Music / Harmonics [duplicate]

Question

I am trying to understand the physics behind waves, sound, and music and feel as if I'm missing some important conceptual components.

How is it even possible for a string to vibrate at multiple frequencies at the same time? I understand that harmonics exist and how they work but I have yet to find any information on why this even happens. To me it seems like if you were to pluck something like a guitar string, it would simply oscillate at its natural frequency based on the length and tension of the string? So, I guess my question is, how can it have more than one natural frequency? How are multiple waves with different wavelengths being generated from a single pluck?

Any insight is greatly appreciated but please provide answers without equations, conceptual understanding is more important to me than exact numbers.

Answer 1

The concept you need to understand is superposition. If you have a wave traveling in one direction, you can have another wave traveling through the same medium in another direction - and they can keep going. This is why you could have four people standing on the corners of a square, and talking to the person diagonally across from them - and the sound of their voices arrives at their partner's ears even through it had to cross through the sound of the other person speaking at the same time.

A string can vibrate at its fundamental frequency, or at harmonics of that frequency. When you pluck a string, you send an impulse of sound to both ends of the string. There, it gets reflected and comes back. "Plucking" is something we consider a "wide band excitation" - it contains "all frequencies". But after reflection, only waves that "fit nicely" on the string continue to exist - the others tend to extinguish themselves very quickly.

A picture might help:

From top to bottom: a string about to be plucked, the same string a moment later, and the first four fundamental modes of the string.

Answer 2

The maths/physics that will truly help you get to the bottom of this is Fourier analysis, but as you have requested for a minimum of equations, I'll not follow that route.

How can a string vibrate at multiple frequencies at once?

What is truly happening is it is vibrating in a complicated waveform. But you could construct this complicated waveform, by adding together simpler (sinusoidal) waveforms. These are harmonics, the building blocks of the more complicated wave.

Why this necessarily has meaning to us, somewhat depends on how the human ear works. Inside the ear, where the sound waves are converted into impulses for the brain to decode, there are a series of tiny hairs. Each of these tiny hairs has a slightly different length. This means that each hair will vibrate at its own, slightly different natural frequency. Hence if you play two notes at once, the tiny hair for one note will vibrate, and the tiny hair for the other note will vibrate too. Your brain can decode this, and you can understand there are two notes playing at the same time.

It's similar with the complicated waveform coming from say, a piano. Each of the simpler waveforms will excite the tiny hair corresponding to that frequency*, and you can piece together the sound in your mind.

*You could ask why sine waves are the building blocks for these sounds. My guess, would be that with small vibrations, the small hairs are approximately like simple harmonic oscillators, and so a sinusoidal frequency at their resonant frequency will excite them efficiently.

To me it seems like if you were to pluck something like a guitar string, it would simply oscillate at its natural frequency based on the length and tension of the string?

Again, Fourier analysis is the best way to go on this. But essentially, when you pull the string back, you establish a very complicated shape on the string. (e.g. roughly a triangle shape.) As you release the pluck, you give a velocity profile along the string too, which will also be complicated. You can understand how the string will evolve by looking at it in terms of the fundamental modes of oscillation, and adding them together (owing to linearity of the problem). So again, it's based on how similar the initial conditions are to a sum of the modes of oscillation. This is all made clear and exact in Fourier analysis.

Answer 3

How is it even possible for a string to vibrate at multiple frequencies at the same time? I understand that harmonics exist and how they work but I have yet to find any information on why this even happens.

It's actually very difficult to avoid producing the these extra vibrations.

If you visit the National Institute of Standards website, I think there are related links, that demonstrate the lengths labs need to go to, to achieve purity of frequency, if I can put it like that.

For example, the tension in the string is never absolutely the same at every point along it's length (normally we just measure from both ends and get an average), the composition of the wood of guitar neck varies, the air coming out of the soundbox bounces around and vibrates the stings.

All this from, to you and I, a "simple" pluck of the string.

These factors all contribute to different vibration rates, but the harmonics stand out, because of their math relationship to the fundamental note, and because they can "even" each other out, as in the straight lines in the first picture produced from lots of curved lines.

Here is a graph composed of straight lines.

In 1807, Joseph Fourier said (in effect, not literally) that he could make a copy of the above graph, but using curved lines, like these below.

His "trick" was to start with the basic curve above, and then add harmonics of it, waves that resembled the original curved line above, but with various fractions, (harmonics) that, when added together in the right way, would eventually result in this below:

Image Source: Computing Harmonics

If you keep adding more and more waves, you eventually get your straight-ish graph back again. Magic.....

The music you hear when you pluck the guitar string is composed of these waves, although you may not hear them separately, you do hear the overall note they create.