Why do harmonics occur

The diagram below depicts this length-wavelength relationship for the fundamental frequency of a guitar string.

The second harmonic of a guitar string is produced by adding one more node between the ends of the guitar string. And of course, if a node is added to the pattern, then an antinode must be added as well in order to maintain an alternating pattern of nodes and antinodes.

In order to create a regular and repeating pattern, that node must be located midway between the ends of the guitar string. This additional node gives the second harmonic a total of three nodes and two antinodes. The standing wave pattern for the second harmonic is shown at the right. A careful investigation of the pattern reveals that there is exactly one full wave within the length of the guitar string.

For this reason, the length of the string is equal to the length of the wave. The third harmonic of a guitar string is produced by adding two nodes between the ends of the guitar string. And of course, if two nodes are added to the pattern, then two antinodes must be added as well in order to maintain an alternating pattern of nodes and antinodes. In order to create a regular and repeating pattern for this harmonic, the two additional nodes must be evenly spaced between the ends of the guitar string.

This places them at the one-third mark and the two-thirds mark along the string. These additional nodes give the third harmonic a total of four nodes and three antinodes.

The standing wave pattern for the third harmonic is shown at the right. A careful investigation of the pattern reveals that there is more than one full wave within the length of the guitar string.

In fact, there are three-halves of a wave within the length of the guitar string. For this reason, the length of the string is equal to three-halves the length of the wave. After a discussion of the first three harmonics, a pattern can be recognized. Each harmonic results in an additional node and antinode, and an additional half of a wave within the string.

If the number of waves in a string is known, then an equation relating the wavelength of the standing wave pattern to the length of the string can be algebraically derived. The above discussion develops the mathematical relationship between the length of a guitar string and the wavelength of the standing wave patterns for the various harmonics that could be established within the string. Now these length-wavelength relationships will be used to develop relationships for the ratio of the wavelengths and the ratio of the frequencies for the various harmonics played by a string instrument such as a guitar string.

Consider an cm long guitar string that has a fundamental frequency 1st harmonic of Hz. For the first harmonic, the wavelength of the wave pattern would be two times the length of the string see table above ; thus, the wavelength is cm or 1. The speed of the standing wave can now be determined from the wavelength and the frequency.

The speed of the standing wave is. Since the speed of a wave is dependent upon the properties of the medium and not upon the properties of the wave , every wave will have the same speed in this string regardless of its frequency and its wavelength. So the standing wave pattern associated with the second harmonic, third harmonic, fourth harmonic, etc. A change in frequency or wavelength will NOT cause a change in speed.

Now the wave equation can be used to determine the frequency of the second harmonic denoted by the symbol f 2. This same process can be repeated for the third harmonic.

Now the wave equation can be used to determine the frequency of the third harmonic denoted by the symbol f 3. Now if you have been following along, you will have recognized a pattern. The frequency of the second harmonic is two times the frequency of the first harmonic. The frequency of the third harmonic is three times the frequency of the first harmonic. The frequency of the nth harmonic where n represents the harmonic of any of the harmonics is n times the frequency of the first harmonic.

In equation form, this can be written as. The inverse of this pattern exists for the wavelength values of the various harmonics. These relationships between wavelengths and frequencies of the various harmonics for a guitar string are summarized in the table below.

By contrast, if you pluck the guitar string very near the center of the string, you put very little energy into the 1st, 3rd, 5th, harmonics, which have a node at the middle of the string. This gives a sort of rounder, "oooo" sound to the strings. Give it a try! When you pluck a guitar string the potential you apply to the string is approximately a Dirac delta function.

That is to say, the release of the string is a near instantaneous kick. One of the beautiful properties of the delta function is that its Fourier transform is unity. This means that it is made up of equal components of all frequencies. So, when you pluck the string you excite every single resonant mode equally in the delta function limit. What determines the different sounds of different instruments is how long each resonant frequency can be sustained, i.

When you pluck a string it does not start out like the fundamental above. The string is pulled into a bent shape of two straight lines and an angle and it may not be bent at the middle.

Releasing the bent string causes a bunch of harmonics of various amplitudes depending on how far off-center it was bent. It can not return to the bent angle shape and the energy has to go somewhere.

