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## Wave Equations

##### - what is required for an equation to be a wave equation? -

Mathematically we can define waves in several ways:

- As a
**shifting sinusoid**mathematically expressed in real form as:

where*A*and*B*are real constants,

i.e. amplitudes,(1)

or in complex form as:

where*A*is a complex amplitude,

i.e. an amplitude and phase shift.(2) In both the above forms *κ*is the wavenumber in units of radians per meter and*ω*is the angular frequency in units of radians per second. The independent variables*x*and*t*are position and time. In the complex form only the real part of the complex expression is considered to describe actual physical attributes of a wave. Because of this we are free to put a minus sign in front of the*i*in (2) without changing the physical attributes of the wave represented as long as we stay consistent in our use of a minus sign or not throughout the calculation at hand.Using a negative

*κ*in either of the above equations represents waves propagating in the negative*x*direction.

- A wave can also be an
**arbitrary function**that keeps its shape and moves along as time progresses. You might notice that the sinusoids in (1) and (2) are functions of*κx*−*ωt*. Alternatively they could be written as functions of*x*−*ct*where*c*=*ω*/*κ*is the wave velocity. Note that*κx*−*ωt*can be factored as*κx*−*ωt*=*κ*×(*x*−*ωt*/*κ*) =*κ*×(*x*−*ct*) where*κ*is the wavenumber for the sinusoids of (1) and (2).

This method of using a function of*x*−*ct*can work for any arbitrary function. As an example at the right we see a Gaussian bell shaped function converted into a function that moves with velocity*c*by replacing the*x*with an*x−ct*. - A different way to mathematically define a wave is with a
**differential equation**which has as its solution functions such as those above. The most cited example of such a defining differential equation is "the wave equation":

. (5)

The advantage of differential equations is that they allow the calculation of waves in different coordinate systems and varied boundary conditions such as for the waves inside a spherical resonator. Differential equations can also be modified to accommodate changing properties of the wave media such as varying wave velocities as a function of position.

Differential equations other than (5) may also serve as wave equations, even though (5) has become known as the official "wave equation".

### The wave equation - non-dispersive waves

Equation (5) has solutions which are non-dispersive, that is, waves whose
velocity of propagation does not depend on the frequency of the wave. Examples
of such waves are a acoustical waves of small amplitude and waves on a string, also of small amplitude. Another wave type
which is non-dispersive is electromagnetic waves such as light, radio waves and x-rays. Such electromagnetic waves have a wave
variable ( the *ψ* in (5) ) that is a vector in the general case;
however the wave variable can be a scaler as in (5) in certain cases, the cases that
are often treated in electromagnetic textbooks.

One property of a medium governed by a non-dispersive wave equation
is that waves of arbitrary shape keep their shape as they propagate. One
way to understand this is that any
arbitrary function of *x−ct* is
a solution to the wave equation (5). That is to say direct substitution of
*f*(*x−ct*) into (5) maintains the equality shown in (5). This type
of solution of "the wave equation" was first proposed by
Jean le Rond d'Alembert
in 1746.

To understand this we use the chain rule of differentiation:
so that substituting
*ψ*(*x−ct*) = *ψ*(*u*) into (5) yields:

where *u*=*x−ct*.

Another way to understand that waves of arbitrary shape propagate and maintain their shape is to understand that any function can be broken down into a sum of sinusoids of varying frequencies. Because these frequency components all travel at the same velocity, the shape will maintain itself.

In contrast in a medium with a dispersive wave equation, the various frequency components of an arbitrary shape will travel at different speeds and so add up differently as time progresses. This results in a wave that changes its shape as time progresses.

Another interesting exercise is to factor (5) which yields:

One solution of this equation occurs when the right most factor operating
on *ψ* is zero:

Putting *ψ* = *x−ct* into the left side of (8)
yields zero and thus satisfies (8), (7) and (5). Inverting the order of
the two factors in (7) results in an equation like (8) but with a minus
sign. That equation is satisfied by *ψ* = *x+ct* which is
a sinusoidal wave traveling in the negative *x* direction.

We see that (5) is a second order differential wave equation with two solution: a positive going and a negative going wave. Equation (8) is a first order differential wave equation with only one solution, a positive going wave.

### Water waves - dispersive

A common type of waves are those we see on the surface of water, i.e. water waves. These are dispersive. To better understand their dispersive nature we look at one of the defining differential equations, Equation (8) of this link.

