One Dimensional Waves

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One Dimensional Waves.

Three Dimensional Waves.

Harmonic Waves. Complex Representation of Waves.

Plane Waves.

Cylindrical Waves.

Spherical Waves.

Wave Motion.

Before we can understand how light moves from one medium to another and how it

interacts with lenses and mirrors, we must be able to describe its motionmathematically. The most general form of a traveling wave, and the differential equation

it satisfies, can be determined as follows. First consider a one dimensional wave pulse

of arbitrary shape, described by , fixed to a coordinate system O'(x',y')

Now let the O'system, together with the pulse, move to the right along the x-axis at

uniform speed vrelative to a fixed coordinate system O(x,y).
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As it moves, the pulse is assumed to maintain its shape. Any point Pon the pulse can

be described by either of two coordinates xor x', where x' = x - vt. The ycoordinate is

identical in either system. In the stationary coordinate system's frame of reference, the

moving pulse has the mathematical form

If the pulse moves to the left, the sign of vmust be reversed, so that we may write

(2.1)

as the general form of a traveling wave. Notice that we have assumed x = x'at t = 0,

and that the function fis any function whatsoever. We can extend this formalism to

three dimensions by defining the wavefunction, , as a function which requires fourvariables as input (three spatial and one temporal), and returns a single number as the

result. Thus we can write . In particular, we have that

(2.2)

As before, we see that the function fcan be any function whatsoever. The shape of the

wave at any instant, say at t = 0, can be found by holding the time constant at that

value. In this case,

(2.3)

represents the shape, or profile, of the wave at that time.


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Equating the two equations yields our result,

(2.4)

Three Dimensional Waves

This result can be extended to three dimensions by using the same argument for all

three components of the vector. In this case, the derivative is replaced by the

directional derivative, or gradient, operator, which is written as

(2.5)

in Cartesian coordinates. Thus, the wave equation (2.4) becomes

(2.6)

where is called the Laplacian operator, and is defined as

(2.7)

in Cartesian coordinates.

Harmonic Waves

Of special importance are simple harmonic waves that involve the sine or cosinefunctions. These waves can be written in a uniform way as

(2.8)

where A is known as the amplitude of the wave, kis the propagation number, and

is the initial phase, or epoch angle. These are periodic waves, representing smooth


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pulses that repeat themselves endlessly. Such waves are often generated by

undamped oscillators undergoing simple harmonic motion. More importantly, the sine

and cosine functions together form a complete set of functions; that is, a linear

combination of terms like those in (2.8) can be found to represent any actual

periodic wave form. Such a series of terms is called a Fourier series.

How do we interpret (2.8) in a physical manner? Consider the following drawings

The top drawing shows the wave at a fixed time, as in a snapshot. The maximum

displacement of the wave is the amplitude A, and the repetitive spatial unit of the waveis shown as the wavelength, . Because of this periodicity, increasing all xby

should reproduce the same wave. Mathematically, the wave is reproduced because the

argument of the sine function is advanced by . Symbolically,


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It follows that , so the propagation constant contains information regarding the

wavelength

(2.9)

Alternatively, if the wave is viewed from a fixed position, as in the bottom figure, it is

periodic in time with a repetitive temporal unit called the period, . Increasing all tby

, the wave form is exactly reproduced, so that

Clearly, , and we have an expression that relates the period to the

propagation constant kand the wave velocity v. The same information is included in the

relation

(2.10)

where we have used (2.9) together with the reciprocal relation between the period

and the frequency ,

(2.11)

Two other parameters are also frequently used. The combination

(2.12)

is called the angular frequency, and the reciprocal of the wavelength

(2.13)

is called the wave number.

