Introduction to Traveling Light Waves

In this, and in all reading assignments, Discussion Questions and Activities are meant to be completed when they are reached in the reading, before continuing.  Getting the "correct" answer to Discussion Questions is not important.  Instead, the purpose of Discussion Questions is to address the issues, start you thinking about the material, and identify your preconceptions.  Completing these assignments before continuing with the reading will aid you greatly in the learning process.
 

Information transfer depends highly on the properties of waves, since information is often sent using light, radio waves, or other wave carriers.  This reading assignment will introduce you to the properties of waves.

Definition of a Wave

When trying to come up with a good definition of the wave, I found the following statement in a new physics textbook by Reese: "You all know what a wave is . . . at least until you have to write a paragraph defining one."  I have seldom agreed more with an author.  A precise definition of a wave requires mathematics, and any verbal definition (short of repeating the mathematical definition) will have some shortcomings.  Nevertheless, I will present my best effort at a verbal definition, hoping to give you a general understanding of waves.

Waves come in many different shapes and forms.  Water waves move across the sea; sound waves move through air; and sports fans create waves that move through stadiums.  In each of these situations, information or energy moves without an individual particle carrying it.  For example, the wave sports fans create in a stadium moves around the stadium while each person stays in his or her seat.  Information (the wave) moves between two points, but no individual physical object moves between those two points.  This is one defining characteristic of waves.

Water molecules and sports fans move up and down while the wave moves sideways.  Such waves in which the motion of the wave is perpendicular to the motion of the objects creating the wave are called transverse waves.  Sound waves are compression waves, or longitudinal waves, in which the motion of the air is in the same direction as the motion of the wave.  Note, however, that the motion of the individual regions of air is on a much smaller scale than the motion of the wave.  Air is compressed a fraction of a fraction of a centimeter when I speak, but the sound wave travels many meters.  Another example of a compression wave would be people shoving in a long line.  If a rude person at the end of the line shoves the next person forward, that person will fall into the next person who will fall into the next person . . . .  The motion of the people is in the direction of the motion of the wave, so it is longitudinal.  Still, no one person moves from the rear of the line to the front of the line, but the "signal" of a person falling forward moves all the way through the line, so it is a wave
 

Describing a Wave

For example, consider the function  y(x) = f(x) = (1 + x2)-1, which is plotted to the right in red.   If this pulse were to move to the right (toward "positive x"), it would look like the green dotted line at some later time t.  This new function can be described by  y(x,t) = f(x-vt) = (1 + (x-vt)2)-1, where v is the velocity of the pulse, found by dividing the distance one point on the pulse moves by the time.  The minus sign between the x and the vt indicates that the wave is moving toward positive x:  for a larger t, x must be bigger if y is to have the same value as before.  To obtain the expression for a wave after it has moved toward positive x with speed v for a time t, we merely replace x in the function with x-vt.

The wavelength l is the length of one cycle, or the distance between identical points in the cycle.  We often measure it from peak to peak, but we could use any point in the cycle to determine the wavelength.  Figure (a) above shows two determinations of the wavelength, one using the peak of the cycle and the other using an intermediate point.  The amplitude y_m of the wave in the figures is the distance from the x-axis to the minimum or maximum values.  This distance is not well-defined in waves that aren't symmetric about the axis, so we often discuss the "peak-to-peak amplitude," equal to the difference between the maximum and minimum values (2y_m for our wave).  The period T of the wave is the time required for the wave to complete one cycle.  Figure (b) below shows our wave at three different times; at time t₃, after one-half period has elapsed, the wave has moved half a wavelength to the right.  When an entire period has elapsed, the wave will have moved an entire wavelength to the right, reproducing the situation of Figure (a).  Alternatively, we can use a plot of y vs. t to define the period., shown in Figure (c)  Consider "Fred," who is standing at the origin.  At t=0, Fred sees the wave decreasing quickly to -y_m.  For half a period, the amplitude stays at -y_m, before steadily rising over the next half period to +y_m.  Thus the y vs. t graph of Figure (c) has the shape of the y vs. x graph but is reflected.
 

Perhaps the most familiar mathematical form for a wave is a sine wave.  Consider the red sine wave in the graph to the left.  It is described by  y(x) = ym sin (2px/l), where again l is the wavelength and ym is the amplitude of the wave.  After a time t has elapsed, the wave has moved to the right, as indicated by the green dashed line.  Following our definition of waves, we can describe this traveling sine wave by  y(x,t) = ym sin [(2p/l)(x-vt)].

where k is called the wave number, and w is called the angular frequency, or angular speed.  From comparison with the original equation we see that

The wave number and angular speed relate the physical quantities of distance and time to the cycle measurements of radians.  Remember that one complete cycle has 2p radians.  Wave number k is the number of radians completed in one meter.  The angular speed w is the number of radians per second.  It is related to the more familiar concept of period T.

