Explanation of the Equation for Sound from a String - Succeed in Physical Science. Also refer to physics, wire, vibration, tension, length, mass, wavelength, Ron Kurtus, School for Champions. Copyright © Restrictions
Equation for Sound from a String
by Ron Kurtus (8 January 2008)
When a string or wire is stretched between two posts and it plucked, it will vibrate and create a sound or musical note. The vibration of the string will create a fundamental frequency, according to the tension, length, and mass of the string. There is a general equation that calculates the frequency. Changing the parameters results in changing the frequency of the vibration and thus the sound.
Questions you may have include:
- What is the string frequency equation?
- How can you change the parameters?
- What are equations for solving for the various parameters?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
String frequency equation
The equation for the fundamental frequency of an ideal taut string is:
f = √(TL/m)/2L
where
- f is the frequency in Hertz (Hz)
- T is the string tension in Newtons (N)
- L is the length of the string in meters (m)
- m is the mass of the string in kilograms (kg)
- √(TL/m) is the square root of T times L divided by m
Note: Typically, you would measure the string in centimeters, but since the tension is in Newtons, you have to convert centimeters to meters. Likewise, grams would be more appropriate for the mass of the string, but you have to convert grams to kilograms for this equation. If you are working with pounds, inches and mass-pounds, you would make the conversion to the metric or SI system of measurement.
Ideal string
The equation is actually an approximation for an ideal string. Factions such as elasticity and characteristics of the material are not really taken into account. Although they can affect the outcome, the string frequency equation is a good approximation.
Music
If the parameters of the string or wire—the length, tension and mass—are at certain values, the sound made from plucking the string will be a musical note that is pleasing to the ear. But if they are slightly different, the sound may not be musical and just be a sound.
Note that what is pleasing in one culture or nationality may not be considered musical in another culture.
Changing the parameters
If the frequency for a given string—and the resulting sound—is a specific value, and you change one parameter of the string but keep everything the same, the frequency will change accordingly. For example, if the frequency of he string is 500 Hz for a given configuration and if you:
- Double the tension of the string, the frequency goes up to 707 Hz
- Shorten the length of the string by 1/2, the frequency goes up to 707 Hz
- Reduce the mass of the string by 1/2, the frequency goes up to 707 Hz
This is determined by substituting the values in the f = √(TL/m)/2L equation, as follows:
√(2TL/m)/2L = 1.414√(TL/m)/2L = 1.414f = 1.414 * 500 Hz = 707 Hz
√(½TL/m)/½2L = 1.414f = 707 Hz
√(TL/½m)/2L = 707 Hz
(The math is tricky, so you must be careful doing it.)
You can change the mass of the string by keeping the same material and making the string thicker or thinner. Or you can keep the thickness the same and change the material. A metal wire would be heavier than an cotton string of the same thickness and length.
Solving for other parameters
You can solve for the other parameters by squaring each side of the equation—or multiplying each item by itself—and rearranging them. Squaring a square root, gets rid of the square root sign and just leaves the number or variable.
Squaring both sides of the equation f = √(TL/m)/2L, results in:
f² = (TL/m)/4L² or
f² = TL/4mL² = T/4mL
Find tension
Thus, if you know the mass of the string, the length and the frequency of the sound, you can find the tension of the string: T = 4mLf².
Find length
If you know the tension, the mass of the string and the frequency of the sound, you can find the length of the string: L = T/4mf².
Find mass
If you know the tension, the length of the string and the frequency of the sound, you can find the mass of the string: m = T/4Lf².
Summary
A stretched string or wire will vibrate and create a sound or musical note when plucked. The vibration will be a fundamental frequency, according to the tension, length, and mass of the string. There is a general equation that calculates the frequency. Changing the parameters results in changing the frequency of the vibration and thus the sound.
Create your own music
Resources
The following resources provide information on this subject:
Websites
Books
Top-rated books on Physical Science
Mini-quiz to check your understanding
1. Why isn't the equation f = √(TL/m)/2L exact?
2. If you increase the tension on the string 4 times, what happens to the fundamental frequency?
3. If you know the mass and length of the string, how do you determine the tension to achieve a given frequency?
If you got all three correct, you are on your way to becoming a Champion in Physical Science. If you had problems, you had better look over the material again.
What do you think?
Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. We will try to get back to you as soon as possible.
Share link
Feel free to establish a link from your website to pages in this site.
Or use our form to send this link to yourself or a friend.
Students and researchers
The Web address of this page is
www.school-for-champions.com/science/sound_string_equation.htm.
Please include it as a reference in your report, document, or thesis.
Where can you go from here?
Equation for Sound from a String