The result from that shape is all harmonics and sounds like a "rich" sine wave from the odd harmonics dominating. A guitar or violin is plucked very off-center so it is more like a sawtooth and gets all harmonics, odd and even with a set of amplitudes distinctive to the instrument. This was first? The response can be derived mathematically. If we pinch the string at the middle, this corresponds to a condition on the configuration of the string at the initial time, i.

Furthermore, we must impose Dirichlet boundary conditions as the string is fixed at either end, i. Solving the wave equation via Fourier series is tedious but easily doable. Eventually, we obtain the wave harmonics, images of which are available in the OP.

A couple things play in here. First, the string is "closed" at both ends, meaning the ends are locked down and can't move. This means any resonant wavelength must have "nodes" , which is a contraction of "no displacement," at the ends.

By comparison, the acoustic waves in an open-ended tube may have a node at one end, but the other end is unrestrained and could be a maximum. Next, the resonant "standing wave" for each frequency is actually the combination of travelling waves moving in phase and in opposited directions along the string.

And so on for all higher harmonics. Note that you can suppress, e. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why do harmonics occur when you pluck a string? But its waveform may be decomposed into a sum of sine waves of different frequencies.

Now why doesn't a guitar string vibrate as a sine? As mentioned by others, this is controlled by the constraints applied to the string. The contact with the pluck, where the string is struck, the stiffness of the string, the connections to the guitar body, the body itself, the room, your fingers It all has to do with overtones. In a nutshell, sound is a compression wave. It's usually drawn as a standing wave for simplicity.

Every pitch is at a set frequency, so the high point in the wave occurs every so often. An overtone, which is what a harmonic is, happens when you have two sound waves whose high points overlap at certain intervals. For instance, an octave above any given note is twice that note's frequency, so the high points of the upper note will overlap the high points in the lower note every other time.

Similar effects occur for most overtones. A guitar string really does only vibrate at a single frequency, which is determined by its length and its tension. The overtones line up with other frequencies, which causes any appropriately tuned strings nearby to resonate with the string if they match one of the harmonics.

This is a gross oversimplification of course. This youtube video is the best explanation of the whole process I've seen in a while. An ideal one carefully plucked at its middle would, but real-world guitar strings are not idealized strings. They are not massless, they have thickness, they're often twisted bundles of metal, inconstant tension, gauge, etc. And, probably most importantly, they're plucked somewhere close to one end of the string, which is against the natural motion an ideal string would like to take.

Thus, more than one mode frequency of the string will be vibrating; these are the harmonics. The colors are the different modes overtones or harmonics of the string's vibration. Any of these colored "strings" is a natural motion the black string would like to take. Since the red "string" has the largest amplitude, its frequency is the most prominent heard coming from the vibrating string.

All these colors, when superimposed, create the non-"pure" vibration of a plucked sting. You can see the shape of the black string isn't symmetric, is "bent," unlike the colored "strings. Plucking at the middle of the string is one way to minimize the harmonics. If you do this, you'll hear a more pure sound. This is because it's not as against the natural motion of a string as plucking near an end of the string. One way to understand harmonics is to look at mathematical operations, like Fourier transforms , or other transforms.

These operations convert transform an integral equation of some quantity, typically amplitude vs. Another way is to look at how non-linearity create harmonics.

This is what I'll elaborate on here. Non-linearity is not something unknown for musicians, as soon as an amplifier or a microphone is non-linear, it creates harmonic distortion , which is just parasitic harmonic frequencies added to the amplified copy of the audio input. Harmonic distortion in music is also called So many different words for one physical effect! As an example of linearity, imagine a spring. If one elongates the spring they sense a restoring force, the larger the elongation and the larger the force, maybe to the point the spring cannot be extended further.

In general a helical spring develops a restoring force exactly proportional to the elongation:. Such system is said linear as regards to its response to a perturbation. For more information on the linearity of springs, and some applications, a good read is the Wikipedia article on Hooke's law. The diapason is an interesting instrument because it oscillates mostly without harmonic. Prongs oscillation occurs in the linear domain of metal elasticity, where the restoring force is proportional to the current distance from the rest position.

This quasi- linear elasticity exists for metallic material but only for small displacements, which means small energy transmitted to air and limited sound intensity. If we tried to create higher sounds, we would leave the linear domain and harmonics would appear.

