Nodes on a Vibrating String

[MyNewPony]

Prove that there are n-1 nodes on a string fixed at both ends for the nth harmonic.

It is simple to show this using a diagram.

[PLAIN]http://www.space-matters.info/img/nodesandmodes.jpg

However, is there a way to show this mathematically?

 

[Borek]

You probably have to go through the harmonic frequencies and harmonic wavelengths - once you have wavelength and you know how it depends on the initial length of the string, the rest should be obvious.

 

[Thaakisfox]

Think of the properties of the sine function (which is the shape of the string with the given boundary conditions.)

 

[MyNewPony]

You probably have to go through the harmonic frequencies and harmonic wavelengths - once you have wavelength and you know how it depends on the initial length of the string, the rest should be obvious.

Wavelength = (2/n)*length of string

Can I get a hint on what to do next?

 

[rwisz]

Yes, there is a mathematical proof for the number of nodes on a vibrating string fixed at both ends for the nth harmonic.

First, we need to understand that a harmonic is a multiple of the fundamental frequency of the string. For example, the second harmonic is twice the frequency of the fundamental, the third harmonic is three times the frequency, and so on.

Now, for a string fixed at both ends, the fundamental frequency is determined by the length of the string and the tension applied. This is known as the first harmonic or the fundamental mode.

For the nth harmonic, the frequency can be expressed as n times the fundamental frequency. Mathematically, this can be written as:

f_n = n*f_1

Since the number of nodes on a string is directly related to its frequency, we can use this equation to determine the number of nodes for the nth harmonic.

Let's take the example of the second harmonic, which has a frequency of 2*f_1. Using the equation above, we can see that there are two nodes on the string. Similarly, for the third harmonic, which has a frequency of 3*f_1, there are three nodes on the string.

Therefore, for the nth harmonic, there will be n nodes on the string. However, since the string is fixed at both ends, the endpoints will not be considered as nodes. This means that the total number of nodes will be n-1.

In conclusion, the mathematical proof for the number of nodes on a vibrating string fixed at both ends for the nth harmonic is n-1. This can be derived by understanding the relationship between frequency and number of nodes and using the equation f_n = n*f_1.