Solving The Wave Equation

In physics, following the example of James Clerk Maxwell, we can combine the third and fourth of Maxwell’s Equations under the conditions that occur in vacuum and get an equation that has as its solution a description of an electric or a magnetic field. In the case of an electric field that equation has it that

(Eq’n 1)

We now want to tease a proper algebraic description of **E** out of that
equation.

Begin with Euclid’s first common notion: things that stand
equal to the same thing also stand equal to each other. We can say that in
reverse: things that stand equal to each other also stand equal, both of them,
to some other thing. We assume that neither of the terms in Equation 1 equals a
constant, unvarying zero (we don’t want to consider uniform, stationary electric
fields), so that some other thing also does not equal a constant, unvarying
zero. The terms in Equation 1 involve differentiating **E** with respect to
mutually exclusive coordinates, so that some other thing does not involve either
of those coordinates. And both terms involve the first power of **E**, so
that some other thing must also contain the first power of **E**. Thus we
infer that each of the terms in Equation 1 stands equal to **E** multiplied
by some constant;

(Eq’n 2)

and

(Eq’n 3)

In those equations k represents a constant.

In those equations I have tacitly assumed that the second
term opposes the first term; that is, when the second derivative grows larger
the second term grows larger in the negative sense and when the second
derivative diminishes the second term diminishes in the negative sense. The k^{2}
term thus acts as a negative feedback on the derivative. That mathematical fact
ensures that **E** remains finite and bounded, which satisfies the
requirement that the energy manifested in the combined electric and magnetic
fields remain conserved.

To simplify our analysis let’s assume that **E** varies
only with the elapse of time and in the x-direction. Under that assumption
Equation 2 becomes

(Eq’n 4)

We can now "factor" the operator that acts on **E**, so Equation 4 becomes

(Eq’n 5)

If we apply the operator ∂_{x}-ik to that equation, we get

(Eq’n 6)

which tells us, when we compare it with Equation 4, that **E**’=(∂_{x}-ik)**E**
represents a solution of the wave equation. And that fact means that we can
rewrite Equation 5 as

(Eq’n 7)

which we can solve readily for **E**’ to get

(Eq’n 8)

in which **E**_{0}, the amplitude of the solution, represents the
antilogarithm of the constant of integration.

If we use Equation 8 to substitute into the definition of
**E**’, we can determine the algebraic form of **E**, which we get as

(Eq’n 9)

In the realm of mathematics that stands as a legitimate solution of Equation 2, but in the realm of physics it represents a forcefield that increases or decreases without bound, which represents a violation of the conservation of energy theorem. Thus, in physics we dismiss Equation 9 as a legitimate description of the electric field and use only Equation 8 with both positive and negative arguments.

We can use the same technique to solve Equation 3, getting

(Eq’n 10)

In this case we can delete the negative-signed argument from the solution on the basis of the statement that time does not elapse backward. Again we get an expression analogous to what we see in Equation 9 and again we dismiss it from the realm of physics.

Equation 1 tells us that applying the second derivative with respect to distance to the proper description of the electric field yields the same result as does applying the second derivative with respect to time and dividing by the speed of light. In order for that statement to stand true to mathematical reality in all circumstances we must have

(Eq’n 11)

that is, our description of the electric field must consist of two factors, each of which encodes only one of the variables in Equation 1. In addition, Equation 1 tells us, because the differential operators have the same form, that S(x) and T(t) must both have the same algebraic form.

If we hold the value of x constant and let the value of t increase, Equation 11 will become a constant fractional multiple of Equation 10 and if we hold the value of t constant and let the value of x vary, Equation 11 will become a constant fractional multiple of Equation 8. We thus infer that

(Eq’n 12)

For convenience I have removed the prime from **E**: since we have
dismissed the expression in Equation 9 as a solution in physically real
situations, we can do this without causing confusion. Having the product of two
exponential functions, we can see that if we hold the value of x constant, its
exponential provides the fraction to multiply Equation 10 and that if we hold
the value of t constant, its exponential provides the fraction to multiply
Equation 8. Indeed, strictly speaking, I should have shown the constant
fractions explicitly in Equations 8 and 10 as unknowns.

That solution describes a sinusoidal pattern, both in
space and in time. If we hold the value of t constant, then as the value of x
changes the value of **E** rises and falls in a wave-like pattern. On the
other hand, if we hold the value of x constant and let the value of t increase,
then the value of **E** vibrates like the motion of an harmonic oscillator.
If the values of both variables change, then the wave pattern moves: the
equation

(Eq’n 13)

describes a given value of **E** moving in either the positive or negative
x-direction at the speed c. Because its solution describes a wave-like
phenomenon, we call Equation 1 the wave equation.

But Equation 1 has applicability beyond describing electric fields propagating through vacuum. It also describes magnetic fields propagating through vacuum: indeed it must do so because Maxwell’s Equations necessitate that magnetic fields accompany electric fields that do not emanate directly from charged bodies. The wave equation also describes vibrations propagating through uniform elastic media; that is, it describes sound. Any phenomenon that we can describe in analogy with sound we can describe with the appropriate form of the wave equation.

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