Waves, the Wave Equation, and Phase Velocity
Waves, the Wave Equation, and Phase Velocity
What is a wave?
Forward [f(x-vt)] and
backward [f(x+vt)]
propagating waves
The one-dimensional wave equation
Harmonic waves
Wavelength, frequency, period, etc.
Phase velocity Complex numbers Plane waves and laser beams
Photons and photon statistics
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What is a wave?
A wave is anything that moves.
To displace any function f(x) to the right, just change its argument from x to x-a, where a is a positive number.
If we let a = v t, where v is positive and t is time, then the displacement will increase with time.
So represents a rightward, or forward, propagating wave.
Similarly, represents a leftward, or backward, propagating wave.
v will be the velocity of the wave.
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The one-dimensional wave equation and its solution
We’ll derive the wave equation from Maxwell’s equations next class.
Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f:
Light waves (actually the electric fields of light waves) will be a solution to this equation. And v will be the velocity of light.
The wave equation has the simple solution
where f (u) can be any twice-differentiable function.
Proof that f (x ± vt) solves the wave equation
Write f (x ± vt) as f (u), where u = x ± vt. So and
Now, use the chain rule:
so
and
Substituting into the wave equation:
The 1D wave equation for light waves
We’ll use cosine- and sine-wave solutions:
Or
Where
The speed of light in vacuum, usually called “c”, is 3 x 1010 cm/s.
A simpler equation for a harmonic wave:
E(x,t) = A cos[(kx – wt) – Ө]
Use the trigonometric identity:
cos(z–y) = cos(z) cos(y) + sin(z) sin(y)
where z = k x – w t and y = Ө to obtain:
E(x,t) = A cos(kx – wt) cos (Ө ) + A sin(kx – wt) sin(Ө)
which is the same result as before,
as long as:
A cos (Ө ) = B and A sin(Ө ) = C
The Phase Velocity
How fast is the wave traveling?
Velocity is a reference distance divided by a reference time.
The phase velocity is the wavelength / period: v = λ/ t
Since n = 1/t :
In terms of the k-vector, k = 2p / l,
and the angular frequency, w = 2p / t, this is:
The Phase of a Wave
The phase is everything inside the cosine.
E(x,t) = A cos(φ ), whereφ = k x – w t – θ
φ = φ (x,y,z,t) and is not a constant, like θ!
In terms of the phase, w = –dφ /dt
k =dφ /dx
dφ /dx l v = –dφ /dt
-----------------------
0 1 2 3
x
f(x-1)
f(x-2)
f(x-3)
f(x)
0 1 2 3
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f(x-1)
f(x-2)
f(x-3)
f(x)
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where E is the light electric field
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For simplicity, we’ll just use the forward-propagating wave.
Definitions
Spatial quantities:
Temporal quantities:
This formula is useful when the wave is really complicated.
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