What is the velocity and direction of a transverse wave on a string?

[dwn]

Homework Statement

A transverse wave travels on a string whose mass density is

 kg/m.
The displacement (in meters) is given as a function of time and position as:

    D(x,t) = 1.5sin(3x-24t+90°)


What is the velocity and direction of the wave?
What is the speed of the particle located at x=8 and t=1?

Homework Equations

Derivatives of the former.

$$∂^2D/∂t^2=v^2(∂^2D/∂x^2)$$

The Attempt at a Solution

I took the partial derivatives with respect to both x and t. Applied them to the wave equation. But I don't understand how I'm supposed to get any figures to answer the questions above.

$$v=sqrt(-864sin(-24t)/(-13.5sin(3x))$$ ----> not sure what I do after this point.

 

[SteamKing]

What is the definition of velocity in terms of the position function?

 

[PhysicsandSuch]

I'm not entirely sure the question makes sense. Perhaps a) was looking for the speed and direction of the propagating wave?

In this case you could use the wave equation you mentioned to work out the speed. As far as direction goes, it's a one dimensional transverse wave without boundary conditions, so you just have to state along which axis it's headed.

For b), a particle located on the string would have a motion described by the displacement function directly, so simply differentiating that expression with respect to time and evaluating at the point given should yield the particle speed.

I would double check your math on those partial derivatives though: even without evaluating at any point, all of your trig functions should vanish and leave you with a speed based on the coefficients you pull out using the chain rule.