The speed of a waves on a string in Simple harmonic motion

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Discussion Overview

The discussion centers on the speed of waves in simple harmonic motion on a string, specifically examining the role of tension in determining wave speed. Participants explore the implications of different interpretations of the force denoted as "F" in the wave speed equation, considering both horizontal and resultant forces.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the speed of a wave on a string is given by $$v= \sqrt{\frac{F}{\mu}}$$ and question whether "F" refers to the horizontal force or the resultant force.
  • Others clarify that "F" is the tension force in the string, but debate whether it should be considered in the x direction or as a tangent to the string.
  • Some argue that the distinction between horizontal tension and resultant tension is negligible for the approximation used in deriving the wave equation.
  • A participant notes that the vertical displacement amplitude is assumed to be small, suggesting that the tension in the string remains effectively constant during wave propagation.
  • Another participant provides a quantitative example, stating that for a typical steel string guitar, a small displacement results in a minimal change in tension, supporting the approximation that tension does not significantly vary.

Areas of Agreement / Disagreement

Participants generally agree that "F" represents the tension in the string, but there is disagreement regarding whether this tension should be considered in the horizontal direction or as a resultant force. The discussion remains unresolved regarding the implications of these interpretations on the wave speed equation.

Contextual Notes

Limitations include assumptions about small vertical displacements and the impact of tension changes on wave speed, which are not fully resolved in the discussion.

annamal
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The speed of a wave in simple harmonic motion on a string is $$v= \sqrt{\frac{F}{\mu}}$$ where v= the horizontal velocity of the wave on a string.
Is the F the horizontal force or the resultant force (combination of Fy and Fx)?
 
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annamal said:
The speed of a wave in simple harmonic motion on a string is $$v= \sqrt{\frac{F}{\mu}}$$ where v= the horizontal velocity of the wave on a string.
Is the F the horizontal force or the resultant force (combination of Fy and Fx)?
That "F" is the Tension force:

1663170680261.png

http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/string.html
 
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annamal said:
The speed of a wave in simple harmonic motion on a string is $$v= \sqrt{\frac{F}{\mu}}$$ where v= the horizontal velocity of the wave on a string.
Is the F the horizontal force or the resultant force (combination of Fy and Fx)?
It does not matter. To the approximation that is used to derive the wave equation, the difference is negligible.
 
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Orodruin said:
It does not matter. To the approximation that is used to derive the wave equation, the difference is negligible.
Ok, not sure how the tension in the x direction can be approximated as the resultant tension
 
annamal said:
Ok, not sure how the tension in the x direction can be approximated as the resultant tension
The vertical displacement amplitude is assumed to be small in those derivations, so the tension in the string is not affected (in magnitude) by the wave passing by. Does that help?
 
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For a typical steel string guitar a 2mm displacement at the midway point translates into an increase in string tension on the order of one part in one thousand. So it's a pretty good approximation to just assume that the tension doesn't change. Then you can focus on the vertical force and come up with a straightforward equation for the speed of the wave.
 

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