Is the tension of the string same or not?

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SUMMARY

The discussion centers on the treatment of tension in strings as presented in Halliday's physics book. It establishes that while tension is considered constant along a stationary string, it varies within a wave due to the motion of different points. The tension is maximum at nodes and minimum at antinodes, which is crucial for understanding energy transfer in waves. The confusion arises from the application of equal tensions in deriving wave speed versus energy transfer, highlighting the need for clarity in these concepts.

PREREQUISITES
  • Understanding of wave mechanics and properties of waves
  • Familiarity with the concepts of nodes and antinodes in wave theory
  • Knowledge of tension in strings and its implications in physics
  • Basic grasp of energy transfer in mechanical systems
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  • Study the derivation of wave speed in strings using the wave equation
  • Explore the relationship between tension and energy in oscillating systems
  • Investigate the implications of varying tension in different wave types
  • Review examples of stationary versus dynamic systems in physics
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Students of physics, educators teaching wave mechanics, and anyone interested in the intricacies of tension in wave propagation and energy transfer.

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In halliday physics book when we are going to derive the speed or the equation of a wave we consider the tension of the string to be same along the string, but when we are going to derive the energy transferred by the string we consider the tension to be maximum in nodes and minimum in antinodes, what is the reason? Am I making some mistakes?
 
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The tension is constant in a stationary string. It varies inside the wave.
 
Every point in a wave is moving, so it is not stationary, as a result the tension is not same at different points. so why does halliday have considered the tension in a wave same for calculating the wave speed?
 
No, what I meant was that the still areas of the wire in front of and behind the wave are at a constant tension. You need to know this tension to know the tension in the wave (it is additive).
 
Thanx, but it seems that I couldn't explain what I meant. Let me explain with attached picture (pls. see it). In halliday book the 1st picture is used for deriving the speed of a wave, he has considered the specified two tensions (which are shown with F) to be equal.
The 2nd picture has been used for deriving the general equation of a wave; Halliday has considered the specified two tensions (which are shown with F) to be equal, just like the previous part.
Well, here (3rd picture) the problem comes in; Halliday has explained that the tension in part one is not equal with the tension in part two, as a result the potential energies differ from each other.
Now I have mixed up completely, really I don’t know where the tensions are equal and where they aren't.
 

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