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## Equation for Sound Created from a Wire

by Ron Kurtus (revised 28 July 2012)

A wire that is stretched between two posts and is plucked will vibrate and create a sound or musical note. The vibration of the wire will create a fundamental frequency, which has its nodes at the end points.

The equation or formula to find the frequency of the sound as a function of the wire tension, length, diameter and density of the material, is based on the *Equation for Sound Created from a 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 wire 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

## Wire frequency equation

The equation for the fundamental frequency of a taut wire as a function of tension, length, diameter and density of the wire material is:

f = (1/Ld)*√(T/πδ)

where

**f**is the frequency in hertz (Hz) or cycles per second**L**is the length of the wire in centimeters (cm)**d**is the diameter of the wire in cm**T**is the tension on the wire in gm-cm/s²**π**is the Greek letter pi = 3.14**δ**is the density of the wire in gm/cm³ (Greek letter small delta)**√(T/πδ)**is the square root of**T**divided by**π****δ**

### Derivation

The way this equation was derived is by starting with the general string equation:

f = (1/2L)*√(T/μ)(

See Equation for Sound Created from a String for more information on the subject.)

Since the linear density **μ = m/L**, the equation can be rewritten as:

f = (1/2L)*√(TL/m)

where **m** is the mass of the string in grams (gm).

The mass of the wire can be written in terms of its density and volume:

m = δV

where **V** is the volume of the wire in cm³.

The volume is the length times the area of the cross section of the wire:

V = Lπd²/4

where **πd²/4** is the area of the cross section in cm².

Substituting **δV** for **m** in **f = (1/2L)*√(TL/m)**, you get:

f = (1/2L)*√(TL/δV)

Substituting **Lπd²/4** for **V**, you get:

f = (1/2L)*√(4TL/δLπd²)

Simplify to get:

f = (1/Ld)*√(T/πδ)

### Approximation

This equation is an approximation for an ideal one-dimension string. Adding the factors of diameter and density can reduce the accuracy a slight amount, provided the diameter is very small compared to the length.

The equation typically only applies to wires, because the density of a string or non-metal material has too much variation. You can select a wire material—such as copper—and find its density, while you seldom can find the density of a string.

## Examples of changing the parameters

If the frequency from a given wire configuration has a specific value, you change one parameter of the wire, keeping everything else the same, to change the frequency. This can be stated as a percentage or proportional change.

### Double the diameter

Changing the diameter of the wire changes its fundamental frequency. This can be seen on a guitar or piano, where the thicker wires have a lower frequency.

Let **f _{1}** be the initial frequency of the wire:

f_{1}= (1/Ld)*√(T/πδ)

Keeping everything else the same, double the diameter **d** to become **2d**:

f_{2}= (1/L2d)*√(T/πδ)

f=_{2}f_{1}/2

Thus, if **f _{1}** = 800 Hz, doubling the diameter of the wire will result in

**f**= 400 Hz.

_{2}### Changing the material

If you change the material of the wire, keeping everything else the same, you can compare the effect of the different densities. For example, the density of copper wire is 8.94 gm/cm³, while the density of steel wire is 7.8 gm/cm³.

Let **f _{C}** be the frequency of the copper wire and

**f**be the frequency of the steel wire.

_{S}The frequency for a given copper wire is:

f8.94_{C}= (1/Ld)*√(T/π)

f= 0.334_{C}*(1/Ld)*√(T/π)

Keeping everything else the same except the material, the frequency for a similar steel wire is:

f7.8_{S}= (1/Ld)*√(T/π)

f= 0.358_{S}*(1/Ld)*√(T/π)

Comparing the two frequencies, you get:

f= (0.334/0.358)_{C}f_{S}

f= 0.933_{C}f_{S}

In other words, changing the wire from copper to the less-dense steel will result in a frequency that is 0.933 of the original. Thus, if the frequency of the steel wire was 800 Hz, the frequency of the copper wire will be 746 Hz.

## Solving for other parameters

You can solve for the other parameters by squaring each side of the equation and rearranging them.

f = (1/Ld)*√(T/πδ)

f² = (1/L²d²)*(T/πδ)

Rearrange the variables:

f² =(T/πL²d²δ)

### Solve for tension

You can rearrange the equation to solve for the tension:

T = πf²L²d²δ

Thus, if you know the other factors, you can find the tension on the wire.

### Solve for length

You can rearrange the equation and take the square root of each side of the equation to solve for the length:

L² = T/f²d²πδ

L = √(T/f²d²πδ)

Simplify by taking the square root of **1/f²d²**:

L = (1/fd )*√(T/πδ)

Thus, if you know the other factors, you can find the length of the wire.

### Solve for diameter

You can rearrange the equation and take the square root of each side of the equation to solve for the diameter of the wire:

d² = T/f²L²πδ

d = √(T/f²L²πδ)

d =(1/fL)*√(T/πδ)

Thus, if you know the other factors, you can find the diameter of the wire.

### Solve for density

You can rearrange the equation to solve for the density of the wire:

δ = T/f²L²d²π

Thus, if you know the other factors, you can find the density of the wire and thus its material.

## Summary

A stretched wire that is between two posts and is plucked will vibrate at a fundamental frequency. The equation or formula for the frequency of the sound as a function of the wire tension, length, diameter and density of the material is **
f = (1/Ld)*√(T/πδ)**.

Changing the various parameters result in changing the frequency of the vibration. You can also rearrange the equation to solve for the various parameters.

Strive for success

## Resources and references

### Websites

**Vibrating String** - Hyperphysics

**Vibration of Stretched Strings** - TutorVista.com

**Vibrating string** - Wikipedia

### Books

**Top-rated books on Sound Waves**

## Questions and comments

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## Equation for Sound Created from a Wire