Hello. In this video we are going to characterize one aspect of light waves by looking at what frequency and wavelength are and how they are related to the speed of light.
First we need to understand what frequency is, frequency’s relationship to wavelength, how frequency is measured, and finally looking at how frequency and wavelength are related to the speed of light.
And after all that we should be able to see why the speed of light
is always equal to its wavelength times its frequency.
And so we have the added bonus of gaining insight into
the beauty of the ability of mathematics to describe nature.
So, first question, what is frequency. The symbol for frequency is the Greek letter nu, which unfortunately bears a striking resemblance to the letter v, but it is a nu.
We define frequency as how often a wave cycle passes through a given point per second. A wave cycle is a single up/down motion, so we can think of the wave as consisting of a series of wave cycles.
And we will put a line at the wave front to indicate a point in space. So you can see that a moving wave has some amount of wave cycles passing a given point per unit time.
Now let’s take a look at the relationship between frequency and wavelength.
We know frequency has the symbol nu.
And here we can see the symbol for frequency, the Greek letter lambda.
Wavelength is defined as the distance from one wave cycle to the next.
You can see there are two different waves.
The top wave has a longer wavelength, the bottom wave has a shorter wavelength.
Both waves will move at the same speed.
So how will the different wavelengths affect their frequency,
the amount of wave cycles passing through the yellow line?
To bring the point home, each wave will flash as it passes the yellow line.
You can see that the shorter wavelength flashes more frequently than
the longer wavelength.
But before we connect this with light waves,
we need to understand how frequency is measured.
Frequency is defined as:
the number of cycles passing a given point per second.
Here we put a counter on both the wave cycles and on time.
For the top wave, we see that six cycles passed through in 7 seconds.
Dividing 6 cycles by 7 seconds gives a frequency, or nu, of 0.86 cycles per second.
The shorter wavelength at the bottom gives a frequency of 2.1 cycles per second.
So here we get a better look at the relationship between
frequency and wavelength.
The longer the wavelength the smaller the frequency,
and a shorter wavelength has a larger frequency.
So, frequency and wavelength have an inverse relationship.
It is important to notice here that the unit for frequency is simply per second.
We say cycles per second,
but mathematically it is only expressed as per second,
which is often expressed as s to the negative one.
Finally, hertz is another way to express cycles per second.
All of these are in common use.
Now let’s talk about the speed of light so we can tie it to frequency and wavelength. Speed is how long it takes to get from one place to another. Let’s say you have a friend standing 300 million meters away from earth, or about 23 ½ x the earth’s diameter, waiting with great anticipation for a signal from you standing on earth.
How long would it take your friend to see a single wave of light emitted from your red laser? Well, the speed of light happens to be 300 million meters per second, and so it takes one second for the flash of red light to reach your friend.
Interestingly, when your friend sees the light they actually are seeing what you did one second in the past.
Let’s look at something slightly more realistic... the moon, it would take 1.23 seconds for your light to go 370 million meters to the moon.
The sunlight we see reflected from the moon’s surface takes 1.23 seconds to reach the earth.
And so, when we look at the moon, we see what happened on the moon’s surface 1.23 seconds in the past.
The speed of light is normally given in scientific notation, 3 x 10⁸ m/s.
And yes, it has its own symbol, which is lower case c, so c, the speed of light, is 3 times 10⁸ meters per second, or 300 million meters per second.
Knowing that the speed of light is a constant will give us access to the mathematical relationship it has with wavelength and frequency. Let’s derive that relationship.
First you should know that the wavelengths of light cover a very wide spectrum, from greater than 100 meters to less than a trillionth of a meter. A narrow band within that range are light waves that we have evolved to detect with our nervous system, and so this is the visible spectrum, with wavelengths between 400 and 700 nanometers, with each color corresponding to a different wavelength.
Other wavelengths correspond to a variety of different waves such as radio waves, microwaves, infrared, ultraviolet, x-rays, and gamma rays.
These are simply names given to different ranges of wavelengths.
However it is important to remember that all are moving at the same speed, 3 x 10⁸ m/s, the speed of light.
Before we get to the final lap, let’s make sure we have our symbols and units straight, which can be confusing.
Each symbol simply represents a specific measure.
It is really the unit that is most important, which gives meaning to the number of the measurement.
So how are wavelength and frequency related to the speed of light?
If we look at the units of speed and wavelength,
we can see that dividing meters per second by meters will cancel meters,
and we get per second, the unit of frequency.
Let’s say we have light with a wavelength of one meter. Its speed is 3 x 10⁸ m/s, and the division gives us a frequency of 3 x 10⁸ cycles per second.
So a wavelength of one meter
corresponds to a frequency of 3 x 10⁸ cycles per second.
If we have a wavelength of 2 meters, we get a frequency of 1.5 x 10⁸ cycles per second.
And if we have a wavelength of 0.5 meters we get a frequency of 6 x 10⁸ cycles per second.
So, any wavelength has a corresponding frequency, as we can see here. Now we can start putting it all together using symbols to arrive at our equation relating wavelength, frequency, and the speed of light.
So, the speed of light, divided by wavelength equals frequency. We can algebraically rearrange to c = Lambda nu, but it is generally represented as lambda nu = c, and so we will keep that.
If we rearrange our original calculation to lambda nu = c, we can see that the units work out correctly, meters times per second equals meters per second.
So lambda, wavelength, times nu, frequency, equals c, the speed of light.
and therefore each wavelength has a corresponding frequency, and any wavelength,
multiplied by its corresponding frequency, will always give the speed of light.
You now should be able to answer our last question: Why is the speed of light always equal to wavelength times frequency.
Well I hope you noticed in this illustration that wavelength decreases going toward the right, while frequency increases going toward the right.
What you should also notice is that this inverse relationship is an exact proportion.
Because the speed of light is constant, any decrease in wavelength is accompanied by a proportionate increase in frequency.
It is a simple relationship, shown by the equation itself, but the equation comes out of the nature of light.
A simple example of the beauty of mathematics giving a description of a natural phenomenon, the relationship of frequency, wavelength, and the speed of light.
So that should be the end of the video but I have another illustration to show you. I originally was going to show the speed of light by showing a light wave traveling from the sun to the earth. The problem is it takes 8 minutes and 19 seconds, traveling at 300 million meters per second.