Harmonics: Plucking & Damping Good Vibrations: Standing waves can be produced on a string fixed at both ends if the length of the string L is a multiple of half the wavelength l, so that the ends of the string are nodes. These waves are called “harmonics” and are identified by “n”, the number of half-waves they contain. l / 2 antinode node 4th harmonic ; n = 4 ; L = 4 ✕ (l / 2)

Harmonics: Plucking & Damping Materials: Your principal tool will be a one-string guitar thingy. Here are the names of the parts: Tuner Nut Fretboard Saddle You will also use: Guitar pick Q-Tips (2) Chromatic tuner app Laptop & mic Meter stick • Be sure the computer mic is at maximum sensitivity. • Prepare the guitar by running a piece of masking tape along the fret board under the string. • Use the chromatic tuner app to tune the pitch of the string (the 1st harmonic) to “C3” (about 130 Hz). Record the value.

Harmonics: Plucking & Damping Basic Method: In this lab, you will explore the first 8 harmonics on a simple guitar string. For each, you will (more or less) … • Sketch the wave using a line in the Notes section to the right to represent the string at rest. Keep the length the constant. • Mark two nodes (other than the ends) and two antinodes on the “fret board” • Excite that harmonic by plucking the string at an antinode while immediately damping the string with a Q-Tip at a node. • Measure the frequency of each harmonic, and compile and plot fn vs n in Logger Pro. • Show that once the note is started, the string can then be damped at any node without affecting the note. • Show that the note can be killed by damping at any antinode. • And other stuff along the way …

Harmonics: Plucking & Damping 1st Harmonic or “Fundamental”: Sketch the wave for the first harmonic “f1”. It has only one half- cycle. Calculate the location of the only antinode on the fret board. Mark it and label it “1a”. The only nodes, of course are at the ends.  Excite f1 by plucking the string at the antinode. Measure and record the frequency.  Kill the note by damping it on the same antinode. Describe the sound that remains.  Now, pluck the string near the nut or saddle and then kill f1. Describe the sound that remains.  Do you get it? If you pluck at the antinode of f1, you mainly get f1, which you can kill almost entirely. Pluck near the end and you get lots of other frequencies that survive after you kill the fundamental !

Harmonics: Plucking & Damping 2nd Harmonic or first octave: The harmonics that are powers of two are special because they are multiples of f1. We call these multiples “octaves”.  Sketch the wave for f2.  Mark the only node (“2n”) and the two antinodes (“2a”).  Excite f2 by plucking the string at either antinode and immediately damping the string with a Q-Tip at the node. Measure f2 and confirm that it is an octave above the fundamental (that is, twice f1).  Show that, once started, the string can then be damped at the node without affecting the note. That’s kinda cool.  Show that damping at an antinode immediately kills the note.  Again, pluck at a far end and damp at the antinode. What do you hear? What note is it?

Harmonics: Plucking & Damping 3rd Harmonic or “Major Fifth”: The harmonics next above the octaves are special in that they are at “half-octaves”. They are all the fifth note of an octave on a standard musical scale. So, if the octave starts at “C” (called the “root”), then the fifth is C-D-E-F … G.  Sketch the wave.  Mark the two nodes and three antinodes.  Excite f3 — you know how. Measure f3 and verify that the 3rd harmonic is a G.  Show that the note survives if damped at either node, and is killed if damped at either antinode.  Use an extra hand and Q-Tip and damp the string at BOTH nodes and pluck at the central antinode. Does the note sound any different? How so?

Harmonics: Plucking & Damping 4th Harmonic: Another octave, another “C” — and you heard this note before.  Sketch the wave.  Mark these specific nodes: L/4 and L/2 (where L = length) Mark these specific antinodes: L/8 and 3L/8  Excite f4. Measure f4 and verify that it’s a higher octave “C”.  Try the following specific sequence: (1) As a reminder, excite f2. (2) Now try to excite f4 by plucking at the L/8 antinode and damping at the L/2 node. What note do you hear instead? (3) You hear f2 but you should have also excited f4 ! Prove that you did by repeating the procedure, but this time damp at L/4 after the note starts. What note remains? Why did f2 disappear? Now you know that you can excite and kill certain combinations of harmonics.

Harmonics: Plucking & Damping 5th Harmonic:  Do all the usual stuff.  Try to make a combination of f3 and f5 the way you made a combination of f2 and f4. Explain with a sketch why you can’t do so. 6th, 7th & 8th Harmonics: Do all the usual stuff.  Play around a bit and see if you can make unusual notes, make notes disappear, etc.

Harmonics: Plucking & Damping Analysis: Recall from the Introduction that the string can only support frequencies that have nodes at the nut and saddle. So: L = n ✕ (ln / 2) for the nth harmonic.  Show that the frequency of the nth harmonic is given by: fn = n ✕ v / 2L where v is the speed of the wave on the string. Plot fn versus n. Use your plot of fn versus n, a linear fit, and the formula you derived to calculate the velocity of the wave.