Question

Shortening a guitar string to one-third its initial length will change its natural frequency by what factor?

Asked by: elva b.

Answer

A guitar string, when stretched within its elastic limits, obeys the wave equation. A little analysis of the motion of elastic strings gives the equation:

v = sqrt(T/m)

Here, v is the speed of waves traveling on the string, T is the tension in the string, and m is the linear mass density (mass per unit length) of the string.

What we call the "natural frequency" in here is actually the lowest natural frequency of the string. Given f is the lowest natural frequency of the string, 2f, 3f, 4f, etc. are also natural resonant frequencies of the string, and these are called the "harmonics". This happens because a string has an infinite number of degrees of freedom.

The natural frequency of a string can be found by looking for a sinusoidal solution where the nodes coincide with the fixed ends of the string. This yields the equation:

f = sqrt(T/m) / 2L

Here, T and m are as before, and L is the length of the string. and f is the frequency of the vibration in Hertz (that is what you hear!).

So, we see that the natural frequency depends on three factors: How much tension you apply, whether or not they are Coated guitar strings, how "thick" and how long they are. It seems the original question is incomplete; we can easily assume the same linear mass density, but how about the tension? If we assume the tension is unchanged as well (thus making the question well-defined), the answer is now straightforward. Since length and frequency are clearly inversely proportional, if the length is shortened to one-third its initial length, the frequency will increase to three times its initial value.

Answered by: Yasar Safkan, Ph.D., Instructor, Yeditepe University, Istanbul, TURKEY

Search

Loading






Science Quote

'After a certain high level of technical skill is achieved, science and art tend to coalesce in esthetics, plasticity, and form. The greatest scientists are always artists as well.'

Albert Einstein
(1879-1955)

Top Selling

Here are our physics & astronomy bestsellers:
Magnetic Levitator - Classic
KonusScience 5 Way Microscope Kit
Periscope
Space Wonder Gyroscope
CHEM C2000 Chemistry SuperKit v2.0
Snap Circuits Jr.
Solar Radiometer
Mini Plasma Ball
Smart Robot 4M Kit
130 Electronics Projects Kit

Sponsors

USC University of Southern California Dornsife College Physics and Astronomy Department McMaster University Physics and Astronomy Department