Differential Equation Help - Vibrating String

[bunkergirl198]

Differential Equation Help -- Vibrating String

Homework Statement

Suppose a tightly stretched vibrating string has a variable density $$\rho$$(x). Assume that the vibration is small and is only in the vertical direction (transverse-vibration). Derive the PDE taking into consideration the gravity and the frictional force.

Homework Equations

Gravity = -mg
Friction force on an object moving with velocity v = -$$\beta$$v.

The Attempt at a Solution

Well. That's the hard part :)

u^tt + $$\frac{mg+Bv+T}{\rho(x)}$$ u_xx=f(x,t)

Initial conditons
u(x,0)=f(x)
u_t(x,0)=g(x)

Boundary conditions
u(0,t)=0
u(l,t)=0

I don't really think my differential equation is right, now do I know how to derive it. It was just my spin on an equation I came across a while ago (if I knew what it was I'd cite it.)

Thanks for ANY help :)

 

[Awatarn]

There are more than one way to tackle this problem.
First of all, you should find wave equation in string (without point mass). This equation will give you a general wave equation.
Second, consider motion of the point mass by using Newton's equation. There are three force relevant to the mass: frictional force, gravitational force, and string tension (on the left and right of the mass). For string tension, you may make approximation:
$$sin\theta \approx tan\theta \approx \frac{dy}{dx}$$
Solve these these equation with initial condition, their boundary conditions, and continuity of string.
You may further assume that wave is propagating from the left to right. Therefore there are two waves on the left of the mass (propagating to right and left) and only one wave on the right (propagating to the right, or transmission)

Hope it would help you