Physics of the Didgeridoo

Contents
   
How the didgeridoo makes sound
    The pitch of the sound you hear is dependent on the frequency
       
Frequencies of different notes
    Calculate frequency from wavelength and velocity
    Velocity of sound depends on air temperature
    The wavelength of sound depends on the length of the tube
    The width of the tube changes the wavelength slightly
   
Formula to calculate frequency of didgeridoo sound
    A chart of didgeridoo lengths for different notes

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How the didgeridoo makes sound
    The basics of didgeridoo physics is pretty simple, it is just an open-ended tube.  By vibrating your lips at the closed end of the tube you create areas of higher and lower air pressure.  These compressed and decompressed bits of air only have one direction to travel so they head down the tube, bounce off the open end, head back up to your mouth, bounce off that end, and then head back down the tube again.  This process causes air molecules at the open end of the tube to start vibrating, which is the sound that you hear.

The pitch of the sound you hear is dependent on the frequency
    Sound is a bunch of vibrations in the air, and the pitch of a note is the frequency of these vibrations.  Frequency is measured as the number of vibrations per second.  One vibration per second is also called one hertz. 
    The international standard is that middle A (the A note that is in the same octave as middle C) is at a frequency of 440 hz.  The A note an octave higher is at twice the frequency (880 hz), and the A note an octave lower is at half the frequency (220 hz).  The notes are set up so that every half step (i.e. an A to an A#, or a B to a C) is the same percentage increase in frequency.  Because there are 12 half steps in an octave, this means that each half step increases by the twelvth root of two, or by a factor of approximately 1.059.  This also means that any note that is exactly one octave above is twice the frequency, and any note that is one octave below is half the frequency.
    Here is a chart of the frequencies of different notes.

Calculate frequency from wavelength and velocity
    The frequency of a note is the the velocity of the sound wave divided by the wave length.  Both the velocity of sound and the length of the sound wave can change based on different factors.

Velocity of sound depends on air temperature
    At 0°C sound travels at about 331 m/s.  As air warms up, sound travels faster.  At room temperature (20°C) sound travels at 343 m/s, and at body temperature (37°C) sound travels at 353 m/s.
    This means that your didgeridoo will change in pitch depending on the air temperature.  A change in temperature from 0°C to 33.4°C will increase the pitch of the didgeridoo by 5.94%, or one half step (i.e. from an A to an A#).  If you started playing your didgeridoo at room temperature, the air inside it would slowly warm up to close to body temperature as it was filled with air from your lungs.  The change in temperature from room temperature to body temperature will increase the pitch by 2.89%, or about a quarter step.
    Interestingly, temperature has a slightly smaller effect at warmer temperatures.  An increase in temperature from 0°C to 1°C will increase the pitch by 0.183%, whereas an increase in temperature from 36°C to 37°C will only increase the pitch by 0.162%.
    The velocity of sound can also vary with air humidity, which would undoubtedly change as you spray spit into your didgeridoo, but this is a small factor so I will ignore it.

The wavelength of the sound depends on the length of the tube
    Because of how the didgeridoo makes sound, as described above, the sound wave starts at your mouth, goes down to the end of the tube and back up again, and then ends at your mouth.  So the wavelength of the sound wave is actually twice the length of the tube.

The width of the tube changes the wavelength slightly
    There is a slight complication to this straightforward calculation because the sound wave doesn't actually bounce off the exact end of the tube, it usually extends just a little further out of the tube before it reflects back.  And the wider the tube is, the further out the sound wave reaches.  The distance the sound wave travels outside of the tube before it reflects back is about 0.6 times the radius of the tube.  So for a typical PVC pipe of 4.75 cm diameter, the didgeridoo is effectively 1.425 cm longer than it actually is.
    If you aren't using a piece of PVC pipe, but a stick or something that flares out a lot, the physics can change.  For some reason a big flare can actually reduce the wavelength, but I don't really understand why.

Formula to calculate frequency of didgeridoo sound
    Taking the above mentioned factors into account, the pitch of a didgeridoo could be calculated with the following formula:
    Frequency = ((331 m/s) * SquareRoot(1+(AirTemperature / 273°C))) / (2 * (LengthOfPipe + (0.6     * RadiusOfPipe)))

A chart of didgeridoo lengths for different notes
    Here is a chart of didgeridoo lengths to calculate what length you need for a specific note.  This is calculated for an air temperature of 25°C.  If you want to make the didgeridoo in tune at an air temperature of 37°C, increase all the lengths by 2%.