One of the first acoustics topics most physics classes cover is resonance. Often, physics teachers will show the video clip below of the collapse of the Tacoma Narrows Bridge (also known affectionately as “Gallopin’ Gertie”) to demonstrate the potential effects of resonance.
While the rippling motion of the bridge seems unreal, the physical principle of resonance is quite real. Gaining an understanding and intuition for resonance is foundational for many acoustics applications, especially in understanding musical acoustics. So let’s start with an easy example.
Jump Rope Physics
Let’s say that you and your friend hold opposite ends of a jump rope. If your friend holds their end still and you move your end up and down, you will be able to watch the waves travel along the rope. If you give your end of the rope a sharp up-down you will create what is called a pulse. Notice that when the pulse reaches the end of the rope where your friend is, it “bounces back”, flipping so that parts of the wave that were up are now down and vice versa. In other words, when the wave on the rope reflects, it flips upside down. We call this a 180-degree phase shift.
If you move the rope up and down at just the right frequency (number of up-downs per second), something special happens. The wave on the jump rope will appear to stop in place. The crests and troughs of the wave will stop moving along the rope and instead bob up and down in place. These waves are called standing waves because they do not appear to move. This video shows the creation of standing waves at a few different frequencies.
The name standing wave is a little bit misleading. A standing wave is really a superposition (combination) of two waves moving opposite directions along the length of the rope. However, because the two waves are in sync, the resulting wave appears to stand in place. Resonance, then, is the result of the creation of standing waves. Re-watch the clip of Gallopin’ Gertie and notice how the waves on the bridge appear to stay in place bobbing up and down.
Nodes and Antinodes
Before we talk about any of the math used to find resonant frequencies, we need to talk about two more words: nodes and antinodes. A node is a point on a standing wave between two humps where the rope doesn’t move. In motion, nodes are the “hinge points” of the wave halfway between the highest and lowest points of the wave. Nodes are called fixed points, because they don’t move. In the jump rope example with your friend, your friend’s end of the rope (which is a fixed point) is a node.
Antinodes are just what they sound like—the opposite of nodes. If nodes are the points that move the least, antinodes are the points that move the most. You can find an antinode at the tip of any hump of the wave. Antinodes represent the position of maximum displacement from rest along the rope. In the jump rope example, your end of the rope is an antinode. We call your end of the rope a free or open end.
Notice that you will never have two nodes or two antinodes next to each other—they always alternate, one then the other, along the length of the standing wave. With nodes and antinodes down, we are ready to start calculating resonant frequencies. Continue on here.