We have replaced *Δy* of that link (the vertical
displacement of the water's surface due to the wave) with *δ _{y}*. The quantity

*δ*is the displacement of water in the

_{x}*x*direction due to the passing wave.

Looking at Equations (3a) and (3b) of the the link we see that the ratio of *δ _{y}*/

*δ*= −

_{x}*i*tanh

*κy*or solving for

*δ*this is

_{y}*δ*= −

_{y}*i δ*tanh

_{x}*κy*. Substituting this into (9) yields

where *y* is the depth of the water.

In (10) we have a wave equation expressing the horizontal displacement of the water parcels in response to a passing wave as a function of *x* and *t*.

To find out the constraint on these waves we substitute the complex traveling wave expression *δ _{x}* =

*A e*into (10) to get:

^{i(κx−ωt)}*gκ*tanh

*κy*=

*ω*

^{2}or:

We next solve for the phase velocity, i.e. the velocity of the wave crests. The standard equation for the phase velocity is *v _{p}* =

*ω*/

*κ*which when used with (11) gives us:

Thus the phase velocity is not constant with frequency. Water waves are therefore dispersive, meaning that different frequencies travel at different velocities.

The group velocity, i.e. velocity of energy and pulses such as the Gaussian shown above, has the standard equation *v _{g}* =

*dω*/

*dκ*. Using (11) this yields:

Incidently, for waves which are without dispersion, i.e. all frequencies travel at the same speed as discussed in the previous section, the phase velocity equals the group velocity and both are a constant.

### Waves in a waveguide

Microwaves (electromagnetic waves used for microwave ovens, cell phones and radar) are often channeled down metal pipes called waveguides. These have a
rectangular cross section (width *a* and height *b*).
While there are many modes of propagation, each with its own electric and magnetic
field configuration, these modes can be lumped into two classes. We will
focus on the TE modes which are the commercially most used. We will further
restrict our study of TE modes with non-zero electric fields in the *y*
direction, *E _{y}*.

The electromagnetic wave equation written for the *E _{y}* component of the electric field in free space is:

When confined to a metal waveguide the *y* electric field parallel to the walls
at the walls must be zero. It is also true that *E _{y}* at the top
and bottom of the waveguide must have

*∂E*/

_{y}*∂y*= 0. Due to these two constraints the electric field must assume a solution in the form:

where the *x* and *y* wavenumbers are given by:

The modal indices *m* and *n* are positive integers. For the
modes we are treating *m* must be greater than zero while *n* can be
zero or greater. (The TE_{0,1} mode, i.e. *m* = 0, *n* = 1 mode, has all its electric field in the *x* direction and is zero in the *y* direction while
the TE_{0,0} mode has all zero electromagnetic fields and is not a wave mode
at all.)

Substituting (14) into (13a) yields:

The first two terms in (16) are set by the waveguide dimensions, i.e. by (15), and can be replaced by the cutoff frequence *ω _{c}* given by:

Equation (17) calculates the lowest frequency that a particular mode
will propagate in the *z* direction. We can use (17) with
(16) to get:

Equation (18) is the defining differential wave equation for these fields
propagating in the *z* direction. It is similar to (5) except that it
has an additional term involving *E _{y}* times a constant (the first
term in (18) ).

We next replace *f*(*x*,*t*) in (14) with the standard complex
propagating sinusoid *Ae ^{i(κzz−ωt)}*
and then substitute the resulting

*E*into (18). We finally solve for

_{y}*ω*and get:

From this we can calculate the phase and group velocities:

##### Similar wave media: Electromagnetic waves in a plasma

A plasma consists of ionized gas molecules and free electrons and highly interacts with electromagnetic waves trying to propagate through it. There are a number of types of electromagnetic waves possible (with static magnetic fields or without and other criteria). See this link for details. One of the simpler cases is without a static magnetic field and is governed by this equation:

where *ω _{p}* is the plasma frequency and is the lowest
frequency at which this type of wave will propagate. It is related to
the density of free electrons in the plasma. Alternate source: D.K.Cheng, 2nd Ed. Field and Wave Electromagnetics p.377 Eq.8-73. Equation (22) is basically the
same as (19).

##### Similar wave media: the Klein Gordon Equation in quantum mechanics

The Klein Gordon equation governs the motion of spinless, chargeless particles which are moving at relativistic speeds (within an order of magnitude of the velocity of light):

where *c* is the velocity of light, *m* is the mass of the particles under study, and *ℏ* is a quantum mechanical constant.

