, k constant.
– Phase is constant if r.k is a constant.
– 3D differential wave equation, spherical waves and cylindrical waves.
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