How Do Nodes Form on a Vibrating String for the nth Harmonic?

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Homework Help Overview

The discussion revolves around the formation of nodes on a vibrating string fixed at both ends, specifically for the nth harmonic. Participants are exploring mathematical and conceptual approaches to understand the relationship between harmonic frequencies, wavelengths, and the number of nodes.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are considering the properties of harmonic frequencies and wavelengths, and how these relate to the length of the string. There is a suggestion to use the sine function to describe the shape of the string under given boundary conditions. Some are seeking hints on how to proceed with the mathematical proof.

Discussion Status

The discussion is ongoing, with participants providing insights into the mathematical relationships involved. There is an emphasis on exploring the implications of wavelength in relation to the string's length. No consensus has been reached, but several lines of reasoning are being examined.

Contextual Notes

Participants are working within the constraints of proving the number of nodes for the nth harmonic, and there is a request for hints rather than complete solutions. The original poster has provided a diagram for visual reference, but the focus remains on mathematical justification.

MyNewPony
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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?
 
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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.
 
Think of the properties of the sine function (which is the shape of the string with the given boundary conditions.)
 
Borek said:
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?
 

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