Resources
Acoustics and vibrations animations – Check out these animations of different types of waves (longitudinal, transverse, etc.).
Circular harmonics – Animations of circular harmonics, which are the waves that are created on a circular membrane, such as a drumhead.
PhET Wave on a String Simulation
Change different properties of a string and see how it affects the properties (amplitude and wave speed) of a wave.
Video Transcript
Hello there! Welcome to lecture 19: vibrations and waves!
So many aspects of our lives are dictated by waves: the light waves we use to transmit wifi and cellular signals, the sound waves we use to communicate with others, water waves we travel over in the ocean, and even the motion of a pendulum that keeps time in a grandfather clock.
In this lecture, we’ll talk about different waves and wave properties, and start a fascinating conversation about waves that will unlock the mysteries of sound and light in lectures to come.
Each of the following concepts will be discussed in this video: vibrations and waves, types of waves, interference, standing waves, and the Doppler effect.
Vibrations and waves
A wave is a periodic oscillation that transfers energy from one place to another. Waves travel through space or some other type of medium, whereas vibrations remain fixed in one location. There are many different types of waves, some of which we’ll discuss in more detail in future lectures: light waves, sound waves, water waves, gravitational waves, seismic waves, and more.
The term periodic oscillation refers to the types of motion that waves make. This is a repeating pattern that gives rise to many wave properties. Let’s take a look at a graph of one type of representative wave. This graph shows the amount of energy or intensity, or some other related quantity, of the wave as it travels. Note that as it moves through space, it oscillates up and down, up and down, in a rhythmic manner.
You can think of this wave as looking like a squiggle that moves up and down in space and time. Each of the high points on the wave is known as a crest. The low points of the wave are troughs. The center position is known as the equilibrium value. This would tell us the value of the wave if it were at rest.
Wave amplitude describes the maximum displacement from the equilibrium position. In other words: how far away from the equilibrium position are the crests and troughs? Depending on the wave, the amplitude could describe energy, intensity, or even just height. So we will not necessarily define one particular unit for amplitude, as it could be describing one of many physical parameters.
The wavelength of a wave tells us the distance between two identical points on a wave. We usually measure wavelength from crest to crest, or from trough to trough. Wavelength has a symbol of the lowercase Greek letter lambda. It has units of distance: meters or centimeters.
A ripple tank was used in many of the demo videos shown in this lecture. A ripple tank contains a shallow tank of water, and has one or more moving piece that vibrates up and down, causing ripples in the water. Those ripples are waves. We can adjust the parameters of these waves to study different wave properties. The light and dark areas we see projected on the screen correspond to the troughs and crests of the wave.
When using the ripple tank, I can change the wavelength, which changes the distance between crests. When I increase the wavelength, the crests get spaced farther apart. When I decrease the wavelength, the crests get spaced closer together.
The period of a wave tells us how long it takes for one complete oscillation to occur. An oscillation is a complete motion from crest to crest. In a pendulum, an oscillation relates to the amount of time it takes for the pendulum to swing out and back again. Period has a symbol of the capital letter T, and has units of seconds.
We can time how long it takes for a wave in the ripple tank to complete one oscillation through a single point: from crest to crest or from trough to trough. This is equal to the wave period.
Frequency tells us how many oscillations occur per second in a wave. Frequency has a symbol of lowercase f and has units of Hertz. One hertz is equal to one divided by a second. Frequency and period are related by the equation f equals one divided by T. In other words: frequency is equal to one divided by the period. The higher the frequency of oscillation, the faster it makes it squiggles, the smaller the amount of time each period will be.
Wave speed tells us the speed at which a wave moves. We know that speed is equal to distance divided by time. In this case, the distance described by the wave is the wavelength, and the time is described by the period. This means that the wave speed is equal to wavelength divided by the period. Or, we can rewrite the equation to say that wave speed is equal to wavelength times frequency. If wave speed is constant, then this equation tells us the relationship between wavelength and frequency.
Depending on the wave, there are many different speeds at which waves can travel. We’ll talk about the speed of sound in lecture 20, and the speed of light in lecture 26.
Types of waves
Waves tend to be characterized by the motion that the wave itself makes compared to how the wave travels through space.
In these videos, I use a slinky to create longitudinal and transverse waves. In a longitudinal wave, the vibrations of the wave travel in the same direction as the wave itself. When I send a longitudinal pulse through the slinky, the pulse travels from left to right, as does any single point on the slinky. I put a piece of pink tape on one point of the slinky to show this more clearly.
In a transverse wave, the vibrations of the wave travel perpendicular to the direction of the wave. When I get the slinky moving in a transverse wave pattern, any single point on the slinky moves up and down, but the wave itself moves left to right.
Light waves are transverse, and we’ll discuss convincing evidence for this in lecture 29. Sound waves are longitudinal. We’ll talk about sound waves in more detail in lecture 20.
There are also some waves that are neither longitudinal nor transverse. Water waves in the ocean, for example, are usually a mixture of both longitudinal and transverse. There is both up and down and side to side motion in ocean waves.
It might be strange to think of traffic as a wave, but I’d like you to think about it next time you’re at a traffic light. At a red light, cars stop and line up. When the light turns green, the motion of the cars will be forward, but the direction that the information in the wave moves is backward. The first car in line moves forward, and then the car BEHIND that one moves, and the car BEHIND that one, and so on.
This animation shows a simulation of cars (each one is represented as a square) stopped at a red light. When the light turns green, each car subsequently starts moving. When a car is red, that means it has started moving. The position of the red square, representing the information transmitted by the wave, moves backward, even though the cars move forward.
Another type of wave is known as a torsional wave. This is a type of twisting back and forth in a periodic manner. A torsional wave is what caused the Tacoma narrows bridge to collapse in 1940. Winds blowing across the bridge created and amplified the torsional wave in the bridge, which eventually overcame the strength of the structure, causing its collapse.
