Lesson Video: Wave Speed | Nagwa Lesson Video: Wave Speed | Nagwa

Lesson Video: Wave Speed Science • Second Year of Preparatory School

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In this video, we will learn how to calculate the speed of a wave from its frequency and wavelength.

11:20

Video Transcript

In this video, we will be learning about wave speed. In particular, we are going to see how we can calculate the speed of a wave from its frequency and wavelength. Before we look at wave speed, we will begin this video by reminding ourselves what a wave is and what some of its properties are.

A wave is a type of disturbance that transfers energy from one point to another. We showed three examples of waves on the opening screen. These include light waves emitted from a light bulb, which transfer light energy from one point to another, or sound waves emitted from a speaker, transferring sound energy from one point to another. Another example of a wave is if we hold on to the left end of a rope and we move that end of the rope up and down continuously, this creates a disturbance in the rope, which transfers energy to the right, like so.

Now let’s take a closer look at this particular wave. We can see that this wave has traveled a certain distance, from its starting position here to its ending position here. So let’s take a snapshot of this wave and plot this wave on a displacement–distance graph as follows. This graph represents the wave at a single instance of time, and we can see how far the wave has traveled.

Now we can recall an important property of a wave, which is the wavelength. The wavelength of a wave is the distance covered by one complete cycle. We often represent wavelength with the Greek letter 𝜆. So, let’s start using it now.

To help us work out what the wavelength of this particular wave is, let’s remind ourselves of what a wave cycle is. Let’s start at this beginning point here. And let’s follow this path along the wave until this point here. It starts by going up and reaching some maximum value. It then goes down to zero and goes down further to some minimum value before it goes back up to zero again at this endpoint. What we have outlined here is one complete wave cycle. Now on this particular graph, this is the distance covered by one complete cycle. So, this length here is the wavelength for this wave.

The wavelength is a length, so this means the standard unit used to measure wavelength is the meter. So looking at this graph, we can read off a value of five meters for the wavelength of this wave. Now the wavelength can be measured between any point on the wave and the location of that point in the next cycle. So, we could have measured the wavelength of this wave as the distance from one crest to the next crest, like so. Or we could measure it from one trough to the next trough, like so.

We will now recall a second important property of a wave, which is the frequency. The frequency of a wave is the number of wave cycles that pass through a certain point per second. We often represent frequency with lowercase 𝑓. So, let’s consider this example of a wave plotted on a displacement–time graph. We notice here that we have time on the horizontal axis instead of distance. We can think of this graph as representing a single point in space, and it measures the changes in its displacement over time.

To determine the frequency of this wave, let’s look at this time interval here. We see that this wave goes through two cycles in this time interval of one second. So, this wave would have a frequency of two cycles per second. This ratio of units, number of cycles per second, can be written as a different unit. One cycle per second can be written as a hertz, which is written as Hz. So, this wave will have a frequency of two hertz.

Now that we have recalled the meaning of the terms wavelength and frequency, we can now focus on the wave speed. The wave speed is the distance traveled by a given point on the wave in a given time interval. We often use lowercase 𝑣 to represent speed. But how would we calculate the wave speed from its frequency and wavelength? Let’s first recall that speed in general can be represented by the equation speed equals distance over time. We know that the wavelength of a wave is the distance covered by one complete cycle and is measured in meters.

We can also recall that the period of a wave is the amount of time that it takes the wave to go through one cycle, which is usually measured in seconds. So we can replace distance with wavelength in this equation, like so, and replace time with period. So, we now have this equation for wave speed. Now we can recall that the period of a wave is equal to one over the frequency of a wave, where frequency is measured in hertz. We can substitute this expression for the period into our equation, like so. Simplifying this, we find that the speed of a wave is equal to the frequency multiplied by the wavelength. So in symbols, we can write this as 𝑣 equals 𝑓𝜆. And this is our equation for wave speed.

If we have a quick look at our units, we see that we are multiplying the units of hertz and the units of meters together. But we can recall that the units of hertz are the same as one divided by seconds. So actually we are multiplying the units of one divided by seconds with meters. This will give us units of meters per second, which is the standard unit of speed. Let’s take a moment now to get some practice with these ideas through an example exercise.

A wave has a wavelength of 0.5 meters and a frequency of 30 hertz. How far does the wave travel in 10 seconds?

We will begin by calculating the speed of this wave using its wavelength and frequency. We can recall that the speed of a wave is given by the formula 𝑣 equals 𝑓𝜆. Here, 𝑣 is the wave speed, 𝑓 is the frequency of the wave, and 𝜆 is the wavelength of the wave. We are told in the question that the wave has a wavelength of 0.5 meters. We are also told the wave has a frequency of 30 hertz. Substituting these values into the wave speed formula, we find that 𝑣 is equal to 30 hertz multiplied by 0.5 meters. This equals 15 meters per second. So, the speed of this wave is 15 meters per second.

We can now recall the formula speed equals distance over time. We want to determine how far the wave has traveled in 10 seconds. So, let’s rearrange this equation to make distance the subject. We can do this by multiplying both sides of the equation by time. This leaves us with distance equals speed multiplied by time. We calculated the speed of the wave as 15 meters per second. And we want to know how far the wave travels in 10 seconds. Substituting these values into this equation, we find that distance is equal to 15 meters per second multiplied by 10 seconds. This gives us a value of 150 meters. So, we have found that the wave travels a distance of 150 meters after 10 seconds.

Let’s now consider a second example.

An oscillating object produces mechanical waves with a frequency of 750 hertz that travel at 500 meters per second. The object then starts to produce mechanical waves with a frequency of 800 hertz that also travel at 500 meters per second. Do all the waves produced by the object have the same wavelength?

We begin by recalling the wave speed formula: 𝑣 equals 𝑓𝜆. Here, 𝑣 is the speed of the wave, 𝑓 is the frequency of the wave, and 𝜆 is the wavelength of the wave. We want to calculate the wavelength of the mechanical waves produced by this oscillating object. So, we should rearrange this equation to make the wavelength, 𝜆, the subject. We can do this by dividing both sides by the frequency, 𝑓. This will leave us with the equation 𝜆 equals 𝑣 over 𝑓.

Now we can go ahead and calculate the wavelengths of the mechanical waves. At the start, the mechanical waves have a frequency of 750 hertz and travel at a speed of 500 meters per second. Substituting these values into this equation, we find that the wavelength of these waves is equal to 500 meters per second over 750 hertz. This gives us a value of 0.666 meters. Afterwards, the mechanical waves have a frequency of 800 hertz but still travel at the same speed of 500 meters per second. Substituting these values into the equation, we find that the wavelength of these waves is equal to 500 meters per second over 800 hertz. This gives us a value of 0.625 meters.

We can now compare these two wavelengths. We can see that these two values are different. So this means that the waves produced by the object do not have the same wavelength.

Now let’s review what we have learned in this video on wave speed. We learned that the wavelength of a wave is the distance covered by one complete cycle and is represented by the Greek letter 𝜆. We saw that the frequency of a wave is the number of wave cycles that pass through a certain point per second. And lastly, we learned that the speed of a wave is equal to its frequency multiplied by its wavelength. In symbols, this is written as 𝑣 equals 𝑓𝜆.

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