standing waves on a string and music

1. Standing waves and music
The harmonics of a guitar string is the natural frequencies at which it naturally vibrates. A
standing wave pattern can be seen in each harmonic played with a guitar.
These frequencies depend on the tension, linear density, and length of the string:
 Frequency=wave speed/wavelength
 Wave speed=sqrt(tension/linear density)
The four lowest harmonics of a guitar sting are shown below:
Two nodes can be seen at each end of the fixed ends string of each harmonic. These nodes
represent stationary points. While the antinodes represent points of greatest motion at the
maximum amplitude. These antinodes occur midway between the nodes. Each pattern has a
different number of nodes and antinodes. A guitarist changes the position of the end node
through frets on a guitar. By doing so, the length of the vibrating string changes and therefore
the note played.
Equations:

2. The wavelength of a harmonic equals 2L/m, where m is the mode number (m=1,2,3,..). The
frequency equals the wave speed/wavelength. Therefore, the speed of a standing wave on a
guitar string equals 2L/n *f.
Example:
The second string on a guitar corresponds to B which has a first harmonic of 247Hz and is 55.0
cm in length. Suppose you press down on the first fret so that the string is shortened and plays
a C note with frequency of 267Hz. Calculate how far the fret is from the fixed end of the string.
Solution:
Using the Equation derived above we can calculate the speed of the wave on the string using
the first harmonic frequency where m=1:
v=2L/m*f=2(0.55m)/1*(247Hz)=271.7m/s
Since the tension remains the same, the wave speed will not change and therefore we can use
it to calculate the new length:
L=mv/2f=(1)(271m/s)/2(267Hz)=0.5088m
Finally, the difference of 0.55-0.5088=0.0412m=4.12cm is the distance from the fret to the fixed
end of the string.