Interference
Unlike matter, where two objects cannot occupy the same space at the same time, waves, being energy, CAN exist in the same place at the same time. When waves overlap with each other they interfere. That is, the value of one wave at that point adds to the value of the second wave. The total amount of energy present is equal to the sum of those two values.
In a ripple tank, when two ripples are generated close to each other, the two waves will interfere. We see the effect in the pattern the waves generate, called an interference pattern. The shape of the pattern depends on the wavelength and how close together the two individual waves are formed.
If two waves interfere where the crest overlaps with crest, and trough overlaps with trough, then the situation is called constructive interference. The result is a wave with greater overall amplitude than either one individually. If the two waves have identical amplitudes, the constructive interference will yield a wave with twice the amplitude.
In a ripple tank, the light areas correspond to the wave troughs. When two troughs come together, we get constructive interference. The dark areas correspond to the wave crests. When two crests come together, we get constructive interference.
If two waves interfere where crest overlaps with trough, and trough overlaps with crest, then the situation is called destructive interference. If the two waves have the same amplitude, then if we sum up the contributions of each wave, they cancel out. The result is a wave with zero amplitude. This condition of destruction is temporary… as the waves continue to propagate, as they stop overlapping with each other, the interference will end.
In a ripple tank, the light areas correspond to the wave troughs. When a trough meets a crest, we will get destructive interference.
It is also possible to have many other types of interference that are neither purely constructive nor purely destructive.
Interference happens all the time, when pebbles are dropped in a pond, when light waves interact, and when sound waves come together. We’ll talk about things that happen when sound waves interfere in more detail in lecture 20. The interference of light waves was used to prove that the speed of light is constant, which was one of the foundational experiments of the 19th century.
Standing waves
Standing waves are waves that oscillate in time but that do not move at all in space. They occur due to interference between two waves that have equal amplitude and frequency that travel in opposite directions.
I can generate standing waves in a coiled rope by shaking one end up and down. This creates one wave, which reflects off of the back end of the coiled rope and travels back toward my arm. By shaking at just the right frequency, the two waves: the forward wave and the backward wave: will interfere just right to create a standing wave.
Locations on a standing wave that are fixed are known as nodes. Locations where the wave oscillates between minimum and maximum values are called antinodes.
Many different standing waves can be generated on this coiled rope, each one is known as a harmonic. The first harmonic is the wave with one antinode and two nodes. The first harmonic is also sometimes referred to as the fundamental mode. I calculated the time between oscillations, the period of the wave, to be 0.63 seconds. Using the equation that relates frequency and period, f equals one over T, we see that the frequency of the first harmonic is 1.58 hertz.
We can also measure the wavelength of the standing waves. The two ends of the coiled rope are held 1.4 meters apart. Because the first harmonic is only one half of a wavelength, that means the wavelength of the first harmonic is 2.8 meters. Using the equation v equals lambda divided by t, we can calculate the wave speed as 4.4 meters per second.
Next, I generated the second harmonic. This harmonic has two antinodes and three nodes. The period of this harmonic was 0.33 seconds, giving a frequency of 3 hertz. The wavelength of the second harmonic is 1.4 meters. The wave speed is therefore 4.2 meters per second.
The third harmonic has three antinodes and four nodes. The period is 0.2 seconds and the corresponding frequency is 5 hertz. The wavelength of the third harmonic is 0.93 meters. The wave speed is therefore 4.7 meters per second.
Note that the wave speed is relatively constant for all of the standing waves. The speed of the wave has to do with the properties of the coiled rope that I used to generate the standing waves.
What we can determine from this is that the harmonic has to do with the number of antinodes. Also, as the harmonic number increases, so does the frequency. In fact, the frequency of a harmonic is related to the number of the harmonic and the frequency of the first harmonic. In equation form: the frequency of the nth harmonic is equal to n, the harmonic number, times the frequency of the first harmonic.
The eighth harmonic would be difficult for me to generate, but I can determine what frequency it would require by using this equation. The first harmonic has a frequency of 1.58 hertz. Multiplying by eight, we can see that the frequency of the eighth harmonic would be 12.7 hertz.
The Doppler effect
Have you ever noticed that when an ambulance or police car drives by, the pitch of the sound of the siren seems to change? This is due to the Doppler effect.
When a wave is traveling at a certain speed relative to an observer, the wavefronts become bunched up in the direction of motion. This shortens the wavelength, and given a constant wave speed, increases the frequency. Frequency and pitch are related in sound waves, which is why a siren sounds higher pitch when an ambulance is approaching your location.
As the wave travels away from the observer, the wavefronts are spread farther apart. This increases the wavelength and decreases the frequency. This means that a siren sounds lower pitch when it is moving away from your location.
Depending on the wave, the speed of motion can have profound applications due to the Doppler effect causing the wavefronts to bunch together. In a boat, water is the physical medium in which waves propagate. If we drop a pebble into a pond, the ripple pattern is even and uniform.
If, however, we drop a pebble in motion, the Doppler effect will cause the wavefronts to bunch together. At a certain point, when the pebble moves at the same speed as the speed of the water waves, a wake will form. This happens with boats, which generate bow and stern waves when they travel through water.
In sound waves, when the speed of an object travels faster than the speed of sound, this disturbance creates a sonic boom. This is a shockwave created due to the overlapping of wavefronts at a speed faster than the air can meaningfully respond to the changes in pressure caused by the wave. This compresses the air and creates the shock wave we call a sonic boom. The sonic boom travels in a cone shape whose angle corresponds to the exact speed of the supersonic object.
Thanks for taking the time to learn about vibrations and waves! Until next time, stay well.