We'll be back to the diapason later. Let's first see a truly non-linear system: The guitar string! An oscillating system like a vibrating string has also a rest position. When moved away from this position it develops a force, in the form of a tension, tending to restore the rest state, the larger the transverse distance from the rest position, the larger the longitudinal tension. However the guitar string doesn't work in the small linear elasticity range of the diapason, it needs to produce powerful sounds, the string is "excited" with large inputs, to which the material is not able to respond in a linear way.

The tension is not proportional to the transverse distance at a given point of the string:. Note: The figure above has been updated after user comment on wrong value for amplitude x. Other factors play a role, including the response is not time independent, the response also depends on previous perturbation of the string. The result is the restoring force is not a scaled copy of the current string distance from its rest position and, adding complexity, at a given time, the scaling factor is not the same for all segments of the string.

This non-linearity between displacement amplitude and restoring tension is the origin of the harmonics. The actual mechanism is complex, but we'll see a simple case by looking again at the diapason, which after all is not totally linear Saying a diapason has no harmonics was an approximation.

The diapason usually develops the second harmonic, and the detail of how this happens is a good example of the extreme sensibility of physical oscillating devices to asymmetry and non-linearity which is seen in action with strings. Basically the prongs of a tuning fork oscillate in their common plane like cantilever beams, and the center of mass, seen from the top is kept motionless due to the symmetry of the displacements.

However this is not the case for its vertical position. When the prongs oscillate, their individual center of mass moves up and down by a small distance, following a circular arc. It also occurs at Hz or whatever frequency the fork is tuned for.

This displacement of the mass induces a reaction in the vertical direction, the stem goes up and down by a very small amount. The diapason is usually held against another support, e. When doing so, prongs vibrations are transmitted to the support acting as an amplifier. It appears the stem is more efficient to transmit vertical vibrations, and the table surface is more easily bent vertically than moved horizontally.

Due to this selective amplification by the table, the very small vertical vibration has now more relative importance. The frequency of the transverse wave and the center of mass wave being the same, this could still goes without consequences, however what is problematic is their waveform is different, one is a distorted sinusoid.

And guess the reason for this distortion Here we are: Non-linearity! This graph is part of a study that is good to read. It shows the two oscillations scales are not of the same order. While the transverse wave is almost sinusoidal, the vertical wave due to the mass displacement has peaks and lows of different shapes.

The cause is the vertical oscillation alternates tension and compression forces in the stem, to which metal responds in a different way, with different velocities. We are left with two waves which naturally interfere, creating the Hz harmonic:.

This is a simple example of harmonics created by a small non-linearity, the principle is the same for other materials and vibrating devices, including guitar strings, though more elements are involved.

Technically we say it's an homomorphism, from homos- same and -morphe shape. That's big words, in practical here is a linear transformation:. If input x produces output u, then for any number k, k. This means the output must be proportional to the input. If the input is increased by a factor k, then the output is increased by the same factor. If input x produces u, and input y produces v, then input k1. It means the output produced by the sum is the same to the sum of the outputs from individual inputs.

That's all. It can happens for amplifiers, for vibrating devices, for cosmic waves, or the electrical grid. Harmonics are usually unwanted, but they can be difficult to remove. You asked "Why doesn't a guitar string vibrate at one frequency only? Let's look at it from the other perspective: Every kind of musical instrument makes sounds that have overtones in them, and every pitch played on every kind of instrument has multiple frequencies in it--not just on the guitar.

There is no repeating, oscillating sound made by any musical instrument that has absolutely no overtones. The only sound that can exist that has no overtones would be a completely pure sine wave. You could only create a pure sine wave with an electronic oscillator.

No acoustic or electro-acoustic musical instrument can create a sound that is similar to a pure sine wave. It would vibrate only at the fundamental frequency if you excited it exactly at the middle and there were no losses in the material.

However, if you excite it randomly, it will go to an "equilibrium" status in which only standing waves multiple of fundamental frequency survive. That is, in an ideal scenario you could generate only the desired harmonics if you excite an ideal string in the correct points. In all of the examples you'll want to let the t variable change by pressing the play button next to the variable.