Earthquakes are one of the most devastating natural disasters - they can cause huge amounts of damage. If an earthquake strikes under an ocean, it can cause a tsunami, which is even more destructive. It is important to be able to predict when earthquakes are going to occur and to measure their strength in order to make preparations to prevent their harmful effects. Seismometers are devices used to analyse the seismic waves produced by earthquakes. Waves are characterized by several different quantities and in this article, we will explore these quantities as well as some methods for measuring them.
Measuring waves science
Physicists use the concept of waves as a model for explaining various other phenomena, such as light and sound.
A wave is often represented by a wavy line, like the one shown below. This line is actually a sine wave (or a cosine wave) - the equation for the curve is \( y=\sin{x} \).
Fig. 1 - A wave is often represented as a sine or cosine curve.
If you compare this curve to a water wave moving through the sea, you should be able to see where its characteristic shape comes from. The sine graph follows the displacement from the rest position of the water wave at each point as it moves - the vertical \( y \) axis shows the height surface of the water and the horizontal \( x \) axis shows the distance travelled by the wave. The highest points are called the crests of the wave and the lowest points are called the troughs of the wave.
The wave crest (or peak) is the highest point of oscillation above the rest position, while the trough is the lowest point of oscillation below the equilibrium position.
Wave measurement definition
Waves are characterized by different physical quantities that can be measured.
Wavelength
One full wavelength \(\lambda\) is the length of one complete cycle of the wave, most easily measured either from crest-to-crest or trough-to-trough.
Fig. 2 - The wavelength of a wave is the distance from one crest to the next.
Wavelength is a distance and hence it is measured in meters, \( \mathrm m \). A common mistake is measuring the wavelength from the crest to the trough, giving a value equal to half of the correct wavelength.
Amplitude
The amplitude, \( A \), of a wave is the maximum displacement of a point on a wave between the crest (or the trough) and its equilibrium position.
Fig, 3 - The amplitude of a wave is measured from the position of zero displacement to a crest.
A wave's amplitude is measured parallel to the \( y \) axis from the \( x=0 \) to a crest (or a trough), as shown above. Amplitude is a distance, like a wavelength, so it is also measured in meters, \( m\). Note that amplitude is not measured from trough to crest, which is another common mistake. This would result in an obtained amplitude that is double the actual amplitude.
The wave amplitude can help inform us about how much energy is in a wave. For example, big (tall) water waves carry more energy than little waves, as you might have experienced yourself. Another example is that an electromagnetic (light) wave with a high amplitude will be brighter than a dimmer, low-amplitude wave. Similarly, a sound wave with a high amplitude will be louder than a wave with a lower amplitude.
The total vertical displacement between the crest and the trough of a wave is actually a wave property known as wave height \(h\), which we measure in \(\mathrm{m}\). Wave height is a particularly useful concept in coastal science. It is equal to twice the wave amplitude \(A\),
\[h=2A.\]
Frequency
To understand the meaning of the frequency of a wave, we should consider the following scenario. Imagine a buoy floating in the sea, with waves steadily passing by it, causing it to bob up and down, but stay in the same place. The graph below shows how the height of the buoy changes as time passes.
Fig. 4 - A displacement against time for a buoy bobbing up and down due to water waves.
Be careful! This looks very similar to the graph for a point on a water wave, but here the \( x \) axis represents time instead of distance. In this case, the distance between two crests of the wave is not the wavelength, but the time period.
The time period can be thought of as the equivalent of wavelength when working in time instead of distance.
The time period of a wave is the time taken for one complete wave cycle to pass a point.
The time period is measured in seconds, \( s \). This can be seen from how the time period is measured along the \( x \) axis and the units of the \( x \) axis in seconds.
Fig. 5 - The time period of a wave is the time taken for the wave to return to the same point of the cycle.
The frequency of a wave is related to its time period by the equation,
$$f=\frac 1T.$$
The units for frequency are Hertz (\( \mathrm{Hz} \)), which is equivalent to \(\frac 1 {\mathrm{s}}\) or \( \mathrm{s^{-1}} \).
The frequency of a wave is the number of complete wave cycles passing a given point per second.
Fig. 6 - A lower frequency wave will have a longer time period than a higher frequency wave.Waves with shorter periods will have higher frequencies, as more waves can pass through a point every second. On the other hand, waves with a longer period would have lower frequencies, because fewer waves can pass through a point every second.
Question
A boat sails across a lake at a constant speed, producing small waves as it moves. A nearby swimmer looks at her watch and notices that \( 2 \) waves pass her every second. What is the time period for the waves coming from the boat?
Solution
As the boat is moving at a constant velocity, all of the incoming waves should have the same wavelength and frequency relative to the swimmer, so we can treat them as cycles of the same wave. The statement that \( 2 \) waves pass the swimmer every second is the same as saying that the frequency of the waves is \( 2\,\mathrm{Hz} \). To find the time period, we must use the formula
$$f=\frac 1T,$$
which can be rearranged to
$$T=\frac 1f.$$
Substituting in a frequency of \( 2\,\mathrm{Hz} \) gives a time period of
$$T=\frac {1}{2\,\mathrm{Hz}}=0.5\,\mathrm s.$$
You would be correct if you had assumed that the amplitude of the incoming waves increased or diminished depending on the distance between the boat and the swimmer at any given point in time. However, the amplitude of the wave is not relevant to the time period or frequency of the waves.
Relationship between wavelength and frequency
We have now explored many of the quantities that describe the measurement of waves, but how do they fit together? In particular, what is the link between wavelength and frequency?