The argument of the sine, which is an angle that depends on space and time, is called

the phase, . So, in (2.8) we have

(2.14)


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The magnitude of z, symbolized by r, also called its absolute value or modulus, isgiven by the Pythagorean theorem as

Combining this with the diagram, we see that and . Combining these

in (2.17), we get

which, by Euler's formula, is

(2.18)

where

(2.19)

The complex conjugate z*is simply the complex number zwith ireplaced by -i. Thus, ifz = x + iy,

Using Euler's formula, it is possible to express a simple harmonic wave by


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(2.20)

where

(2.21)

and

(2.22)

Expressed in the form of equation (2.20), the harmonic wave function thus includes both

the sine and cosine waves as its real and imaginary parts. Calculations employing the

complex form implicitly carry correct results for both sine and cosine waves. At any pointin such calculations, appropriate expressions for either form can be extracted by taking

the real or imaginary parts of both sides of the equation. Because the mathematics with

exponential functions is usually simpler than with trigonometric functions, it is often

convenient to deal with harmonic waves written in the form of equation (2.20).

Plane Waves

Can we write the wavefunction for more complicated waves? We can, if we take

advantage of the symmetries inherent in the wave form. For example, let's consider a

wave which exhibits rectangular symmetry. In other words, consider a wave moving inthe kdirection such that, at a fixed time, the phase is a constant. Then the

surfaces of constant phase form a family of planes at right angles to the vector k. These

planes are called the wavefronts of the disturbance. Mathematically, we can write

(2.23)

where k is now called the propagation vector and denotes the direction of motion for the

wave. The wave given by (2.23) can easily be seen to satisfy a wave equation of the

form

(2.24)

A wave which satisfies the wave equation (2.24) is called a plane wave. Notice that by

an appropriate rotation of our coordinate system, we can orient the wave so that it is

propagating solely in the new xdirection. In this case, the wavefunction becomes


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which supports our claim of rectangular symmetry.

Cylindrical WavesWhat other types of coordinate symmetry can we use? There are many different ways

to construct a three dimensional orthonormal coordinate set, and each one can be used

to define a particular wavefunction. Two of the more common curvilinear coordinates

are cylindrical and spherical coordinates. Recall that cylindrical coordinates are defined

by

Thus,

In this coordinate system, the Laplacian operator becomes

(2.25)

The requirement of cylindrical symmetry means that


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The -independence means that a plane perpendicular to the z-axis will intersect the

wavefront in a circle, which may vary in r, at different values of z. In addition, the z-

independence further restricts the wavefront to a right circular cylinder centered on the

zaxis and having infinite length. The differential wave equation is accordingly

(2.26)

Let us assume a solution of the form

Substituting this back into the wave equation, we get

Notice that the right hand side is purely a function of t, while the left hand side is purely

a function of r. This means that each side must be equal to an arbitrary constant

independent of rand t. This technique is known as separation of variables. The right

hand side can immediately be solved to yield

where kis a constant. The left hand side is known as Bessel's equation, and is solved

in terms of special functions known as Bessel functions. The important thing to know

about Bessel functions are that for large values of rthey can be approximated as

Thus, for large r, the cylindrical wavefunction becomes

(2.27)


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Spherical Waves

Finally, let's consider the case of spherical symmetry. Spherical coordinates are defined

by

so

In this coordinate system, the Laplacian operator becomes

(2.28)

The requirement of spherical symmetry means that

Thus, the wave equation becomes

(2.29)


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As before, let's assume that the solution can be separated, . Then

separation of variables yields the two equations

(2.30)

and

(2.31)

Equation (2.31) is known as the modified Bessel's equation. We can gain an intuitive

idea of the form of the solution to (2.29) without having to solve (2.30) and (2.31)

explicitly. Notice that the left hand side of (2.29) is equivalent to

so the wave equation becomes

(2.32)

where we multiplied through by r. Notice that (2.32) is just the one-dimensional wave

equation with the wavefunction replaced with . Thus, we know that the generalsolution to (2.32) is of the form

(2.33)

When fand gare simple harmonic functions, this simplifies down to


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(2.34)

Notice that each wavefront is given by

kr = constant

so that the amplitude of the spherical wave decreases as it moves away from its source.

Last updated: June 14, 1997

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