Normal modes of a string thought experiment

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SUMMARY

A standing wave on a stretched string occurs when the free end is moved at one of the string's normal-mode frequencies. If the end is moved at a different frequency, the resulting wave will be a moving wave that represents a superposition of the string's normal modes. The phenomenon can be understood through the concept of a "driven harmonic oscillator," where the system's behavior becomes complex. The discussion emphasizes the relationship between normal modes and traveling waves in string dynamics.

PREREQUISITES
  • Understanding of normal modes in string theory
  • Familiarity with harmonic oscillators
  • Knowledge of wave superposition principles
  • Basic concepts of wave mechanics
NEXT STEPS
  • Explore the mathematical derivation of normal modes for a string of length L
  • Study the principles of driven harmonic oscillators in physics
  • Investigate the superposition of waves and its applications in string dynamics
  • Learn about the complexities of real string behavior beyond ideal models
USEFUL FOR

Students and educators in physics, particularly those focusing on wave mechanics, string theory, and harmonic oscillators. This discussion is beneficial for anyone seeking to deepen their understanding of wave behavior in physical systems.

Nikitin
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Hey! So If I have a stretched string of length L fastened at one end, and I am moving the other end sinusoidally, will a standing wave appear ONLY if I move the other end with one of the normal-mode frequencies of the string? If not, will the resulting wave be a moving wave which is a superposition of the string's normal modes?
 
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Do it and see :)

Shaking the string from one end is a bit different from plucking it.
You get a node a little distance x from your fingers - and you will have excited a normal mode of a string length L-x which is fixed at both ends. Otherwise you have to move your hand with the frequency of the normal modes of your string (work them out for a string length L with one end free).

When you drive a string like that - the system is a "driven harmonic oscillator" and the solutions can get quite complicated. You can represent the result as a superposition of normal modes - just as the normal modes can be represented as a superposition of traveling waves.

Real strings can get more complicated still.
 

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