With the standard complex
propagating sinusoid *Ae ^{i(κzz−ωt)}*
substituted in for

*ψ*this equation becomes:

Again we see the same dependence for the frequency *ω* on the wavenumber *κ* as we see in (19). All three waves have differential
equations that have second order derivatives in space and time plus an extra
term that is a constant times the wave parameter. All three display a minimum
frequency of wave propagation, i.e. a cutoff frequency. All three approach
dispersionless propagation at very high frequencies but show dispersion
at frequencies just above the cutoff frequency.

Equation (24) is the relativistic equivalent of the Schrodinger equation which we will discuss next.

### The Schrodinger equation

The Schrodinger equation governs the motion of spinless particles which are moving at non-relativistic speeds (much slower than the velocity of light):

where *V* is the potential the particle experiences and can vary with position.

With the standard complex
propagating sinusoid *Ae ^{i(κzz−ωt)}*
substituted in for

*ψ*this equation becomes:

In (26) we see a very different dispersion relation than we saw in the previous
wave media. If potential *V* is zero, the frequency depends on the square
of the wavenumber. If *V* is not zero but positive then there is an offset,
and there will be a minimum propagating frequency; this occurs when a particle
has sufficient energy (frequency and energy are proportional in quantum mechanics)
similar to the cutoff frequencies in the previously discussed media.

*p*=

*mv*=

*h/λ*remembering that

*κ*= 2

*π/λ*and that

*ℏ = h/*2

*π*. The constant

*h*is Planck's constant.

### A comparison of the discussed wave types

The *a*'s in the above equations are real constants. The differential equations (30) through (33) are taken from the preceding text with the constants replaced by *a*'s.

Looking at the above differential wave equations in Table 1 we see that all the equations relate derivatives with respect to position *x* to derivatives with respect to time *t*. All the derivatives are operating on a wave parameter labelled *ψ* for the purposes of comparison.

What differs between the four equations are the derivative orders. That is, in (30) we see second derivatives with respect to both *x* and *t* whereas in (31) we see a first derivative with repect to *x* and second derivative with respect to *t*. Eqs. (32) and (33) have a third term which has no derivative.

Eqs. (31) and (33) have the complex constant *i* in them which signals that they are written for complex solutions. The *i* is also necessary so that the standard complex sinusoidal wave, i.e. *Ae ^{i(κx−ωt)}*, be a solution to the equation.

### Requirements for a general differential wave equation

It would seem that we could write differential wave equations containing derivatives of any order, even fractional orders, as long as we included the required complex constant to insure that *Ae ^{i(κx−ωt)}* be a solution.

Could we write a differential wave equation that did not have *Ae ^{i(κx−ωt)}* as a solution but instead had some other solution that moved along in time? If that solution had a fixed shape as it moved along then this shape could be decomposed into sinusoidal Fourier components which would need to move along in unison so that the shape be preserved in time. This would indicate a non-dispersive differential equation of the type of (30) which of course has sinusoidal solutions.

So for a non-sinusoidal wave equation we are left with solutions which are non-sinusoidal and that do not maintain their shape as they moved along. Such "wave" solutions might be addressed by a differential equation which lacks the required complex constant to make *Ae ^{i(κx−ωt)}* a solution. It is not clear that these would be classified as waves.

### Example of creating a diferential wave equation

Suppose we want to create a differential wave equation for a media that supports sinusoidal waves and has a dispersion relation of
. To avoid fractional order derivatives we square both sides to have integer powers of *κ* and *ω*:
where *a* is the proportionality constant. We now write a differential equation that includes a second derivative of *ψ* with respect to time *t* and a third derivative with respect to position *x*:
where *b* is a complex constant to be determined. We substitute our standard wave
into that and arrive at:
.
This means
or
.
Substituting *b* into our differential equation we have our final wave equation:
.

### Non-linear wave equation?

All the above work assumes that we are dealing with a linear wave medium. That is to say if *ψ*_{1} and *ψ*_{2} are both solutions of the wave equation for the medium in question then the sum of these, i.e.
*ψ*_{3} = *ψ*_{1} + *ψ*_{2}, will also be a solution. Note for example that our derived wave equation just above is first order in *ψ* even though it involves higher order derivatives of *ψ*.

In reality, many wave media are not completely linear, especially for large amplitude waves. We will not address such waves here, except to say they are more complicated to deal with.

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