I was having trouble finding an example of a standing wave that I wanted to work with. I thought of the vibration that my phone produces when my alarm goes off every morning. I thought of the vibration that my electric razor produces every time I shave. I tried these out and wasn't too excited. I also couldn't properly replicate the visuals of the standing wave with these objects.
Yesterday I was playing tennis, and that's when it hit me: tennis racquets! Tennis racquets are loaded with vibration every time you hit a ball with them. Every string in the racquet is a taut, vibrating element, and the frame (often made of graphite) acts as a resonator. In a sense, it's almost like a multi-string instrument. A symphony of tennis balls getting whacked across the court sounds quite nice to me.
I tried to replicate the visual of the standing wave, by tying string to the racquet and hitting it with my hand or with a ball, but the vibrations were too small to notice. I tried with a thicker yarn, but also no luck. It's hard to replicate for many reasons: the vibrations are damped when I hold the racquet; the strings are very tight so the vibrations are there, just hard for the naked eye to see; the full-force impact of hitting a ball when playing tennis was difficult to replicate in my apartment.
I dived deeper into the physics of tennis racquets, how they function, when vibrations occur and why. What I learned is that the whole racquet makes standing waves when it hits a ball. Once a ball hits a racquet, part of the racquet moves back, and the other part of the racquet moves forward. This happens over and over, causing it to vibrate.
Something interesting happens when you hit the ball with the racquet's "sweet spot". I learned over many tennis lessons that the racquet's "sweet spot" - which is near the center of the racquet strings - is what I had to try to hit the ball with every time. It also produces a very satisfying "POP" sound when you successfully hit with your racquet's "sweet spot". Aside from auditory feedback, you also get physical feedback: it feels like the racquet doesn't vibrate, and you almost feel the full power of your swing. Nothing feels or sounds more satisfying.
What really blew my mind is that there's actually three "sweet spots" in the racquet: the vibration node, the center of percussion (COP), and the power point. The "sweet spot" where you feel like there's no vibration is called the vibration node. When this occurs, the vibration transmitted to your hand is so minimal that it almost feels like you didn't hit the ball. If the opposite happens (hitting the ball NOT with the "sweet spot") you feel a much stronger vibration.
There isn't only one node because there's actually three motions taking place when a ball hits a racquet: vibration, translation, and rotation. The motion of vibration we already covered: it is zero at the vibration node.
The second "sweet spot", the COP, is the point where the total force and torque translated to the hand becomes zero. Thus, all force is used to push the ball rather than to move the arm or racket. This is actually the "sweet spot" that tennis players are trained to aim for, and the one that produces the greatest, smoothest sensation.
The third "sweet spot", the power point, is the node where the ball bounces the most. This point is closer to the 'throat' of the racquet, which leads to greater stiffness from the racquet's frame, reducing the energy spent on the racquet's deformation. If the ball hits closer to the top of the frame, there is greater deformation of the racquet, causing greater absorption of energy and thus absorption of the ball's speed. This is called the "dead spot".
To change the subject a little, this led me down the rabbit hole of "hearing" a tennis racquet vibrate. The different sounds a racquet can produce are fascinating, from the ball hitting the string at different points and angles, to the ball hitting the frame. A racquet's vibration frequency depends on its stiffness: the stiffer the racquet, the higher its frequency.
I found some sound clips of the sound a tennis racquet makes when you hold the tip of the grip near your ear and tap the grip. These range in frequency from 120Hz to 220Hz and you can really tell the difference in stiffness as it relates to frequency.
Then I started playing around with these sounds and trying out different combinations:
All in all it was very interesting to learn about this and hear these combinations in action!