As for the title:Amplitude of Transmitted Wave in a Two-Part String

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Discussion Overview

The discussion revolves around the amplitude of a wave transmitted from a dense part of a string to a light part, focusing on the effects of differing mass densities. The scope includes theoretical considerations and mathematical reasoning related to wave behavior at the boundary between two media with contrasting properties.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant notes that the amplitude of the transmitted wave will depend on the mass densities of the two parts and suggests that solving a double wave equation system is necessary to determine the exact amplitude ratio.
  • Another participant expresses interest in the limiting case where the ratio of the densities is very large, implying that a simpler solution might exist without resorting to differential equations.
  • A different viewpoint suggests that the amplitude may not change, but the frequency will, questioning the relationship between energy and frequency in this context.
  • A subsequent reply clarifies that the energy-frequency relationship referenced is not applicable to this problem, indicating that the scenario resembles a wave equation with a free end boundary condition.

Areas of Agreement / Disagreement

Participants express differing views on whether the amplitude of the transmitted wave changes, and there is no consensus on the implications of frequency changes or the necessity of solving differential equations.

Contextual Notes

There are unresolved assumptions regarding the specific values of the mass densities and the boundary conditions applied in the wave equations. The discussion also reflects varying interpretations of energy relationships in the context of wave transmission.

salsero
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A string has two parts: one with a very high mass density (per unit length), and the other with a very low mass density. A wave with amplitude A moves from the dense part toward the light part. What will be the amplitude of the wave which is transmitted to the light part?
 
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This will depend on the different densities of the strings. There will be some transmission and some reflected energy. To get the exact ratio it would be necessary to solve a double wave equation system. with a boundary condition at connection point be that their displacement be the same for all time.

This problem may be solved is some textbooks. If you have access to a university liberary you may want to search the Partial Differential Equation text for a solution.
 
I am interested in the limit where the ratio of the densities of the heavy part and the light part is very large (infinite).

Probably the answer to this limiting case can be found without solving differential equations...
 
I may be wrong but I don't think the amplitude will change, but the frequency between the two part will. Isn't energy a product of frequency? E=hv
 
Originally posted by birdus
I may be wrong but I don't think the amplitude will change, but the frequency between the two part will. Isn't energy a product of frequency? E=hv

This is not a Quantum Mechanics problem, hν has nothing to do with it.

In the limit you speak of the result would be the same as a wave equation with a free end boundary condtion. This should be a easy solution to find in most PDE texts.
 

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