Harmonics revisited.

The above picture represents "harmonics" on a string - waves that form and give the maximum possible amplitude.  A string can vibrate at any frequency, but certain frequencies will make the wave BEST vibrate - with the greatest height/amplitude.  These waves are called harmonics and they occur at frequencies which are integer multiples of the lowest possible "resonant" frequency.  For example, if the frequency will produce a simple 1 "hump" pattern (n = 1) at 10 Hz, the next harmonic (2 "humps" or n = 2) will be at 20 Hz.  The next harmonic (n = 3) is at 30 Hz, and so on.

Another way to put it:

Harmonics are a specific case of waves "interfering" with each other – making “standing waves” or “harmonics.” Here we see that certain frequencies produce larger amplitudes than other frequencies.  There is a lowest possible frequency (the resonant frequency) that gives a “half wave” or “single hump”. Every other harmonic has a frequency that is an integer multiple of the resonant frequency. So, if the lowest frequency is 25 Hz, the next harmonic will be found at 50 Hz – note that that is 1 octave higher than 25 Hz. Guitar players find this by hitting the 12^th fret on the neck of the guitar. The next harmonics in this series are at 75 Hz, 100 Hz and so on.

Sound waves are mechanical - that is, they require a medium.  There can be NO sound where there is no medium through which to travel.  That makes them very different from light (or any e/m waves).

Also note that it is easy to calculate the wavelength of a harmonic.  The formula is:

wavelength = 2 L / n

That means, it is twice the length of the string divided by the harmonic number.


Useful terms in music:

In music, the concept of “octave” is defined as doubling the frequency. For example, a concert A is defined as 440 Hz. The next A on the piano would have a frequency of 880 Hz. The A after that? 1760 Hz. The A below concert A? 220 Hz. Finding the other notes that exist is trickier and we’ll get to that later.

Interference:

Waves can “interfere” with each other – run into each other. This is true for both mechanical and e/m waves, but it is easiest to visualize with mechanical waves. When this happens, they instantaneously “add”, producing a new wave. This new wave may be bigger, smaller or simply the mathematical sum of the 2 (or more) waves. For example, 2 identical sine waves add to produce a new sine wave that is twice as tall as one alone. Most cases are more complicated.

In music, waves can add nicely to produce chords, as long as the frequencies are in particular ratios. For example, a major chord is produced when a note is played simultaneously with 2 other notes of ratios 5/4 and 3/2. (In a C chord, that requires the C, E and G to be played simultaneously.) Of course, there are many types of chords (major, minor, 7ths, 6ths,…..) but all have similar rules. In general, musicians don’t remember the ratios, but remember that a major chord is made from the 1 (DO), the 3 (MI) and the 5 (SO). It gets complicated pretty quickly.

Harmonics in tubes - also, transverse and longitudinal waves:

I will demonstrate this in class.  Here are relevant sites which show longitudinal waves (as well as transverse waves).  You know the difference between mechanical and EM waves - mechanical waves need a medium, but EM waves do not (though they can travel in many different mediums, but at a speed less than the speed of light).  Waves can also be classified according to shape:  transverse (standard-looking waves, like sine waves, which move perpendicular to the direction of wave travel.  Longitudinal waves move parallel to the direction of wave travel.  The animations below show this better than words can express:

http://www.acs.psu.edu/drussell/Demos/waves/wavemotion.html

http://www.animatedscience.co.uk/blog/wp-content/uploads/focus_waves/tl-wave.html

Note that light is a transverse wave, but sound is a longitudinal (or compression) wave.