How is equation 15.14 derived in the wave speed derivation?

[Fluxthroughme]

With reference to this diagram, my textbook tells me the following:

I am able to follow the rest of the derivation after this point, but I do not understand where equation 15.14 came from? I'm trying to think in terms of small angle approximations, but nothing is really coming of that. Any help is appreciated.

Thanks.

 

[tms]

The equation just says mathematically what the words say.

 

[Fluxthroughme]

The equation just says mathematically what the words say.

I know what the equation says. I don't know where it came from, I.E, I don't know why it is true.

 

[SteamKing]

The rope can only experience tension. Therefore, the slope of the rope at any point must also coincide with the direction of the tension force in the rope.

 

[Nickie]

Equation 15.14 is derived by applying the small angle approximation to the trigonometric functions in the equation. In this case, we are dealing with a transverse wave propagating along a string, and the small angle approximation allows us to simplify the trigonometric functions involved in the derivation. The equation is derived by considering the forces acting on a small section of the string and using Newton's second law to equate the sum of these forces to the mass of the section times its acceleration. This leads to the equation 15.14, which relates the wave speed to the tension and mass per unit length of the string. The small angle approximation is necessary because it allows us to simplify the equations and make them easier to solve, while still providing a good approximation of the actual values.