String hanging from the ceiling

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The discussion centers on calculating the time it takes for a transverse pulse to travel along a string of length L and mass m, which is supporting an object of mass M. The relevant equation for the time interval is Δt=2√(L/(mg))(√(M+m)-√M). There is uncertainty about whether the waiting time is simply twice Δt. The origin of the equation is explained as being due to varying wave speeds at different points along the string. Understanding this concept is crucial for accurately determining the time for the reflected pulse to return.
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Homework Statement


A string of length L and mass m, hanging an object of mass M from the ceiling. After sending a transverse pulse from the top of the string , how much time we have to wait until the reflected pulse returned to the top?


Homework Equations


The time interval for a transverse pulse to travel the legth of the rope is ##Δt=2\sqrt{\frac{L}{mg}}(\sqrt{M+m}-\sqrt M)##.


The Attempt at a Solution


I think the time for waiting is just twice of Δt, but I still concern that maybe I miss something.

Sincerely.
 
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Where does that equation come from?
 
It is from that different points on the string have posses different wave speeds.
 

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