Video Transcript
In this video, we’re going to learn about the production of standing waves. We’ll learn what standing waves are. And we’ll see what the formation of these waves means for systems’ dynamics. To get started, imagine that you have created a musical instrument which has never been seen before. This stringed instrument uses one string whose frequency is tuned by changing the material properties of the instrument it’s attached to. You want to figure out how to set up this instrument so that the string, when it’s plucked, creates a steady note. To learn how best to design this instrument, it will be helpful to know about the production of standing waves.
A standing wave is a special type of wave that it’s possible to produce using easily available materials. Say you take a lightweight elastic cord and you fix one end of the cord to the wall. You then pull the cord taut and start moving the free end you’re holding in your hand up and down quickly. By doing this, you create a wave on the string. And the wave moves towards the wall. As you keep moving your hand to generate these peaks and troughs, eventually the wave reaches the wall. And then something interesting happens.
When the wave runs into the wall, being fixed and placed, the wall has nothing to do but to reflect the wave back where it came from. This reflected wave is an inverted version of the wave that ran into the wall. It has the same amplitude and frequency as the incoming wave, but it’s flipped upside down. At this point, things get a bit confusing for the cord that’s carrying these waves. It has one wave moving to the right and another wave moving along the same cord to the left. When these two oppositely moving waves overlap like they now are along the entire length of the cord, a combined or resultant wave is formed.
The principle of superposition tells us that if we wanna find the height of the resulting wave formed by combining the incoming and outgoing wave, then we simply add together the height value of the incoming wave to the height value of the outgoing wave. Their sum will equal the value of the resultant wave at that point in space. Looking at our incoming and outgoing waves at this snapshot in time, we see that no matter where in space we pick our point, the sum of the incoming and outgoing wave will be zero. That is, these two waves perfectly cancel one another out. At this instant in time then, our two waves perfectly destructively interfere so that the combined wave is nulled out.
But remember that these two waves are in motion. The incoming wave is moving towards the wall and the outgoing wave is moving the other way. If we let time advance a bit so that our incoming wave moves a bit to the right and our outgoing or reflected wave moves a bit to the left, now we see the two waves overlap one another perfectly. At this instant in time, if we compute the resultant wave from adding these two individual waves together, we find that the resultant wave is not zero but actually has an amplitude twice that of the incoming and outgoing waves, respectively.
This is when the waves constructively interfere. And of course, this too is just at one particular snapshot in time. The incoming wave keeps moving to the right and the reflected wave keeps moving to the left as we move our hand up and down on the free end of the rope. What ends up happening is our system oscillates between constructive and destructive interference. So that if we tracked the resulting wave over time, the places where it has zero amplitude stay in place even as this resultant wave oscillates up and down.
Unlike our incoming and outgoing waves then, this resultant wave is standing in place. It’s called a standing wave. The hallmark of a standing wave is these locations that don’t move. These locations where there is no net displacement are called nodes. When a standing wave is formed, that indicates that a system, in this case the cord and the wall, is in resonance, meaning that the frequency of the incoming wave is the same as the frequency of the outgoing wave. We’ve all had experience with standing waves when we, for example, pluck one of the strings on a guitar. Standings waves can be fun to watch because they truly do stand in place, not moving to the left or to the right.
That said, there’s still motion in a standing wave. Notice that on this standing wave directly in between the nodes where the wave doesn’t move are the locations where the wave moves the most. Each one of these points directly between the nodes will move from a maximum height to a minimum height and then back over and over. Since these points do the opposite of what the nodes do, that is, they move as much as any point along the wave, we call them antinodes. In a standing wave, the nodes don’t move at all. And the antinodes move the most.
By way of example, say that we have a standing wave that’s formed on a string fastened at both ends. The ends of the string are a distance 𝐿 apart. We count that our string has one, two, three, four, five antinodes to it. And we’ll say further that this particular string has a linear mass density of 𝜇, that is, the string’s mass per unit length. In addition, we’ll assume that the string is under a tension force 𝐹 sub 𝑇, measured in newtons.
Given all this information, we want to solve for the speed of the standing wave 𝑣, the wavelength of the standing wave, its frequency, and its period. It may seem strange to think that a standing wave has a speed, but indeed it does. It’s equal to the speed of the moving waves that create the standing wave. On that basis, we can recall that the speed of a wave on a string is equal to the square root of the tension in the string divided by the string’s linear mass density, that’s the speed of the moving waves and therefore the speed of the standing wave.
Since the standing wave, even though it doesn’t move left or right, does have antinodes which move up and down. When we go to solve for the wavelength of the standing wave, we can look at the number of antinodes the wave has, stretched out over the given length 𝐿. We count five antinodes on this standing wave which means that there are two and one-half full wavelengths of the wave stretched out over the length 𝐿. This means that if we divide 𝐿 up into five equal pieces, one wavelength is two times one of those pieces.
To solve for the frequency 𝑓, we can recall that wave speed 𝑣 is equal to the product of wavelength times frequency. This means that frequency is equal to speed over wavelength or the square root of the tension force over 𝜇 times five over two times 𝐿. And finally, to solve for the wave period 𝑇, we recall that it’s equal to the inverse of the frequency or two 𝐿 over five times the square root of 𝜇 over 𝐹 sub 𝑇, the tension in the string. So just like other waves, standing waves have a speed, a wavelength, a frequency, and a period. We’ll see more standing waves when we study normal vibrational modes. But for now, let’s summarize what we’ve learned so far about standing waves.
We’ve seen that when two waves are travelling in opposite directions, both having the same frequency, when they combine they form a standing wave. We’ve also seen that standing waves have nodes which do not move and antinodes which move up and down maximally to the full amplitude of the standing wave. And finally, we saw that standing waves are created when a system resonates. That is when the natural frequency of the system is matched by a driving frequency. Resonance is often created by a reflected wave whether a wall reflecting a wave on a string or an end cap reflecting a column of air. When the reflected wave frequency matches the incoming wave frequency, a standing wave is created.
