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