Equation for Sound Created from a String
by Ron Kurtus
When a string or wire is stretched between two posts and is plucked, it will vibrate and create a sound or musical note. The vibration of the string will create a fundamental frequency, which has its nodes at the end points. Harmonics, with nodes in regular positions along the length of the string, are also possible.
There is a general equation or formula that calculates the fundamental frequency, according to the tension, length, and mass of the string. Changing the various parameters result in changing the frequency of the vibration and thus the sound. You can also rearrange the equation to solve for the parameters.
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. Useful tool: Units Conversion
String frequency equation
The equation for the fundamental frequency of an ideal taut string is:
f = (1/2L)*√(T/μ)
- f is the frequency in hertz (Hz) or cycles per second
- T is the string tension in gm-cm/s²
- L is the length of the string in centimeters (cm)
- μ is the linear density or mass per unit length of the string in gm/cm
- √(T/μ) is the square root of T divided by μ in seconds
Note: Typically, tension would be in newtons, length in meters and linear density in kg/m, but those units are inconvenient for calculations with strings. Thus, the smaller units are used.
Linear density is the mass per unit length: μ = m/L, where m is the mass of the string or wire in gm.
The reason μ is used instead of m/L is because when you use the equation to determine the frequency for a string of a different length, you must also adjust the mass to correspond to the different length. The situation where you change the length but keep the mass constant is seldom used.
The equation is actually an approximation for an ideal one-dimension string. Factors such as elasticity, material characteristics and diameter of the string or wire are not taken into account.
(See Equation for Sound Created from a Wire for more information on that subject.)
The equation holds well if the amplitude of the string is small. The equation falls apart for strings plucked too vigorously.
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.
Examples of 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 else the same, the frequency will change accordingly.
Doubling the tension
If you double the tension, the frequency will go up 1.414 times the original frequency, provided all other parameters remain the same. Consider the frequency for a given configuration:
If T is doubled, then the new frequency f2 is:
You can take the square root of 2 to get:
f2= 1.414*(1/2L)*√(T/μ) = 1.414f1
Thus, if the frequency of the string is 500 Hz for a given configuration, and you double the tension of the string, the frequency goes up to 707 Hz.
f2 = 1.414f1= 1.414*500 Hz = 707 Hz
Shortening the length
If you shorten the length of the string by 1/2, while keeping all the other parameters constant, the frequency also goes up 2 times the original frequency.
f1 = (1/2L)*√(T/μ)
Replace L with L/2:
Again, if f1 = 400 Hz, then f2= 800 Hz.
Note: Since reducing the length by 1/2 also reduces the mass by 1/2, μ remains constant.
Reduce the mass
If you change the material of the string, reducing its mass by 1/2 but keeping the same length, the new frequency is 1.414 times the original frequency.
Substitute μ = m/L in f1 = (1/2L)*√(T/μ):
f1 = (1/2L)*√(TL/m)
Keeping other parameters constant, replace m with ½m:
f2 = (1/2L)*√(2TL/m)
f2 = 1.414f1
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 = (1/2L)*√(T/μ), results in:
f² = (T/μ)/4L² or
f² = T/4μL²
Thus, if you know the frequency of the sound and the mass and length of the string, you can find the tension of the string:
T = 4μL²f²
If you know the tension and mass of the string and the frequency of the sound, you can find the length of the string:
L² = T/4μf²
L = (1/2f)*√(T/μ)
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
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 result in changing the frequency of the vibration and thus the sound.
Create your own music
Resources and references
Vibrating String - Hyperphysics
Vibration of Stretched Strings - TutorVista.com
Vibrating string - Wikipedia
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Equation for Sound Created from a String