MHB Show energy of the wave eqn is conserved from bc's

[skate_nerd]

I have this problem that seemed pretty straight forward at first, but now I'm getting kind of stuck. I was given the expression for the energy of the 1D wave equation where the wave function is denoted as \(u(x,t)\). The energy is
\[E(t) = \int_{0}^{L}\frac{1}{2}\frac{d}{dt}{u^2}(x,t) + \frac{c^2}{2}\frac{d}{dx}{u^2}(x,t)dx\]
At first I needed to find the formula for the time derivative of energy, which I got:
\[\frac{dE}{dt}=c^2\frac{d}{dt}u(L,t)\frac{d}{dx}u(L,t)-c^2\frac{d}{dt}u(0,t)\frac{d}{dx}u(0,t)\]
Now the last two parts are to show that given certain boundary conditions, the energy must be conserved. So basically I take two boundary conditions and show that the above formula has to end up being equal to zero.

The second pair of boundary conditions are straight forward, because they say
\[\frac{d}{dx}u(L,t)=\frac{d}{dt}u(0,t)=0\] so that obviously makes the expression for \(\frac{dE}{dt}=0\).

The first pair is confusing me though. They say that
\[u(0,t)=u(L,t)=0\]
I'm pretty sure it is not the case that if a function has a zero at a certain \(x\) value, then so does its derivative(s). So what am I supposed to do with this information? I've been stuck on this for a while, my brain can't come up with anything. Any help would be awesome!

 

[I like Serena]

Hey skatenerd! :)

I have this problem that seemed pretty straight forward at first, but now I'm getting kind of stuck. I was given the expression for the energy of the 1D wave equation where the wave function is denoted as \(u(x,t)\). The energy is
\[E(t) = \int_{0}^{L}\frac{1}{2}\frac{d}{dt}{u^2}(x,t) + \frac{c^2}{2}\frac{d}{dx}{u^2}(x,t)dx\]

Starting at the beginning, I think you have the wrong equation.
Might it be the following?
\[E(t) = \int_{0}^{L}\frac{1}{2} \Big(\frac{d}{dt}{u}(x,t)\Big)^2 + \frac{c^2}{2}\Big(\frac{d}{dx}{u}(x,t)\Big)^2 dx\]

 

[skate_nerd]

Sorry, that is what I meant to write. And the expression I arrived at for \(\frac{dE}{dt}\) is the correct expression that they expected us to arrive at.

 

[I like Serena]

Now the last two parts are to show that given certain boundary conditions, the energy must be conserved. So basically I take two boundary conditions and show that the above formula has to end up being equal to zero.

It is not clear to me what your boundary conditions are...

The second pair of boundary conditions are straight forward, because they say
\[\frac{d}{dx}u(L,t)=\frac{d}{dt}u(0,t)=0\] so that obviously makes the expression for \(\frac{dE}{dt}=0\).

Can it be that this contains a typo?
It seems more likely to me it should be:
\[\frac{d}{dt}u(L,t)=\frac{d}{dt}u(0,t)=0\]

That is, waves can go forward and backward, but the end points are fixed and do not change in time.

The first pair is confusing me though. They say that
\[u(0,t)=u(L,t)=0\]
I'm pretty sure it is not the case that if a function has a zero at a certain \(x\) value, then so does its derivative(s). So what am I supposed to do with this information? I've been stuck on this for a while, my brain can't come up with anything. Any help would be awesome!

That depends on the derivative you're talking about.

If a function is zero for all values of t, its derivative with respect to t is also zero.
In other words, that "second pair of boundary conditions" can be deduced from this first pair (assuming you have indeed a typo in the second pair).
On the other hand, you can't say much about the derivatives with respect to x, but then, you don't need those.Btw, I'm moving this thread to Applied Mathematics, since it is mostly about the interpretation of the wave equation.

 

[skate_nerd]

Ahhhhh okay that totally makes sense, I didn't think of it that way. Also, I made a typo in both the boundary conditions, but not in the way you guessed. It should have been for the first pair:
\[u(0,T)=u(L,t)=0\]
(I didn't notice on the left side that it was a capital \(T\) instead of lower case, meaning one period and not all values of \(t\)) and the second pair is
\[\frac{d}{dx}u(0,t)=u(L,t)=0\]
Of course I think you could tell, but I never really clarified that all the \(\frac{d}{dx}\)'s and \(\frac{d}{dt}\)'s should be partial derivatives (except for the time derivative of \(E(t)\)).
Thanks for the help though, I think I should be able to answer both the problems now. Appreciate it!