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
f(x)
f(x-3)
f(x-2)
f(x-1)
x0 1 2 3
Introduction
• Wave propagation in a medium is described mathematically in terms of a wave equation; this is a differential equation relating the dynamics and static of small displacements of the medium, and whose solution may be a propagating disturbance.
• Once an equation has been set up, to call it a wave equation we require it to have propagation solutions. This means that if we supply initial conditions in the form of a disturbance which is centered around some given position at time zero, then we shall find a disturbance of similar type centered around a new position at a later time.
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.
f(x)f(x-3)
f(x-2)f(x-1)
x0 1 2 3
f(x - v t)
f(x + v t)
The non-dispersive wave equation in one dimension
The most important elementary wave equation in one dimension can be derived from the requirement that any solution:
• Propagates in either direction (± x) at a constant velocity v,
• Does not change with time, when referred to a center which moving at this velocity.
The solution to the one-dimensional wave equation
where f (u) can be any twice-differentiable function.
( , ) ( v )f x t f x t
The wave equation has the simple solution:
uvtx )( Is called the phase of the wave.
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
1u
x
vu
t
f f u
x u x
f f u
t u t
f f
x u
vf f
t u
2 2
22 2
vf f
t u
2 2
2 2
f f
x u
01
1
2
2
22
2
2
2
22
2
t
f
vx
f
ort
f
vx
f
The wave equation
The one-dimensional wave equation
2 2
2 2 2
10
v
f f
x t
We’ll derive the wave equation from Maxwell’s equations. 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 1D wave equation for light waves
We’ll use cosine- and sine-wave solutions
or
where:
2 2
2 20
E E
x t
( , ) cos[ ( v )] sin[ ( v )]E x t B k x t C k x t
( , ) cos( ) sin( )E x t B kx t C kx t
1v
k
( v)kx k t
where E is the light electric field
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 – t) – ]
Use the trigonometric identity:
cos(z–y) = cos(z) cos(y) + sin(z) sin(y)
where z = kx – t and y = to obtain:
E(x,t) = A cos(kx – t) cos() + A sin(kx – t) sin()
which is the same result as before,
as long as:
A cos() = B and A sin() = C
( , ) cos( ) sin( )E x t B kx t C kx t For simplicity, we’ll just use the forward-propagating wave.
Definitions: Amplitude and Absolute phase
E(x,t) = A cos[(k x – t ) – ]
A = Amplitude
= Absolute phase (or initial phase)
Definitions
Spatial quantities:
Temporal quantities:
The Phase Velocity
How fast is the wave traveling?
Velocity is a reference distancedivided by a reference time.
The phase velocity is the wavelength / period: v = /
In terms of the k-vector, k = 2/ , and the angular frequency, = 2/ , this is: v = / k
Human wave
A typical human wave has a phase velocity of about 20 seats per second.
The Phase of a Wave
This formula is useful when the wave is really complicated.
The phase is everything inside the cosine.
E(t) = A cos(), where = k x – t –
=(x,y,z,t) and is not a constant, like !
In terms of the phase,
= – /t
k = /xAnd
– /t v = –––––––
/x
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