cn 1 wave equation

8/10/2019 Cn 1 Wave Equation

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Equations describing waves

A moving wave must be described as a function of

space and time that gives the amplitude at each point foreach moment in time:

ψ (x,t)

ψ ψψ ψ : amplitude of the wave

The shape of the wave at constant time, for example

t=0 depends only on the position:)(),(

0 x f t x

t =

=ψ

This represents the profile of the wave at a instant of

time.

Example of a wave at t=0:

f( )x exp x2

4 2 0 2 40

0.2

0.4

0.6

0.8

1

f( )x

x



Travelling Waves

Assuming that the wave does not change shape with

time (non-dispersive):

The wave at a later time will have the same shape but a

different position

)(),( x f t xt t

′=′=ψ

4 2 0 2 40

0.2

0.4

0.6

0.8

11

0

f( )x

g( )x

55 x

If the wave is travelling at speed v and the time is t

then the wave has travelled in x a distance of vt .

So at a later time t the function that describes the wave

is

vt x x −=′

and )(),( vt x f t x −=ψ



ψ ψψ ψ (x,t)=f(x−vt) is the most general form of the

travelling wave equation.

If the wave is travelling in the other direction then the

sign of the time part is reversed:

)(),( vt x f t x +=ψ with v>0

For our example wave the wave function is then:2

)(exp),( vt xt x −−=ψ



Harmonic Waves

Harmonic waves have sine (or cosine) wave functions:

)sin()(),(0

kx A x f t xt

===

ψ

kx is in radians {10=2π /360 radians}

k : propagation number [radians/m]

Maximum and minimum values of ψ ψψ ψ are +A and −A

A : amplitude of the wave

If the wave is moving with speed v in +x direction then

the equation becomes:

))(sin()(),( vt xk Avt x f t x −=−=ψ [1.1]



Plots of waves a fixed times and fixed

postions

A 1. k 1 v 0.5

ψ ( ),x t .A sin( ).k ( )x .v t

5 0 5 10 151

0.5

0

0.5

1

ψ ( ),x 0

ψ ( ),x 1

ψ ( ),x 2

x

5 0 5 10 151

0.5

0

0.5

1

ψ ( ),0 t

ψ ( ),1 t

ψ ( ),2 t

t



Wave notation:

Another way to write the wave equation is:

)sin(),( t kx At xψ ±= [1.2]

k=2π/λ π/λ π/λ π/λ k : propagation number [rad m-1]

λ λλ λ : wavelength [m]

κ=1/λ= κ=1/λ= κ=1/λ= κ=1/λ= k/2π /2π /2π /2π κ κκ κ : wave number [m-1], used in some

areas

ω ωω ω =2π ππ π f ω ωω ω : angular frequency [rad s-1]

f : frequency [Hz or s-1]

v p=ω ωω ω /k v p : phase velocity [ms-1]

φ= φ= φ= φ= kx-ω ωω ω t φ φφ φ : phase of the wave [rad]

If φ is constant ψ (x,t) is constant, so vp is the velocity

of points of constant ψ .

Two conventions for a wave travelling in the +x

direction: ψ (x,t)=A sin(kx − ω t ) or ψ (x,t)=A sin(ω t − kx)

I will use the first as this is the convention in quantum

mechanics, Hecht’s book Optics uses the second!



Wave equation

This formula obeys the wave equation:

2

2

2

2

2

1

xt v p ∂

∂=

∂

∂ ψ ψ [1.3]

This is not the only wave equation, but it is the

simplest with wave solutions.

Prove equation [1.2] satisfies equation [1.3]…

In fact any equation of the form [1.1] satisfies [1.3], so

any arbitrary wave can propagate

Only if v p is independent of the wavelength (or

frequency) is the wave shape unchanged.



Generalising the wave equation

Using the equation ψ (x,t)=A sin(kx ± ω t ) implies

ψ (0,0)=0

In the general case this is not always true. A phase can

be added to allow ψ (0,0)≠ 0

ψ (x,t)=A sin(kx ± ω t+ε ) [1.4]

Where ε is the phase at x=0, t=0 in radians.

Remember sin(φ−π /2) = cos(φ) so if the wave has

ψ (0,0)= A then the solution is

ψ (x,t)=A cos(kx ± ω t )

Choose whichever formula is easiest to work with.

The most general solution to the wave equation [1.3] is

)()(),( t v xgt v x f t x p p ++−=ψ

where f(x) and g(x) are arbitrary functions.



Principle of superposition

The sum of any two solutions to the wave equation

is another solution to the wave equation

More complicated waveforms can be built by adding

many simpler wave functions.

Complex number representations of

travelling waves

Reminder of the Euler formula:

θ θ θ sincos ie

i +=

So another way of writing the travelling wave

equations is:

)cos(Re),( )( ε ω ψ ε +±=⋅= +±t kx Ae At x

t kxi

Where Re{ x} means the real part of the complex

number x. The symbol may not be written but is always

assumed to be there are so that ψ (x,t) is always a real

number.



Why use complex notation?

Waves have an amplitude (A) and a phase (φ) at each

point, so do complex numbers. Complex numbers can be

written as:

z=a+ib or z=reiφ

r = (a2+b2)

tan(φ)=b/a

The algebra of complex numbers behave in the same

way as the waves they describe. This makes combining

waves in interference calculations for example much

simpler.

Examples of complex numbers in waves

The complex number representations of sine and

cosine are also used in electrical circuits:

Z

Ve

e Z

Ve I

t i

i

t i )( φ

φ

−==

The REAL current flowing is||

)cos(

Z

t V φ −



In quantum mechanics the amplitude may be complex,

this is possible as all measurable quantities depend on

terms similar to:

ψ *(x,t)ψ (x,t)= Real number

For example the probability of finding a particle with a

wavefunction ψ in the volume dV is

ψ *ψ dV



3 dimensional plane waves

In 3D if the points on a wave with the same phase form

a plane the wave is a plane wave. This is the simplest

form of 3D wave to consider.

Equation of a plane

r ≡ [x,y,z] and k ≡ [k x , k y , k z]

r is a point on a plane (in 3D) and k is a

vector (in 3D), for a plane passing through

point r0 and perpendicular to k:

0)( =⋅− krr 0

or 0)()()( 000 =−+−+− z zk y yk x xk z y x

or a zk yk xk z y x =++

where a zk yk xk z y x =++ 000 = constant

So the equation of a plane can be written as

a=⋅rk



Equation of a plane wave

So at constant time (e.g. t=0) if the phase of the wave is

constant over the plane:

{ }rkr ⋅−= i Aexp)(ψ [1.5]

Now for this wave in a distance λ the phase of the

wave must repeat in phase:

) / .()( k krr λ ψ ψ −= [1.6]

k is the magnitude of k so k =|k|

Combining [1.5] and [1.6] we get

k iik ii e Ae Ae Ae λ λ rkkrkrk⋅−+⋅−⋅− == ) / (

so λ 21

ink iee == for integer n

Solution with n=0 means λ or k = 0, so take n=1 then

find

λk = 2π and k = 2π/λ

So as before k is the propagation number and k is the

propagation vector.



The time dependence is just like the 1D wave, the phase

must advance at a fixed point, r, at a rate ωt so the full

equation is

)()( t i Aeψ −⋅−= rk

r

The planes of constant phase are known as wavefronts

and these move with the phase velocity v p=ω /k , just as

the constant phase points on 1D waves do.

cn 1 wave equation

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