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.
