Equation of velocity on transverse wave

[Joza]

I am deriving the equation for a transverse wave velocity from the difference in the transverse forces acting on a string. ie. v=(F/Greek letter mu)^(1/2)

First of all, can I clear up that this refers to transverse velocity yes, and not phase velocity? (My book isn't clear).These are all partial derivatives by the way, so i presume it's all to do with transverse velocity and not phase since we keep x constant?


I can't really right the entire equation out, so I'll do my best.

So, I end up with an equation with (d^2y/dx^2) = (F/(mu)) (d^2y/dt^2)
and then you compare this to the wave equation.

I don't understand where the left side comes from. The limit as the length goes to 0 is taken of the net force acting on the string. But how do we end up with the second derivative (curvature of string) of y/x when we do this? What is the logic behind it?


Thank you guys!





Sorry for any mistakes, I don't have my book handy.

 

[haruspex]

don't understand where the left side comes from.

Consider two adjacent segments of string, length Δx, tension T. The components normal to the string of the forces this exerts at their junction are Tdy/dx, the slopes being a little different in each and the signs being opposite. The net force is therefore Td²y/dx²Δx.