34 Waves and the Electromagnetic Spectrum (M7Q1)

A wave is an oscillation or periodic movement that can transport energy from one point in space to another. Common examples of waves are all around us. Shaking the end of a rope transfers energy from your hand to the other end of the rope, dropping a pebble into a pond causes waves to ripple outward along the water’s surface, and the expansion of air that accompanies a lightning strike generates sound waves (thunder) that can travel outward for several miles. In each of these cases, kinetic energy is transferred through matter (the rope, water, or air) while the matter remains essentially in place.

All waves are characterized by a wavelength (denoted by λ, the lowercase Greek letter lambda), a frequency (denoted by ν, the lowercase Greek letter nu), and an amplitude. As can be seen in Figure 1, the wavelength is the distance between two consecutive peaks or troughs in a wave (measured in meters in the SI system). The frequency is the number of wave cycles that pass a specified point in space in a specified amount of time (in the SI system, this time is measured in seconds). A cycle corresponds to one complete wavelength. The unit for frequency, therefore, is expressed as cycles per second (s⁻¹), also referred to as hertz (Hz). The amplitude corresponds to the magnitude of the wave’s displacement and so, in Figure 1, this corresponds to one-half the height between the peaks and troughs.

As can be seen in Figure 2, the properties of a wave are related. On the left side of Figure 2, all waves have the same amplitude, but the wave on top only completes 3 cycles per second, and therefore has a frequency of 3 Hz. The wave on the bottom completes 12 cycles in the same amount of time (1 second), and therefore has a frequency of 12 Hz. In order to complete 4 times as many cycles in the same timeframe, the wavelength of the bottom wave must be shorter than the wavelength of the top wave (in fact, it must be 4 times shorter) and assuming that the waves are traveling at the same speed. This produces an inverse relationship between wavelength and frequency—as wavelength decreases, frequency increases. Note that this relationship does not depend on the amplitude of the wave. On the right side of Figure 2, you can see that waves with the same wavelength and frequency can have different amplitudes. The amplitude is related to the intensity of the wave, which for light is the brightness, and for sound is the loudness.

Not all waves are traveling waves. Standing waves (also known as stationary waves) remain constrained within some region of space. The simplest example of a standing wave is a one-dimensional wave associated with a vibrating string that is held fixed at its two end points. Figure 3 shows the four lowest-energy standing waves for a vibrating string at a particular amplitude. Although the string’s motion lies mostly within a plane, the wave itself is considered to be one dimensional, since it lies along the length of the string. The motion of string segments in a direction perpendicular to the string length generates the waves and so the amplitude of the waves is visible as the maximum displacement of the curves seen in Figure 3. The key observation from the figure is that only those waves having an integer number, n, of half-wavelengths between the end points can form. A system with fixed end points such as this restricts the number and type of the possible waveforms. Another important observation is that the waves displaying more than one-half wavelength all have one or more points between the two end points that are not in motion. These special points with zero amplitude are called nodes. The energies of the standing waves with a given amplitude in a vibrating string increase with the number of half-wavelengths, n. Since the number of nodes is n – 1, the energy can also be said to depend on the number of nodes, generally increasing as the number of nodes increases.

The Electromagnetic Spectrum

The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms and was able to demonstrate that white light consists of the individual colors of the rainbow combined together. Newton explained his optics findings in terms of a “corpuscular” view of light, in which light was composed of streams of extremely tiny particles traveling at high speeds according to Newton’s laws of motion. Others in the seventeenth century, such as Christiaan Huygens, had shown that optical phenomena such as reflection and refraction could be equally well explained in terms of light as waves traveling at high speed through a medium called “luminiferous aether” that was thought to permeate all space. Early in the nineteenth century, Thomas Young demonstrated that light passing through narrow, closely spaced slits produced interference patterns that could not be explained in terms of Newtonian particles but could be easily explained in terms of waves. Later in the nineteenth century, after James Clerk Maxwell developed his theory of electromagnetic radiation and showed that light was the visible part of a vast spectrum of electromagnetic waves, the particle view of light became thoroughly discredited. By the end of the nineteenth century, scientists viewed the physical universe as roughly comprising two separate domains: matter composed of particles moving according to Newton’s laws of motion, and electromagnetic radiation consisting of waves governed by Maxwell’s equations. Today, these domains are referred to as classical mechanics and classical electrodynamics (or classical electromagnetism), respectively. Although there were a few physical phenomena that could not be explained within this framework, scientists at that time were so confident of the overall soundness of this framework that they viewed these aberrations as puzzling paradoxes that would ultimately be resolved somehow within this framework. As we shall see, these paradoxes led instead to an entirely new framework that intimately connects both particles and waves at a fundamental level called wave-particle duality, which has superseded the classical view and marked the beginning of a revolution in science.

Water waves transmit energy through space by the periodic oscillation of matter (the water). But waves need not be restricted to travel through matter. As Maxwell showed, electromagnetic waves consist of an electric field oscillating in step with a perpendicular magnetic field, both of which are perpendicular to the direction of travel (Figure 4). The properties of waves—λ, ν, and amplitude—that were used to describe waves in matter also apply to forms of electromagnetic radiation. Electromagnetic waves have wavelengths that fall within an enormous range—wavelengths of kilometers (10³ m) to picometers (10⁻¹² m) have been observed. Frequencies commonly have units of megahertz (1 MHz = 1 × 10⁶ Hz) and gigahertz (1 GHz = 1 × 10⁹ Hz), indicating that 1 × 10⁶ or 1 × 10⁹ wavelengths pass a single point in one second.

Electromagnetic waves can travel through a vacuum at a constant speed of 2.998 × 10⁸ m/s, the speed of light (denoted by c). The product of a wave’s wavelength (λ) and its frequency (ν), λν, is the speed of the wave. Thus, for electromagnetic radiation in a vacuum:

c = 2.998 × 10⁸ ms^-1 = λν

Notice that the units of c in this equation are m/s. For this equation, the wavelength must be in units of meters and the frequency must be in units of Hz (or 1/s) so that when you multiply λν you end up with m  ×  [latex]\frac{1}{s}[/latex]  =  [latex]\frac{m}{s}[/latex], which is the unit of the speed of light. Make sure that your units appropriately cancel using dimensional analysis, as always.

This equation is another demonstration of the inverse relationship between wavelength and frequency: As the wavelength increases, the frequency must decrease so that the speed remains constant. The inverse proportionality is illustrated in Figure 5. This figure also shows the electromagnetic spectrum—the range of all types of electromagnetic radiation. Each of the various colors of visible light has specific frequencies and wavelengths associated with them, and you can see that visible light makes up only a small portion of the electromagnetic spectrum. Because the technologies developed to work in various parts of the electromagnetic spectrum are different, for reasons of convenience and historical legacies, different units are typically used for different parts of the spectrum. For example, radio waves are usually specified as frequencies (typically in units of MHz), while the visible region is usually specified in wavelengths (typically in units of nm or angstroms).