## Saturday, December 7, 2013

### Thermal Expansion and Violins

http://en.wikipedia.org/wiki/Violin

I used to play the violin when I was little. I remember that my violin would undergo changes and require more tuning depending on the temperature. I decided to do some calculations to determine how much a violin changes when the temperature changes, comparing winter to spring. In my hometown, Binghamton, NY, the average winter temperature is -9 degrees C in January and 8 degrees C in May (http://www.weather.com/weather/wxclimatology/monthly/graph/13901). I am going to calculate the change in length of a full size violin and the change in volume.

Change in Length=(alpha)(L0)(change in temp.)
Change in temp.= 17 degrees C
L0 of a full size violin= 590 mm= 0.590 m
Most violins are made with spruce and maple for the back but I am just going to use the alpha and beta values for sitka spruce. For alpha I am looking at the longitudinal change in length (not radial or tangential)
Change in L=(1.74*10^-6 degrees C^-1)(0.590m)(17 degrees C)
Change in L=1.75*10^-5 m

Now I want to see how the volume of the hollow part of a violin changes from winter to spring temperatures:

Change in volume= (beta)(V0)(change in temp.)

To calculate the volume of the violin, I am using the length of the corpus of the violin to be 356 mm, and since the width of a violin changes I am going to approximate the violin as a rectangle and I will average the widths of the three parts of the body (upper: 168 mm, middle: 112 mm, and lower: 208 mm) so 162.7 mm or 0.1627 m, and the height of the ribs on a violin are about 30 mm or 0.03 m.
V=(0.356m)(.1627m)(0.03m)
V=0.00174 m^3

Change in volume= (82.61*10^-6 degrees C^-1)(0.00174m^3)(17 degrees C)
Change in volume= 2.44*10^-6 m^3

Although the changes in length and volume may not seem that large, since violins are very delicate instruments any slight changes in the wood has an impact on the sound of the violin. Now it makes sense why violinists are so careful to control the humidity and temperature of their instruments since the size of the instrument changes with changes in temperature.

Violin measurements from:
http://www.curtisviolins.com/setup.html
http://www.alangoldblatt.com/specs/Violin.pdf

Violin wood information from:
http://www.gussetviolins.com/wood.htm

Alpha value for wood from:
http://classes.mst.edu/civeng120/lessons/thermal/thermal_expansion/index.html

Beta value for wood from:
http://ir.library.oregonstate.edu/xmlui/bitstream/handle/1957/1597/FPL_1487ocr.pdf

Violins and humidity:
http://www.grantviolins.com.au/humidifiers.php