Physics of Musical Bowing

Understanding the Science to improve the Art : Notes for the beginner

A bow is drawn across strings causing them to vibrate, and the body amplifies this to produce sound that is hopefully musical. Using a bow gives the musician greater control over the ‘shape’ of the sound – how long a note is sustained, its dynamics, articulation, colour, texture, envelope, etc. Good bowing is required for good tones – appropriate amounts of friction, force, speed, position, posture, etc…. The bow needs to be wielded well (one of the many reasons why playing string instruments like the violin is so tough!)

bowing

You may have seen musicians of the string family rosin their bows with much élan. Why is rosin applied to bows? Friction is the simple answer, but let’s delve deeper. Hair has scales on its surface so the friction felt in different directions is different. When a rosin cake is rubbed on, by ion exchange the hair picks up tiny particles of rosin. The trapped particles are warmed within a few tens of milliseconds by friction when the hair is drawn across the string – this provides better adhesion. When rosin is warmed, it shows some interesting characteristics – it’s coefficient of static friction rises rapidly while its coefficient of dynamic friction drops slightly (see graph). Rosin is used to create this difference in conditions between static and dynamic friction
rosin

When the bow is drawn across the strings, it is in a stick and slip cycle of Helmholtz motion:
[Bowed string in slow motion : https://www.youtube.com/watch?v=KPpBvHXYWz4 ]
–   Owing to a high coefficient of static friction, the hair grips the string and pulls it along in the direction of motion of the bow (perpendicular to the length of the string) – this is the ‘stick’ phase – the string travels with the bow.
–   This motion of a string when it is bowed is called a Helmholtz motion. Rather than having a parabolic shape as it appears to the naked eye, it is considered to be a V shape, with the kink or vertex (called a Helmholtz corner) travelling along the length of the string. The kink gets reflected at the fixed end (nut or finger) and travels back.
–   On returning, when the kink meets the point of contact with the bow once again, it is moving in the opposite direction, and the tension in the string pulls it off the bow (tension on both sides at the point of contact have downward components). It moves into the ‘slip’ phase – owing to a lower coefficient of dynamic or sliding friction, the string slips easily and moves in a direction opposite to the direction of bowing.
–    Each time the Helmholtz corner crosses the bow, it transitions from a stick to slip phase (or vice versa) – between the bow and the nut/finger it is in the stick phase and between the bridge and the bow, it is in the slip phase.
–   As the string slows down further, static friction begins to dominate – the bow once again grips the string and pulls it – and the ‘stick-slip’ cycle repeats. This stick-slip cycle vibrates at the same time period as the string. For instance if you were bowing an open A string, which has a frequency of 440 hz – this sticking and slipping of the string under a moving bow happens 440 times a second! For higher frequencies, it goes into thousands of times a second
[Helmholtz motion animation : https://plus.maths.org/issue31/features/woodhouse/Helm_anim.gif ]
­­­­­

helmholtz

Good bowing such as appropriate speed, force, placement, etc. creates better Helmholtz waves and so better tones. Improper technique will result in distorted Helmholtz patters

Physics ahead : (source: http://newt.phys.unsw.edu.au/jw/Bows.html)
Shape of string is two straight lines joined by a kink that travels around the parabolic segments at a constant speed V.
Let the frequency be f, and the length of the string be L.
Then the kink travels distance 2L in time 1/f.
So V = 2Lf
If we bow at a distance L/n from the bridge,
During the stick phase, the kink travels to the nut and back, i.e a distance D = 2L – 1/n = 2L(n-1)/n
This will take time t = D/V = 2L(n-1)/nV = (n-1)/nf
Let the bow speed be v, so during the stick phase, the bow and string travel together for a distance given by A = vt
A = v(n-1)/nf
This A is the amplitude of motion of the string at the point of bowing

How this knowledge can help our bowing:
-The bow should be moved parallel to the bridge, else n will change, hence changing the frequency and/or loudness (given by A). There should be a steady point of contact between the bow and the string
– Amplitude of motion is proportional to bowing speed i.e. bowing faster produces a louder sound – this is what is meant by ‘use more bow’ Contrary to conventional belief, bowing harder – i.e. using more pressure just makes the frictional force higher and does not change oscillation magnitude.

Bow pressure is also important – for a given bow speed and position there is only a specific range of bow pressure to produce a good sound. Exactly the right combination of bow speed and pressure is needed – if the pressure is too much, it’ll stick too much- resulting in raucous tones, and if the pressure is too little, it’ll slip too much – resulting in airy over-tones. The maximum and minimum bowing force for different conditions to satisfy Helmholtz motion was studied, these conditions are represented in a Schelleng diagram – pressure along the vertical axis and position (1/n) along the horizontal axis (the axes are logarithmic). The in between region marked normal is where Helmholtz motion can be achieved.

schelling

–   We see that it is easier to maintain Helmholtz motion away from the bridge than near it. Furthermore, if you are closer to the bridge ,you’d need to apply greater pressure. The positive about this is that with greater pressure, better higher harmonics are also produced – thus giving a richer tone
–    String crossing and chords become tricky because different strings and frequencies require different bow pressures for Helmholtz motion- especially when played near the bridge where the range of force is very narrow. Alternatively, we can manipulate bow speed – using a slower speed on lower notes and faster speed on higher notes and shorter strings
–   The more pressure you exert on a string, the faster the bow must be, because a higher speed is like shifting the shelling diagram up
–   Added to that there are a variety of articulations used – Detache, Martele, Legato, Staccato, Colle, Spiccato, etc – these need to be done while getting a good Helmholtz motion. No one said it’s going to be easy!

More to be updated later.

Here’s hoping that understanding the Science improves the Art

 

Sources:
The Physics of Musical Instruments By Neville H. Fletcher, Thomas Rossing
http://newt.phys.unsw.edu.au/jw/Bows.html
http://www.theory.physics.ubc.ca/341-current/neg-damp/node4.html
https://plus.maths.org/content/why-violin-so-hard-play
https://mindfulcellist.wordpress.com/2015/02/10/dancing-the-helmholtz-on-a-string-2104-hours/
http://www.thestrad.com/cpt-latests/how-to-master-bow-speed-and-distribution/

Additional literature:
http://knutsacoustics.com/files/The-Sound-Sculpturing-bow-CISM_7opl67i1.pdf
http://knutsacoustics.com/files/actaacustica-on-the-creation.pdf
http://www.zainea.com/bowed.htm
http://stringvisions.ovationpress.com/2011/07/schelleng-diagram/#1

 

 

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