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Following from the hanging rope model here, there was a question I was doing in which the rope is jerked from the bottom , and from the relation between velocity, tension and mass density, we get velocity as:

$$ v(x) = \sqrt{ \frac{T(x)}{\mu}}$$

And using relation, $ T = \frac{Mgx}{L}$,

$$ v \propto \sqrt{x}$$

So, it turns out that as you go up the rope, the velocity of the pulse speeds up.. but why? How can we understand this result intuitively? By the way I know that equation in premise is an approximation (I'm asking how do we understand why the equation is true in the approximated form)

ZeroTheHero
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tryst with freedom
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    It seems like your actual question is why intuetively waves travel faster when the tension is higher, because you do understand why the lower part of the rope feels less tension than the upper part. Did I understand correctly? – Ofek Gillon Feb 23 '21 at 11:17
  • Well yes at bare bones , but I"m trying to understand what speeds it up as it moves up the rope? @OfekGillon – tryst with freedom Feb 23 '21 at 11:18
  • The fact that there is higher tension. Its like asking why light speeds up when emergen from glass to air - nothing accelerates it, the perturbation in the field just travels faster. – Ofek Gillon Feb 23 '21 at 11:29
  • Right, I meant 'how' rather than 'what', my bad. The second one seems to have some complicated answers here :P – tryst with freedom Feb 23 '21 at 11:32

1 Answers1

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The tension is due to the weight of portion of the rope below the point $x$ so as you go up there’s a greater portion of the rope and thus a greater portion of the mass of the rope to pull the little portion of the rope at $x$. Note that since the tension is “local” (it changes at every $x$ rather than being constant throughout), you also need the “local” mass parameter $\mu$ (the linear mass density) rather than the total mass of the rope.

The rest is thus a question of understanding how the tension (here, the local tension) enters in the velocity: the tension is basically the restoring force so greater tension means greater restoring force.

BioPhysicist
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ZeroTheHero
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  • I don't get how the tension being higher means wave speed is higher, the rest I got – tryst with freedom Feb 24 '21 at 17:09
  • The component of the tension perpendicular to the rope is the restoring force thus greater tension means greater restoring force. – ZeroTheHero Feb 24 '21 at 17:43
  • hmm, if I am not mistaken the tension acts along the length of rope / curve right? I don't understand how there is component perpendicular to the length – tryst with freedom Feb 24 '21 at 17:45
  • @Buraian I think Zero means the component perpendicular to the wave propagation direction – BioPhysicist Feb 24 '21 at 18:45
  • For transverse oscillations it is the perpendicular components that is the origin of the restoring force. See Frank Crawford, Berkeley Physics Course vol. 3 Chapter 2. – ZeroTheHero Feb 24 '21 at 19:32