Nusc said:
This is a standard BVP that can be found in a textbook no?
I havn't seen that specific description in a textbook - but it follows a common form.
The trick is usually to translate what you understand into maths.
Here's how I'm reading what you wrote:
There is a rectangle length L and width W (thickness we don't care about), made out of some unspecified material, aligned: ##-\frac{L}{2} \leq x \leq \frac{L}{2}##.
We can represent the width at some position x measured at time t by a function w(x,t).
At ##t=0## the ends are distorted so that the wiidth there is: ##w(-\frac{L}{2},0)=w(\frac{L}{2},0)=W+\epsilon_0##, i.e. both ends are initially stretched by the same amount.
The material properties of the rectangle mean there is a continuous variation of w with x.
But notice that the specific variation can be
anything so long as those initial conditions are satisfied.
If we let go of the ends, we can expect the width of the rectangle at some position x at t>0 to vary according to:
##w(x,t)=W+\epsilon(x,t):\epsilon(-\frac{L}{2},0)=\epsilon(\frac{L}{2},0)=\epsilon_0##
... we can usually say more about the initial conditions than that though ... since we were the ones causing the initial distortion, we also know the precise shape of the distorted rectangle close to the ends ... i.e. we know, at least, how the width varies with x close to the ends. This can be anything we like - we usually pick:
##w'(-\frac{L}{2},0)=w'(\frac{L}{2},0)=0## ... the primed notation indicates differentiation wrt x.
I have to review this material - I haven't touched BVP's for several years now. How can I derive the other solutions?
By solving the DE normally - the DE in question should be chosen using your knowledge of the physics involved.
For your example, it is probably fair to expect the wave equation to be useful.
It is very common to be asked to look for standing wave solutions and express other situations as a superposition of modes. This is what it looks like you were trying for with your proposed solution - but that's unlikely unless the ends are continuously being distorted (i.e. not just at t=0). Your setup could result in ##w(x,t)=W+\epsilon_0\cos\omega t## or in two pulses running into each other, depending on the material properties.
I don't understand when you say, "...the opposite sides of the rectangle, at the same x, will be oscillating 180deg out of phase. "
If the top edge of the rectangle follows ##y_+(x,t)## and the bottom edge follows ##y_-(x,t)##, then ##w(x,t)=y_+(x,t)-y_-(x,t)## ...
##y_+(x,t)=\frac{W}{2}+\frac{1}{2}\epsilon(x,t)\implies y_-(x,t)=y_+(x,t)=\frac{W}{2}-\frac{1}{2}\epsilon(x,t)##
... i.e. when one goes up the other goes down: ##y_+## and ##y_-## are 180deg out of phase.
