I am trying to solve the wave equation with a method similar to the one explored in this question: Namely I would like to solve it using separation of variables and using the method of shifting the data. In this particular problem the boundary conditions are different and the parameters of the problem are modified to represent the propagation of an elastic wave in a bar (see reference):
GOVERNING EQUATION
$$\frac {\partial^2 u}{\partial x^2} =\frac 1 c^2 \frac {\partial^2 u}{\partial t^2}$$ Where $c = \sqrt{\frac {E} \rho}$ and $\sigma = E \frac{\partial u} {\partial x}$
INITIAL CONDITIONS
$$u(x,0) = 0$$ $$u_t(x,0) = 0$$
BOUNDARY CONDITIONS $$u(0,t) = u_t(0,t) = 0$$ $$E u_x (L,t) = \sin t \Rightarrow u_x(L,t) = \frac {\sin t} E $$
ATTEMPT TO SHIFT DATA
According to Strauss - Partial differential equations: an introduction:
Boundary conditions can always be made homogeneous by subtracting any known function satisfying the same BCs.
In other words for a given $v(x,t) = u(x,t) - U(x,t)$, with $U$ satisfying the boundary conditions, an equivalent problem emerges $v_{xx}(x,t) = \frac 1 {c^2} v_{tt}(x,t)$ which can be solved for $v$ and then $v$ can be used to find $u$.
Therefore I have elected to use as my shifting function $U(x,t) = \frac {x^2 \sin t} {EL}$. It can be shown that this function satisfies the boundary conditions.
In this case $$v(x,t) = u(x,t)-U(x,t) = u(x,t)-\frac {x^2 \sin t} {EL}$$
and
$$u(x,t) = v(x,t)+\frac {x^2 \sin t}{EL} $$
Obtaining partial derivatives for $v$:
$$v(x,t) = u(x,t)-\frac {x^2 \sin t} {EL}$$
$$v_x(x,t) = u_x(x,t)-\frac{2 x}{E L} \sin{\left (t \right )}\qquad \qquad v_t(x,t) = u_t(x,t) - \frac{x^{2}}{E L} \cos{\left (t \right )}$$
$$v_{xx}(x,t) = u_{xx}(x,t) - \frac{2}{E L} \sin{\left (t \right )} \qquad \qquad v_{tt}(x,t) = u_{tt}(x,t) + \frac{x^{2}}{E L} \sin{\left (t \right )}$$
I can already see that $v$ will not be satisfying the equivalent problem $v_{xx}(x,t) = \frac 1 {c^2} v_{tt}(x,t)$. Unlike the previous question the 2nd derivatives of $U(x,t)$ do not vanish and this is a problem. This means (at least for this selection of $U$ that Separation of Variables will not work. I have two questions:
- I have a strong preference for using Separation of Variables. Can an adequate selection of $U$ be made such that this method can be used?
- If not (and only if not) must I resort to another technique to solve this PDE? What is the most straightforward method I can turn to?
