RobertPaulson
New member
- Joined
- Jan 9, 2010
- Messages
- 32
I can normally do these problems but this particular one has a constant that I don't really know how to deal with.
It's quite a long question and I think you'll need all the info so I apologise in advance.
Consider the problem of a string being held at both ends and acted on by the force of gravity. Take the coordinate x to be along the string with the ends of the string at x=0 and x=L. Take t as time and the variable u(t,x) as the vertical position of the string. The equation governing the string motion is the wave equation:
[note all d's in this example are partial differentials]
d[sup:3br02eqd]2[/sup:3br02eqd]u/dt[sup:3br02eqd]2[/sup:3br02eqd] = c[sup:3br02eqd]2[/sup:3br02eqd](d[sup:3br02eqd]2[/sup:3br02eqd]u/dx[sup:3br02eqd]2[/sup:3br02eqd]) + G, x is less than or equal to L and greater than or equal to 0, t>0
Where G is a constant due to gravity and c is the constant wave speed. The ends of the string are held at different heights so that the boundary conditions are:
u(t,x)=0, at x=0, t>0
u(t,x)=H, at x=L, t>0
and the string is initially held completely flat and stationary
u(t,x)=0, at t=0, 0<x<L
du/dt =0, at t=0, 0<x<L
Find the solution u(t,x) using separation of variables.
I would usually approach this problem using the ansatz and eigenvalues but the constant G is throwing me off and I don't really know how to approach it. I was given a hint to rewrite the problem in a homogenous form but I haven't had any luck doing that either.
Hope you can help
It's quite a long question and I think you'll need all the info so I apologise in advance.
Consider the problem of a string being held at both ends and acted on by the force of gravity. Take the coordinate x to be along the string with the ends of the string at x=0 and x=L. Take t as time and the variable u(t,x) as the vertical position of the string. The equation governing the string motion is the wave equation:
[note all d's in this example are partial differentials]
d[sup:3br02eqd]2[/sup:3br02eqd]u/dt[sup:3br02eqd]2[/sup:3br02eqd] = c[sup:3br02eqd]2[/sup:3br02eqd](d[sup:3br02eqd]2[/sup:3br02eqd]u/dx[sup:3br02eqd]2[/sup:3br02eqd]) + G, x is less than or equal to L and greater than or equal to 0, t>0
Where G is a constant due to gravity and c is the constant wave speed. The ends of the string are held at different heights so that the boundary conditions are:
u(t,x)=0, at x=0, t>0
u(t,x)=H, at x=L, t>0
and the string is initially held completely flat and stationary
u(t,x)=0, at t=0, 0<x<L
du/dt =0, at t=0, 0<x<L
Find the solution u(t,x) using separation of variables.
I would usually approach this problem using the ansatz and eigenvalues but the constant G is throwing me off and I don't really know how to approach it. I was given a hint to rewrite the problem in a homogenous form but I haven't had any luck doing that either.
Hope you can help