How to Determine the Displacement of a Stretched String?

[tigger88]

Homework Statement

A stretched string has length L and is attaached to a rigid support at either end. The string displacement from equilibrium at t=0 is given by:
y(x,0) = sin(3(pi)x/L)

and the velocity of the string is:

y'(x,0) = (3pi/L)sqrt(T/rho)sin(3(pi)x/L)
where T = tension in the string, and rho = its mass per unit length
and sqrt(T/rho) is the wave velocity of transverse waves on the string.

Give the form of the displacement y for all x, t.

Homework Equations

(This is kind of where I'm stuck). I assumed the general equation of a wave was y = sin(kx)cos(wt), but that didn't work.
I then tried with y = sin(kx-wt), but that also didn't seem to work.

The Attempt at a Solution

I tried to use the general equation for a wave, differentiate it wrt t, plug in t=0 and compare it to the given velocity equation, but whenever I try I end up with a cos term instead of sine. I really can't figure out how it's a sine term in both the displacement and velocity.

If I used y = sin(kx)sin(wt), then at t=0, y=0.
And if I used y = sin(kx)cos(wt), then dy/dt = -wsin(kx)sin(wt), and at t=0, dy/dt = 0.
If I used y = sin(kx-wt), then dy/dt = wcos(kx-wt), which doesn't have the necessary sine term.

Any ideas?

 

[jambaugh]

You should take as your general solutions both sine an cosine terms:
y= sin(kx)[A sin(wt) +B cos(wt)] terms. Or equivalently work with a phase shift sin(kx)sin(wt+p).