Equation of velocity on transverse wave

In summary: This force is equal and opposite to the force on the same segment (Δx) of the next segment, and this in turn is the gradient of the tension force associated with the next segment, so the force on the next segment is T(d2y/dx2)Δx. The net force on the next segment is therefore T(d2y/dx2)Δx-T(d2y/dx2)Δx, which is zero.In summary, the equation for transverse wave velocity is derived by taking the limit as the length of the string goes to 0 and considering the net force acting on the string. This leads to an equation with a second derivative of y with respect to x
  • #1
Joza
139
0
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) :redface:

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.
 
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  • #2
Joza said:
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.
 

1. What is the equation for velocity on a transverse wave?

The equation for velocity on a transverse wave is v = λf, where v represents the velocity, λ represents the wavelength, and f represents the frequency.

2. How does wavelength affect the velocity of a transverse wave?

The velocity of a transverse wave is directly proportional to the wavelength. As the wavelength increases, the velocity also increases.

3. Is frequency a factor in the equation for velocity on a transverse wave?

Yes, frequency is a factor in the equation for velocity on a transverse wave. The higher the frequency, the higher the velocity of the wave.

4. Can the equation for velocity on a transverse wave be used for all types of waves?

No, the equation v = λf is specifically for transverse waves. Different types of waves, such as longitudinal or surface waves, have their own equations for velocity.

5. How is the equation for velocity on a transverse wave derived?

The equation v = λf is derived from the relationship between velocity, wavelength, and frequency. It is based on the properties of waves, such as their speed and the distance they travel in a given time period.

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