# "Modes of vibration" terminology

I've recently been doing some research on modes of vibration. Specifically with regards to music and drum synthesis. I'm quite the physics noob, but I figured that this is the most fitting site on Stack Exchange.

What still confuses me is: What exactly does the term "mode" mean? Do I say

• The second mode of this system lies at 67 Hertz.
• The second mode of vibration is this exact pattern that the system exhibits. Which it just happens to do when it oscillates at 67 Hertz.

I'm sorry if this is a weird question, but I haven't been able to find any resources which explain the topic in a way that's understandable to physics outsiders.

Related question: Do non-integer modes exist?

I've been doing some more research and I would like to add a collection of interesting resources. Maybe it will help someone out in the future:

• "Do non-integer modes of vibration exist?" Yes, for example bells and drums, and for almost every structure that is not a musical instrument. Apr 18, 2021 at 18:58

The term mode is heavily used in physics and for this reason its usage tends to be not very stringent. It generally refers to the 'natural' motions of oscillating systems.

Let's get a feel for it: Take any system with an oscillating motion, be it a stringed instrument, drums, pendulums, radio antennas, etc. Jiggle or kick one of these and see how they react, i.e. pick a string and listen how it reacts. Looking at this closely, one can actually tell apart these reactions into a set of motions that happen at once and are added together to give the reaction. Each of these motions happens at a well defined frequency and is called a mode.

If you pick a string it will oscillate in many of its modes and you hear all their frequencies mixed together. Pick a string in a different position or hit a drum at a different point and notice that it changes its sound. This means that the relative strength of the modes was changed. For example a string picked at its midpoint sounds like a harp and picked closer to its end sounds like guitar.

With this control one can try to pick/jiggle/kick the system in a way to single out one mode only. This is what happens with Chladni figures on metal plates with sand. Similarly, humming next to a string to make it vibrate, singles out the mode matching the humming frequency.

Talking about modes: In the end, your oscillator has a set of modes and each mode is directly related to a frequency, which is the speed at which these motions naturally happen. The term mode is used to refer to the pattern of motion as well as to the frequency. So saying

• "The second mode of this system lies at 67 Hertz." is perfectly fine and
• "The second mode of vibration is this exact pattern that the system exhibits. Which it just happens to do when it oscillates at 67 Hertz." is kind of fine too, but the relationship between modes and their frequency is more direct than this sentence implies.

Also, in case you where wondering, resonances are the same as modes.

Non-integer modes: Actually, only few systems have integer modes only. That's why making good instruments is not easy. While many systems have overtones, whose frequencies are integer multiples, they usually have many other modes with frequencies in between, which would then be non-integer multiples with respect to the overtone series. You have mentioned Chladni figures in the comments and they have soooo many modes.

Notes for completeness: Adding modes together is only possible in harmonic oscillators which is a certain class of well behaved oscillators defined by exactly this: allowing the addition of modes. They are ubiquitous in physics and their modes are also called harmonics. However there are oscillating systems that are not harmonic, actually the most real systems are only harmonic in more or less rough approximations. These systems still have modes, but the reaction of the system to kicks is not simply a number of modes added together.

Also in general there can be multiple modes / patterns of motion with the same frequency. That is called a degeneracy. But this is a special case and, afaik, not common in everyday physics. Think of two Chladni figures that emerge when bowing the metal plate at different points, but both sound the same frequency (which seems to be a thing).

• Wow, thank you. I'm kind of blown away by your answer. I went into a rabbit hole about chladni plates earlier and they really helped me get an understanding. But what just made everything fall into place was this sentence of yours: "Also, in case you where wondering, resonances are the same as modes.". I think I'm gonna play around with chladni figures soon. I still have a lone speaker laying around. Also, off-topic, but I'd like to mention that my name is Hannes as well – funny coincidence. Haven't met a lot of people with that name. Apr 18, 2021 at 23:01

In a typical (non-electronic) musical instrument, something (a string, air column, or membrane) is vibrating within boundaries. The multiply reflected waves resonate at certain frequencies as standing waves. The different allowed frequencies can be referred to as different modes of vibration. (For example: The different notes that can be produced by a bugle.)

• Do chladni patterns correlate to different modes of vibration? So whenever I turn up the frequency and a stable pattern emerges I have found a mode? Apr 18, 2021 at 19:55
• Each stable pattern would be a mode. Apr 19, 2021 at 13:13
• Alright, thank you for the clarification :) Apr 19, 2021 at 14:29

The term 'harmonic' is more usual in physics, it works like this

from this webpage - it might be about right for what you want

https://www.earmaster.com/music-theory-online/ch03/chapter-3-2.html

• I see, so the nth mode is equivalent to nth harmonic? Apr 18, 2021 at 18:43
• Images are not accessible to all users, and links may become broken, so this answer has the potential to become inaccessible to many people. I suggest supplementing the link with a description of the relevant part of the link, and giving an answer beyond an image. Apr 18, 2021 at 18:54
• I disagree. The term "harmonic" implies an integer multiple of the fundamental frequency. Modes of vibration are only "harmonics" for a few special systems (and even then it is only approximately true). In general, "standing waves" do not correspond to "vibration modes". Apr 18, 2021 at 18:55
• @schroffl Overtones are usually interpreted as resonances of the system. For string instruments and wind instruments these often fall very close to exact multiples of the frequency of the fundamental mode (as John Hunter illustrates). For percussion instruments this is usually not the case. In contrast, harmonics are usually taken to be exact multiples of the fundamental frequency. These are created, for example, when a sine wave passes through a nonlinearity such as a 'fuzzbox' that clips or distorts the sine wave. Apr 18, 2021 at 18:59
• @RogerWood Assuming a snare has its fundamental at 220 Hz. If the membrane resonates at 440 Hz it's called a "harmonic overtone" whereas a resonance at 375 Hz would be called "non-harmonic overtone", right? However, even if what I just said is correct I still don't get how this relates to "modes". Are modes just a synonym for "harmonic overtones"? Apr 18, 2021 at 19:33