They are related by the wave speed equation, which can also be measured. The mathematical relationship is given as
$$v=f\lambda,$$
where \( v \) is the wave speed in \( \mathrm m/\mathrm{s^2} \). This is called the wave equation and tells us how these three quantities depend on each other:
- If the wavelength of a wave remains constant and its speed increases, then its frequency must also increase.
- If the frequency of a wave remains constant and its speed increases, then its wavelength must also increase.
- For waves of a given speed, their frequency will be inversely proportional to their wavelength.
If two variables have an inversely proportional relationship, then an increase in one of the variables causes a decrease in the other.
The speed of a wave is constant if the medium the wave travels through is also constant. For instance, the speed of sound in air with a temperature of \(20^{\circ}\mathrm{C}\) at sea level is approximately \(343\,\mathrm{\frac m s}\). This means the only way to change the speed of a wave, is to change the medium it travels through!
Consider a sound wave travelling from the air into the water. The sound waves that aren't reflected at the water's surface will become heavily distorted. The speed of sound in water is approximately \(1480\,\mathrm{\frac m s}\), a factor of about \(4.3\) greater than in air. This is partially due to the increased density of the water medium compared to air. The sound waves can travel faster in a denser medium as it is easier for particles to bump into each other when oscillating/vibrating.
An example that illustrates this relationship is the electromagnetic spectrum. All electromagnetic waves have the same speed - the speed of light, \( c \) - so the higher frequency waves have a shorter wavelength and vice versa. For instance, gamma rays have the highest frequency and the shortest wavelength whereas radio waves have the lowest frequency and the longest wavelength.
Units of wave measurement
The units and the symbols for the different quantities used to characterize waves are summarised in the table below.
Quantity | Symbol | Units |
Wavelength | \( \lambda \) | \( \mathrm m \) |
Amplitude | \( A \) | \( \mathrm m \) |
Time period | \( T \) | \( \mathrm s \) |
Frequency | \( f \) | \( \mathrm{Hz} \) or \( \mathrm{s^{-1}} \) |
Wave speed | \( v \) | \( \mathrm m/\mathrm s \) |
Measurement of waves methods
There are several experiments that can be performed with different types of waves to measure the quantities that we have used to describe them.
Water waves
In order to measure the speed of water waves, you can use a ripple tank. The experimental setup is shown in the diagram below. Illuminate the ripple tank with a lightbulb from above and place a sheet of paper below. You will need to hang a wooden beam so that it just touches the water on one side of the tank. Place a vibrational motor on top of the beam and connect it to a power supply.
Turn on the power supply and the wooden beam should vibrate up and down, producing horizontal waves in the water, which you should be able to see as shadows on the paper below the tank. To find the wave speed, measure the time it takes one of the waves to travel a known distance (such as the length of the tank) and use the equation
$$v=\frac st,$$
where \( s \) is the distance travelled in \( \mathrm m \) and \( t \) is the time taken in \( \mathrm s \). To find the wavelength of the water waves, place a ruler on the paper next to the wave shadows and take a picture to accurately measure the distance between the wave crests. You can then use the rearranged version of the wave equation,
$$f=\frac v\lambda,$$
to find the frequency of the waves.
Sound waves
We can also measure the physical quantities of sound waves. An example of an experimental setup is shown below. Connect a signal generator to an amplifier and a loudspeaker and place two microphones in a straight line at known distances from the loudspeaker.
Set the signal generator to a specific frequency and record the times at which the peak intensities are recorded by the microphones. Then use the distance between the two microphones along with the time between the peaks to find the speed of the sound waves. Then use the frequency of the signal generator along with the wave speed to find the wavelength.
Wave measurement instruments
In this section, we will discuss some other instruments that are used to measure different types of waves.
Ocean waves
We already discussed buoys in order to understand what the time period of a wave is. Buoys sit on top of the water and move up and down as waves pass. They are sometimes fitted with a device called an accelerometer which is used to record the buoy's vertical position over a time period, which will give information about the water waves causing the buoy to move.
Fig. 9 - The motion of a buoy follows the height of the passing water waves.
Seismic waves
Seismic waves are waves produced by earthquakes and they move through the Earth. They are detected by the use of seismometers, which convert ground vibrations due to the seismic waves into electrical signals and display these signals as seismographs, which show the electrical signal as time passes. Using seismometers, seismologists are able to track the movement of Earthquakes and assess their magnitudes, and therefore determine the dangers they pose.
Wave Measurement - Key takeaways
- Physicists use the concept of waves as a model for explaining various other phenomena, such as light and sound.
- A wave is often represented by a sine or cosine curve.
- The wavelength, \( \lambda \), of a wave is the distance from one crest to the next (or from one trough to the next) and is measured in \( \mathrm m \).
- The amplitude, \( A \), of a wave is the maximum displacement of a point on a wave from its equilibrium position and is measured in \( \mathrm m \).
- The time period of a wave is the time taken for one complete wave cycle to pass a point and is measured in \( \mathrm s \).
- The frequency of a wave is the number of complete wave cycles passing a given point per second and is measured in \( \mathrm{Hz} \).
- The frequency of a wave is related to its time period by the equation \( f=\frac 1T \).
- The wave equation states that \( v=f\lambda \).
References
- Fig. 1 - Displacement distance sine wave, StudySmarter Originals
- Fig. 2 - Wavelength, StudySmarter Originals
- Fig. 3 - Amplitude, StudySmarter Originals
- Fig. 4 - Displacement time sine wave, StudySmarter Originals
- Fig. 5 - Time period, StudySmarter Originals
- Fig. 6 - Frequency and time period relationship, StudySmarter Originals
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