# Wave Equation Partial Differential Equation

Basically I got a simple wave equation with an extra twist. The PDE is $$\frac {\partial^2 y}{\partial t^2} = c^2\frac {\partial^2 y}{\partial x^2} + L$$ with homogeneous boundary condition. As usual, I use the ansatz $$Y(x,t) = F(x)G(t)$$ and I have $$\frac {\partial^2 y}{\partial t^2} =F''G$$ and $$\frac {\partial^2 y}{\partial x^2} =FG''$$. I substitute this in, and now I have $$FG''= c^2F''G +L$$ I don't know how to make this separable--obviously if I divide both sides with $$FG$$ as usual, then I will have that annoying constant $$\frac {L}{FG}$$ and I don't know how to make it separable then.

Can anybody help me? I know I have to separate it somehow, but I don't know exactly how to.

{The actual problem is an inhomogeneous boundary condition type of problem, but I have found the steady state solution, so I just have to find the transient solution, thus the homogeneous boundary condition}

The actual question:

• PDE: $$\frac {\partial^2 y}{\partial t^2} = c^2\frac {\partial^2 y}{\partial x^2} + L,\quad 0 \leq x \leq J,\quad t>0$$ where $$L$$ is constant and $$c$$ is constant wave speed.
• Boundary Condition: $$u(0,t)=0$$, $$t>0$$ and $$u(J,t)=h$$, $$t>0$$
• Initial Condition: $$u(x,0) = 0$$, $$0 and $$\frac {\partial u}{\partial t} (x,0)=0$$, $$0

I did the steady state solution bit, and got it as $$w(x)= \frac{-Lx^2}{c} + \frac {Hx}{L} +\frac{GLx}{c}$$. So I modified the boundary condition and initial condition to get the transient part, and I got the boundary condition homogeneous. However when I try to solve it, the question above arises.

## 1 Answer

Let $y(x,t) = w(x,t) + \phi(x)$ where $\phi(x) = a x^2 + b x + c$. From this it is seen that $\phi'(x) = 2 a x + b$, $\phi''(x) = 2a$ and \begin{align} \partial_{t}^{2} w = c^2 \, \partial_{x}^{2} w + 2a c^2 + L. \end{align} In order to cancel the $L$ term let $2 a c^2 + L = 0$ which leads to \begin{align} \phi(x) = - \frac{L \, x^{2}}{2 \, c^{2}} + b \, x + c_{1}. \end{align} Now it is understood that the transformation \begin{align} y(x,t) = w(x,t) - \frac{L \, x^{2}}{2 \, c^{2}} + b \, x + c_{1} \end{align} will make the p.d.e more solvable without "complications".

How this applies to the original problem is the following: \begin{align} u_{tt} = c^{2} u_{xx} + L \hspace{5mm} u(0,t) = 0 , \, u(a,t) = h\\ u(x,0) = 0, \, u_{t}(x,0)= 0 \end{align} The solution form proposed is \begin{align} u(x,t) = w(x,t) - \frac{L \, x^{2}}{2 \, c^{2}} + b \, x + c_{1} \end{align} for which \begin{align} u(0,t) &= 0 = w(0,t) + c_{1} \\ u(a,t) &= h = w(a,t) - \frac{L \, a^{2}}{2 \, c^{2}} + b a + c_{1} \\ u(x,0) &= 0 = w(x,0) - \frac{L \, x(x-a)}{2 \, c^{2}} + \frac{h \, x}{a} = w(x,0) - \phi_{1}(x) \\ u_{t}(x,0) &= 0 = w_{t}(x,0) \end{align} From the first equation it is seen that $w(0,t) = 0$ and $c_{1} = 0$. From the second let $w(a,t) = 0$ to obtain \begin{align} b = \frac{h}{a} + \frac{L \, a}{2 \, c^{2}}. \end{align} From all of this it can now be said that the original differential set is transformed by \begin{align} u(x,t) &= w(x,t) - \frac{L \, x(x-a)}{2 \, c^{2}} + \frac{h \, x}{a} \end{align} into \begin{align} w_{tt} = c^{2} w_{xx} \hspace{5mm} w(0,t) = w(a,t) = 0 , w(x,0) = \phi_{1}(x), w_{t}(x,0) = 0 \end{align} where \begin{align} \phi_{1}(x) = \frac{L \, x(x-a)}{2 \, c^{2}} - \frac{h \, x}{a}. \end{align} The equation for $w(x,t)$ is then an easier equation to solve. Since the boundaries for $x$ are zero at each end it suggests a sine solution and can be stated as \begin{align} w(x,t) = \sum_{n=1}^{\infty} \left( A_{n} \cos\left( \frac{n \pi c t}{a} \right) + B_{n} \sin\left( \frac{n \pi c t}{a} \right) \right) \, \sin\left( \frac{n \pi x}{a} \right) \end{align}

Note: the boundary conditions $w(x,0) = w_{t}(x,0) = 0$ leads to $A_{1}$ and $B_{1}$ being zero. This suggests that $w_{t}(x,0) \neq 0$.

Returning to the original form the solution is of the form \begin{align} u(x,t) &= \frac{h \, x}{a} - \frac{L \, x(x-a)}{2 \, c^{2}} + \sum_{n=1}^{\infty} \left( A_{n} \cos\left( \frac{n \pi c t}{a} \right) + B_{n} \sin\left( \frac{n \pi c t}{a} \right) \right) \, \sin\left( \frac{n \pi x}{a} \right) \end{align}

• I'm very lost. So with the ϕ(x) known, do we just essentially do the w(x,t) part of the pde? Do we just ignore the L constant and then just plugging in it back once we got into the solution? – Skipe Apr 25 '15 at 18:11
• @user234395 yes. Since the p.d.e. depends on $x$ and $t$ the remaining $w(x,t)$ would satisfy $w_{tt} = c^2 w_{xx}$. – Leucippus Apr 25 '15 at 18:33
• So, when we do the w(x,t) pde part, does the boundary condition changes? I do kind of understand your method, but it is still quite fuzzy to me. Thanks a lot for your help by the way! – Skipe Apr 25 '15 at 19:56
• Or does the boundary condition changes--like do we subtract the 0 boundary condition with ϕ(0)=c? If it is, then it would be pretty complicated wouldn't it? I would have to do the steady state again and things like that? And also does the initial condition changes? – Skipe Apr 25 '15 at 21:18
• @Skipe edit the question/problem to include the boundary conditions of the problem and I'll use them to show how to utilize them. – Leucippus Apr 25 '15 at 22:47