If you have taken a basic introductory physics course, you may have learned about vibrational modes. If you haven’t - let me plagerize the wikipedia’s entry on normal modes and let it describe what they are:
Normal modes in an oscillating system are special solutions where all the parts of the system are oscillating with the same frequency (called normal frequencies or allowed frequencies). The concept of normal modes is of vital importance in wave theory, optics and quantum mechanics.
On a string fixed at two points (like a guitar string, for instance) - the normal modes are shaped like sine waves. That’s why when you pluck a guitar string while sitting under flourescent lights - you can see a kind of bulging of the string at certain points - but relatively no motion at other points.
What does all this have to do with a circular drumhead? Well… Circular drumheads don’t have such nice sinusoidal modes. They are governed by much crazier math. The only reason I’m talking about them here is because I just had to make some animations of the different vibrational modes for one of my homework assignments, and because the animated gifs are really big… I’m going to make you click the “more” button to see them and think (”why did I care?”)
Now, obviously a drumhead doesn’t have as much amplitude when it oscillates… but if you ever wondered what kind of vibrational pattern is going on when you strike a drumhead - it’s probably a sum of a handful of those kind of vibrational modes.
Now you may resume the rest of your daily activities and pretend you never read this post.
June 15th, 2006 at 9:34 pm
One thing I should point out is that the natural frequencies of an oscillating drumhead are not in integer multiples of each other. This makes it a less “musical” instrument than a violin or cello… the sound doesn’t reinforce itself in harmonic patterns. It just makes big thuds.
Another thing you can do to both see the node lines, and tick off your drummer friend, is to sprinkle a light powder like sand or flour on top of a drum. Without even tapping the drum, the sand will settle on the node lines. Once you are impressed with how cool it is - you should leave the room quickly. Drummers are known for their tempers.
June 15th, 2006 at 9:43 pm
Drums can be tuned so as to more resemble certain of those patterns more predominantly than others. An even tensioned tuning with a no batter (bottom) head for example, will produce a strong vibration of the first type. The addition of a bottom head or uneven tuner tensions will take it in alot of the other directions. And the sand thing is cool, but it’s easier to see when you coat the whole surface of the drum lightly. Jello power works well. Then tap it once and the Jello will form along those lines very clearly.
June 15th, 2006 at 10:58 pm
Maybe this applet excite you. I was playing with the electrical circuit applet and found some others that were fun, i.e. the Drum one.
http://www.falstad.com/circosc/
Check out the others. You’d probably get a kick out of ‘em.
June 17th, 2006 at 5:07 pm
This was fun. That applet is really cool too.
I did some studying at BYU of electromagnetic wave patterns (modes) in waveguides such as fiber-optic cables. In those cases, usually the equations involved Bessel functions (as well as sine waves).