Homework Helpers: Physics

8 Waves and Light

Electromagnetic waves, such as light, differ from the waves that we are more familiar with in that they don”t require a material medium through which to travel. In order to argue that light is made up of waves, people must have seen some properties or characteristics of waves that apply to light. Let”s look at some of these features of light and waves in general.

Let”s begin by talking about the anatomy of a wave. It may help to keep our example of the rope from Lesson 8–1 in mind as we talk about the “parts” of the wave. Keep in mind that waving a rope up and down produces a transverse wave, as the disturbance travels perpendicular to the displacement of the medium. The maximum displacement of the rope above its rest position, or the highest point on a wave, is called the crest. The maximum displacement of the rope below its rest position, or the lowest point on the wave, is called the trough. In a uniform wave, the trough is just as low as the crest is high, meaning that if you measure the distance from the rest position (equilibrium) of the wave to the trough it will be the same as the distance measured from the rest position to the crest. We call this distance of maximum displacement the amplitude of the wave.

We use the term amplitude, and also the terms that follow for both transverse and longitudinal waves. The compressions are analogous to the troughs and the rarefactions are analogous to the crests. So, for example, measuring the maximum displacement of the molecules in a rarefaction will still give you the amplitude of a longitudinal wave.

One complete waveform, meaning one crest and one trough in transverse wave, or one compression and one rarefaction in a longitudinal wave, makes up one cycle. If you measure the length of one complete cycle or waveform, it is called the wavelength. We will represent the wavelength with the Greek letter lambda (λ). It is important to remember that in a uniform wave train, the length of the crests and troughs are equal, so the length of each crest or trough is half of the wavelength. People will often measure from a certain point on one cycle to the same point on the next wave cycle to get the wavelength.

Figure 8.1

The amount of time it takes for one complete cycle (a crest and a trough) to pass by a given point is called the period (T) of the wave. So, for example, if our duck from our last lesson bobbed up and down through one complete crest and trough in 2.0 s, that would be the period of the wave. We discussed this concept in Chapter 4, when we talked about circular motion. The meaning of the term, period, is the same in this chapter. We are just using a more specific definition.

The number of complete waveforms, or cycles, that pass a given point in a second is called the frequency (f) of the wave. As you might recall from Chapter 4, the frequency of a wave is the inverse of its period, or . So, if the period of the water wave that has been disturbing our poor duck is 2.0 s, then the frequency of the disturbance is .

Try and see past the numbers and units in these calculations to make sense of them. You will only be able to check to see if your answers make logical sense if you go beyond the numbers, and think of their actual meaning. If you take the example of the duck, realize that a period of 2.0 s means that one complete wave will go by every 2 seconds. We would then predict that half of a wave cycle should be able to go by in 1 second. In this way, our answer of 0.50 Hz, or 0.50 cycles/second, makes perfect sense.

Example 1

What is the period of a wave with a frequency of 5.0 Hz?

Does our answer to Example 1 make logical sense? If the wave has a frequency of 5.0 Hz, it means that every second, 5.0 complete cycles go by a given point. How long would it take for each cycle to go by that point? 1/5 of a second, making 0.20 s our period.

Speed of a Wave

The speed of a wave is obviously the amount of distance that a disturbance travels in a given period of time. As always, we could find the speed of a wave by using our old speed formula, . However, we will often have access to the wavelength (λ) and period (T) of the wave, which are simply specific values of distance and time, so we often write our formula for the speed of a wave as .

All electromagnetic waves travel at a speed of 3.00 × 10⁸ m/s in a vacuum, and this constant is often represented with the letter C, making our formula, .

Example 2

In a vacuum, light travels at (C) 3.00 × 10⁸ m/s. What would be the period of a ray of light with a wavelength of 590 nm (nanometers)?

Our answer means that one complete cycle goes by every 0.000 000 000 000 00 20 seconds!

Going back to our example of waves traveling across the rope, what would happen if the person generating the waves moved her hand much faster? Would the speed at which a disturbance travels down the length of the rope be made to increase? The answer is no. The speed that a wave will travel through a medium is actually dictated by the medium, not by how rapidly the disturbances take place. If the person moved her hand faster, she would generate cycles faster by decreasing the wavelength of each waveform, and increasing the frequency of the waves. So, she would generate more, shorter waves, rather than make the waves travel faster. In other words, the frequency would go up, the wavelength would go down, but the speed would be the same.

To find the formula that shows the relationship between the frequency, wavelength, and speed of a wave, we can simply combine the two formulas that we have already used in this lesson:

Starting with and , we substitute the value of T from the second formula into the first:

This formula, v = fλ, is very useful, particularly when dealing with electromagnetic waves, where it is common practice to substitute the letter C (the speed of an electromagnetic wave in a vacuum) for v, making our formula, C = fλ.

Example 3

The operating frequency of a particular radio station is 93.3 × 10⁶ Hz. What is the wavelength of the radio waves generated by this station?

Remember: Radio waves are examples of electromagnetic waves, as they can travel through space. All electromagnetic waves travel at 3.00 × 10⁸ m/s in space. Our atmosphere slows these waves so little that their speed in our atmosphere still rounds to 3.00 × 10⁸ m/s, so we use this value as a constant for the speed of an electromagnetic wave in space or in our atmosphere.

C = fλ

Example 4

The speed of sound waves in air at a temperature of 20.0°C is approximately 340 m/s. Find the wavelength of a sound wave, under these conditions, with a frequency of 265 Hz.

*Note: Sound waves are not electromagnetic waves, so we don”t use the constant C.

Interference

Is it possible to have multiple waves traveling through the same medium? Of course it is. Can you think of some examples from real life? When there is more than one source of sound in an area, there must be multiple sound waves traveling through the air. When more than one source is creating disturbances in the water, there will be multiple waves and ripples. What happens when waves occupy the same space?

When two or more waves move through the same region of space they maintain their individual integrity, and will not change as a result of overlapping. Though they occupy the same space, the individual disturbances will superimpose and produce a well-defined combined effect.

Constructive interference occurs when multiple waves, or parts of multiple waves, work together and generate a greater net disturbance than either of the individual disturbances. In this way, a number of smaller disturbances can add together to form a larger disturbance.

Figure 8.2

Destructive interference occurs when multiple waves, or parts of multiple waves, work against each other and generate a net disturbance that is smaller than the individual disturbances. In this way, a number of disturbances can add together and form a smaller disturbance. If two waves that have the same frequency and wavelength, but are 180° out of phase combine, the result can be complete destructive interference, where the waves cancel each other out in a specific area.

When Waves Strike the Boundary to a Medium

What happens when a wave hits a boundary to another medium? The wave can be reflected, meaning it will be turned back so that it will travel back through the medium in the opposite direction. Depending on the boundary to the medium that the wave strikes, the wave may be reversed or flipped over in such a way that it is 180° out of phase. The wave could also be transmitted or allowed to pass into the next medium. Such a transmission may be associated with a refraction, which means that the wave can bend, or change directions, as it enters the new medium. We will go into more details about these characteristics in future lessons.

Another interesting property of waves, called diffraction, appears when a wave passes through an opening in a boundary to a medium. Waves will tend to spread out as they pass through such an opening. We can hear people speaking in another room, even if the door is closed, partially because the sound waves will spread out as they pass through the gap between the door and the floor.

Lesson 8–2 Review

1. _______________ is the addition of two or more waves resulting in a smaller net disturbance than the individual waves.

2. What is the wavelength of a microwave (in space) with a frequency of 2.40 × 10⁹ Hz?

3. What is the period of an electromagnetic wave (in space) with a wavelength of 3.0 × 10^–10 m?