Since a point on the wave travels one wavelength in one period, the velocity v of the wave is

so our angular speed w is then

Wave number is 2p divided by the wavelength; angular speed is 2p divided by the period.

Another useful parameter for describing waves is the frequency, f.  Please note that the frequency f is not the same as the angular frequency w.  w represents the number of radians per second, while the frequency f is the number of complete cycles per second.  The frequency is merely the inverse of the period and is related to angular frequency as follows:

Frequency is measured in per second, or Hertz.  One Hz is one cycle per second.
 

Example

2.5 seconds is the period, T.  The frequency is given by

Now that we have used our initial condition to find the amplitude, we can find the height of the wave at any other place and time.  For example, we could find the wave's height at the same location x = l/4 at a later time t = 1.0 s:

	(a)  A plane wave moving to the right.	(b)  Wavefronts drawn on crests of the wave.
	(c)  The equivalent 2D graph.	(d)  Wavefront representation of a plane wave.
	(e)  A circular wave, moving outward.	(f)  Wavefront representation of a circular wave.

Consider, for example, a plane wave: a three-dimensional wave that has the same value for every point on a plane (e.g., every value of y and z): 

Wavefronts don't have to be drawn through crests - any point in the wave cycle will do, but crests are used most often, since they are easy to identify.  Sometimes we also identify the location of the troughs of the wave, using a dashed line.  Wavefront representation is not limited to plane waves, but it is most useful for periodic waves with a uniform amplitude. For example, figures (e) and (f) show the 3D plot and the wavefront representation for circular waves.  Circular wavefronts, and the use of dotted lines for crests, will be used in the Interference, Diffraction, and Optical Storage module.

Electromagnetic Waves

The Electromagnetic Spectrum

Since different wavelengths of light have different energies, they can exhibit different properties.  For this reason, scientists did not initially relate such high-energy radiation like X-rays with the lower-energy visible light and the still lower-energy infrared radiation.  We denote these different groupings in the electromagnetic spectrum.  The spectrum starts with short-wavelength, high-energy, high-frequency gamma rays and stretches through the visible portion to low-energy, long-wavelength, low-frequency TV and Radio waves.  A schematic diagram of the spectrum can be found many places on the web, including http://www.geo.mtu.edu/rs/back/spectrum/.  Note that the only difference between the different portions of the electromagnetic spectrum is the wavelength (and therefore frequency and energy).

Within the visible spectrum, different wavelengths (and energies) of light correspond to different colors, with red light having the longest wavelength (and the smallest energy and frequency) and violet having the shortest wavelength (and the highest energy and frequency).  The visible spectrum stretches from about 400 nm to about 700 nm, with corresponding frequencies on the order of 10¹⁴ Hz.  The color white does not appear in the spectrum, since white light does not correspond to a particular wavelength but is a combination of all visible wavelengths.

Electromagnetic Traveling Waves

The electric and magnetic fields vary sinusoidally and travel as waves, so we can express their magnitudes as

The direction of the fields is given by a right-hand rule involving the direction of motion: if you place your thumb in the direction of the electric field and your index finger in the direction of the magnetic field, your middle finger (or palm) will point in the direction of the wave's motion. This relationship is consistent with the figure above.  Note that the red and blue curves do not represent a physical dimension (such as y or z) but represent the amplitude of the field at the particular value of x.  Because these fields have the form of 3D plane waves, their values do not depend upon y and z.  Each field will be uniform (same magnitude and direction) along any plane defined by a value of x.

Electromagnetic waves are generated by a variety of techniques, but these techniques always involve some charge oscillating in time. Usually, this oscillation of charge sets up an oscillating current in an antenna which radiates electromagnetic power. For efficient radiation, the length of the antenna is related to the wavelength of oscillation it will support, as will be discussed below.
 

Any introductory physics text, such as Fundamentals of Physics by Halliday, Resnick and Walker.

The entire wave module from Glenbrook High School's Physics ClassRoom.

http://www-jcsu.jesus.cam.ac.uk/~rpc25/notes/physics/waves/waves.html

http://en.wikipedia.org/wiki/Plane_wave

http://en.wikipedia.org/wiki/Wavefront