Physics

Source

"To make the graph at right, isolated bellies were driven mechanically at the position of the bass foot of the bridge, and the acceleration was measured. The graph gives the ratio for force to acceleration. If we were vibrating a small mass m, the ratio would be that mass, independent of frequency. However, the resonant behaviour of the plate appears here: the acceleration produced by a given force is a strong function of frequency. On this graph, each major resonance is indicated by an inset photograph of its Chladni pattern." - Source

FREQUENCY

"We can put all of this in a simple expression. If the vibrating part of the string has a length L and a mass M, if the tension in the string is F and if you play the nth harmonic, then the resulting frequency is

fn = (n/2L)(FL/M)1/2 = (n/2)(F/LM)1/2. In instruments such as the violin and guitar, the open length and the tension are fairly similar for all strings. This means that, to make a string an octave lower, while maintaining the same length, you must quadruple the ratio M/L. If the strings are made of the same material, this means doubling the diameter. However, the fat strings are usually composite: a thin core wrapped with windings to make them more massive without making them harder to bend.

Let's see where this expression comes from. The wave travels a distance λ in one period T of the vibration, so v = λ/T. The frequency f = 1/T = v/λ. So f = v/λ. We also saw that, for the fundamental frequency f1, the string length is λ/2, so f1 = v/2L. The wave speed is determined by the string tension F and the mass per unit lenght or linear density μ = M/L, v = (F/μ)1/2 = (FL/M)1/2. So f1 = ½(F/LM)1/2. Multiplying both sides by n gives the frequencies of the harmonics quoted above.

We can rearrange this to give the string tension:
F = 4f12LM." - Source

Source

The fundamental or first mode has frequency f1 = v/λ1 = v/2L

The second harmonic has frequency f2 = v/λ2 = 2v/2L = 2f1

The third harmonic has frequency f3 = v/λ3 = 3v/2L = 3f1

The fourth harmonic has frequency f4 = v/λ4 = 4v/2L = 4f1

The nth harmonic has frequency fn = v/λn = nv/2L = nf1.

~VIOLIN~STRINGS~

A side of a string on a violin is rolled around a peg, sits on a slightly elevated surface above the fingerboard, lays across the bridge, and is held on the other side in a tailpiece (usually containing fine tuners). The violin is tuned in 5th’s. The highest note is an E, then “concert A” (440Hz), then D, and lastly G. When tuned properly, the periodic wave heard can be analyzed as a sum of pure tones, similar to when a tuning fork is struck. But the string isn’t doing all off the work.

The bridge and the sound post are very important features to the sound of the violin, and interestingly enough, they are not held on the instrument by glue of any sort, only by pure force. The soundpost, a thin cylinder piece of wood, sits on the inside of the violin. The purpose of the sound post is to anchor the treble foot of the bridge to the top plate. The bridge responds to the vibration of the strings (from bowing) by pivoting, turning the side-by-side vibrations of the bow to up-and-down vibrations of the violin belly. Since each violin is made differently, every classical violin has a bridge made specifically for the violin in order to maximize the sound quality of the instrument.

On the actual fingerboard -- where the individual places their fingers to play notes -- harmonic waves are created. The shorter the length, the higher the frequency. Similarly can be said about the thickness of the string For example, although the cello also tunes to “concert A” they have much thicker strings so their “concert A” is the same note at a lower frequency. This gives a violin a distinct timbre, versus other instruments such as the cello. This timbre is caused by harmonics and each instrument has a different set of harmonics. Because the strings vibrate so fast on a violin, many modes are created.

For a wave, the frequency equals ratio of speed over length of the wave. In Wolfe’s diagram, the string length L is 2L, L, 2L/3, and L/2. Starting with λ/2, the relationship can be written as 2L/n, where n is the number of the harmonic. The lowest frequency is the one that is heard. Since all waves in a string travel at the same speed, different frequencies are on different wavelengths. These modes and the sounds they produce are part of the harmonic series.

Harmonics is used pretty often in violin music, along with other types of instruments. For example, if an individual were to press down on the string halfway while placing their finger one quarter the distance of the string (starting from the scroll side of the violin) a harmonic on the second string (A string) would create four times the fundamental frequency (fourth harmonic) and would sound two octaves higher. This is the beauty of waves in music.