Modeling an Elastic String

Investigating the Wave Equation

Colin Mitchell

We would now like to take a look at the function of the model, particularly, frequencies with respect to our two parameters, , the density of the string, and T, the tension of the string. Earlier, we defined . Let's look at the n-th mode of vibration, namely,

.

If we notice the trigonometric identity

,

we see that we can use it to reduce the harmonic to

Here, and are arbitrary constants that could be defined, but it is not important to where we are going. If we have the solution in this form, we have one trigonometric function in x, and one trigonometric function in t. Since the frequency we seek is the cycle of transverse vibrations in the string over time, we will look at the time function. Doing this, we notice that the frequency of the n-th mode is

Using a known fundamental frequency (), which is used to tune stringed musical instruments, we can find the necessary tension necessary to tune a string with a given density. Solving for T, we get

Now, we should take care of the physical units in this equation. We will measure in grams/meter, L as meters, and as Hertz (). Putting these into the equation, we get

This translates into gram-meters/second². One Newton of force is equal to , so our force will be measured in milliNewtons.

As an example, imagine we would like to tune a guitar string to concert A, which has a frequency of 440Hz. Also, we know that our guitar string has a density of 100gr/m, and that the string is 1m long. Then we have