Free Your Sound with the Help of the Helmholtz Motion
By Hans Jørgen Jensen
There are a number of fundamental principles concerning the sound production of stringed instruments that are both fascinating and important. One of these is the “Helmholtz Motion of the vibrating string.” Understanding the physics behind the instruments we play can make us more thorough and complete musicians. Unfortunately, principles such as this are not always explained very clearly in method books about string playing. In order to better understand these concepts it is helpful to first go back in time and trace the history of string acoustics.
Helmholtz's Vibration Microscope
Hermann Von Helmholtz was an incredible scientist who achieved a staggering number of scientific discoveries during his life. He was the first person to see and explain how the strings of a bowed string instrument vibrate.  In his revolutionary book On the Sensation of Tone, Helmholtz explains what he saw through using his special vibration microscope. In Helmholtz’s own words:
“By applying a peculiar method of observation. I have found it possible to observe the vibrational form of individual points in a violin string, and from this observed form, which is comparatively very simple, to calculate the whole motion of the string and the intensity of the upper partial tones.” 
To the naked eye the vibrating string seems to be moving back and forth in a parabola-like shape. Helmholtz, however, observed that the string in fact moves in very unexpected ways: in a “V”-shape, i.e. the string gets divided into two straight portions which both meet at a sharp corner. The fact that we see a gently curving (parabola-like) outline of the string’s motion with the naked eye is because the sharp corner moves back and forth along this curve. Hence, we only see the “envelope” or outline of the motion of the string. This “V” shape motion became known as the Helmholtz Motion. 
Follow this link to see how sticking and sliding friction works.  The sticking phase is when the string sticks to the bow and moves along at the same velocity as the bow, and the slipping phase is when the string slips back in the opposite direction at a much higher speed.
See this video for the Helmholtz Motion in action with a slow-motion capture of a violin string and bow:
The string does not vibrate back and forth from side to side like it looks to the naked eye, but it vibrates in Helmholtz Motion.
You can also very clearly see how the eye recognizes the parabola-like shape of the vibrating string in this animation:
To really understand the Helmholtz motion of the vibrating string spend some time observing and studying the above video and animation because a picture (or video) is worth a thousand words.
How to apply this information to your playing?
Start by practicing slow notes using the whole bow and try to be sensitive to how the vibration of the string travels quickly up from the bridge to the nut and back to the bridge. Of course, aural sensitivity to the sound being created is super important but being aware of the Helmholtz Motion of the string creates a totally different feeling when pulling the string. After doing this for a short time you will start feeling how the string really vibrates and you can create a very free and ringing sound. Spend five minutes during your next practice session trying to engage the string with a keen awareness of the Helmholtz motion with an open mind and open ears!
Here is a simple exercise from Chapter 20 of PracticeMind: The Complete Practice Model for practicing open strings:
Exercise 20.1 from PracticeMind – Pulling the bow parallel to the bridge
One necessary and fundamental skill for playing a string instrument is being able to draw the bow in a straight line parallel to the bridge.
The following are selections from the new book PracticeMind by Hans Jensen and Oleksander Mycyk (Chapter 20: Sound Production)
 It is assumed that you have some basic knowledge about bowing and the vibrating string. If not, follow this link: http://www.ehow.com/how-does_7238083_violin-bow-work_.html
 Helmholtz, Hermann L. F. On the Sensation of Tone as a Physiological Basis for the Theory of Music. 2nd English Ed. Trans. Alexander J. Ellis. London: Longmans, Green, and Co., 1954. Page 80. For a fascinating read go to chapter 4 “Musical Tones of Bowed Instruments” and read for yourself the whole experiment as explained by Helmholtz. The book can be downloaded from Google Books. This revolutionary experiment was an incredible yet small part of all the amazing discoveries he did in his research.
 Woodhouse, Jim and Paul Galluzzo. “Why is the Violin So Hard to Play?” +Plus Magazine 31, September 1, 2004. http://plus.maths.org/content/why-violin-so-hard-play