PDEs: Inhomogenous wave equation question help.

Marc271

New member
Joined
Apr 29, 2013
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1
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^2 u/d t^2 = c^2(d^2 u/d x^2) + 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

It wants me to find the solution and the steady-state solution as t tends to infinity (It says by setting [d^2 u/d t^2] = 0).

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, and thanks in advance.
 

nasi112

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Joined
Aug 23, 2020
Messages
516
haha:ROFLMAO:

This post will complete its 9 anniversary after a few months, and not a single professor said Hi Marc, or even said show us your thoughts Marc.

Hey Marc, if you are still over there, listen to me. The probability that you will see this post again is 0.0001%.

But because a lot of students face this type of question frequently, I will share the magical touches to handle such a problem. So for Marc and other students who are studying PDE's, pay a good attention to my steps.

The idea to solve this problem is to split the solution into two parts,

\(\displaystyle u(t,x) = w(t,x) + v(x)\)

where \(\displaystyle v(x)\) will take care of the nonhomogeneous parts (G and H).

Now insert the above equation into the original PDE, and you will get,

\(\displaystyle w_{tt} = c^2 w_{xx} + c^2 v_{xx} + G\)

At this point, you want \(\displaystyle c^2 v_{xx} + G = 0\), to make \(\displaystyle w(t,x)\) homogeneous.

Now you have a homogeneous equation that you are used to solve.

\(\displaystyle w_{tt} = c^2 w_{xx}\) and another equation that you can solve easily \(\displaystyle c^2 v_{xx} + G = 0\).

All remain is to satisfy the boundary and initial conditions.

When \(\displaystyle x = 0\),
\(\displaystyle u(t,0) = w(t,0) + v(0) = 0\), and we want \(\displaystyle v(0) = 0\).

When \(\displaystyle x = L\),
\(\displaystyle u(t,L) = w(t,L) + v(L) = H\), and we want \(\displaystyle v(L) = H\).

When \(\displaystyle t = 0\),
\(\displaystyle u(0,x) = w(0,x) + v(x) = 0\), and we have \(\displaystyle w(0,x) = -v(x)\).
\(\displaystyle u_t(0,x) = w_t(0,x) = 0\).

After we satisfied all boundary and initial conditions, we got two easy equations to solve with complete boundary and initial conditions.

\(\displaystyle w_{tt} = c^2 w_{xx}\)
\(\displaystyle w(t,0) = 0\)
\(\displaystyle w(t,L) = 0\)
\(\displaystyle w(0,x) = -v(x)\).
\(\displaystyle w_t(0,x) = 0\).

And

\(\displaystyle c^2 v_{xx} + G = 0\)
\(\displaystyle v(0) = 0\)
\(\displaystyle v(L) = H\)
 
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