Tuning Fork

Clang, clang, clang went the tuning fork

Materials

A tuning fork, long ones are better, I use a 256 Hz tuning fork.
A tuning fork has two tines and a single base.

Optional a sound spectrum analyzer such as a computer with a microphone running the program Audacity that runs on windows and Macintosh systems.

To do and Notice

Hold the tuning fork where the two tines come together. Strike one of the tines rapidly against the bottom edge of your shoe, Move it away from the shoe as fast as possible.

Listen to the tuning fork, hold the two-tine end of the tuning fork close to your ear. Notice two tones: a lower pitch, and higher pitch ones.

The lower pitch tone is the frequency listed on the tuning fork, the higher pitch tone (which is often much easier to hear, is called the clang tone.

The clang tone is a non-integer multiple of the fundamental tone. And is a little over over 6.1 times the frequency of the fundamental, it depends on the construction of the tuning fork.

Place the base of the tuning fork against a surface and the lower pitch tone becomes much louder. The touched surface does not produce the clang tone. Tuning forks are designed so there is an antinode of vibration of the fundamental tone and a node of vibration of the clang tone at the point where the base attaches to the tines.

Math Root

The clang tone is not an integer multiple of the fundamental tone of the tuning fork because the tuning fork is a bending beam. The motion of bending beams is described by a fourth order non-linear differential equation, unlike a vibrating string which is described by a second order linear differential equation. A string has harmonics which are integral multiples of the fundamental frequency, a tuning fork does not.

Going Further

Place the base of the tuning fork against a surface. You hear only one note.

Record this note using a computer running frequency analysis software and loom at the sound sprectrum.

Notice that there are two large peaks in this spectrum, one at the fundamental frequency and another at twice this frequency.

The bottom of the tuning fork is a parametric oscillator. When the ends of the tines spread apart they cause the base to rise up, and when the come together they make the base rise up generating two cycles of the base for every single cycle of the fundamental oscillation.

You cannot hear this higher tone as a different note because it us exactly a multiple of 2 times the oscillation of the fundamental. A factor of two in frequency is known as an octave in music. And if the fundamental is the note C, then a note with twice the frequency is an octave above C and is also C. Two different pitches enter the ear but the ear hears only one note.

Etc.

The location of the nodes of vibration are different for the fundamental and the clang tone

The clang tone has a node closer to the end of the tines than the fundamental, place your ear near this point and you hear the fundamental. Move your ear down the tines and when you come to the node of viration of the fundamental you hear the clang tone.

Go to a bicycle shop and get the front fork of a touring or racing bicycle, not a mountain bike with shock absorbers.

Strike the bike fork against your foot and listen to the fundamental and the clang tone. their loudest points are separated by quite a distance.