Uniqueness of solution to the wave function

[gabbagabbahey]

Now we've showed that the second term of the integrand is worth $$2 (\nabla(\partial_t u_3) \cdot \nabla u_3)$$.

We did?!... Certainly, if $u_3(0,x)=0$, you can also say $\mathbf{\nabla}u_3(0,x)=0$...that's all you need in order to show the second term vanishes.

In order to say that the integral will be zero for all $t$, you also need to show that $u_3$ satisfies the wave equation, so that you can apply the proof of $E$ being constant.

Real and positive I'd say. Real, not really sure why (I just seen in Born's book on Optics that some general time-harmonic fields are complex and that the real parts represent the fields), but energy has to be real... although I wouldn't be so surprised if it's a complex number whose real part represent the energy.
The first term is positive if it's a real number.

I'd say it is is safe to assume that $u_3$ is real-valued (and hence so are its derivatives). However, $(\partial_t u_3)^2$ doesn't necessarily have to be positive, it could also be zero.

Thus the only way for the E to be 0 is that both terms are actually 0.

Right.

Oh... thus $$u_3=0$$

You can't directly conclude that from the fact that $(\partial_t u_3)^2$ and$\mathbf{\nabla}u_3\cdot\mathbf{\nabla}u_3$ are both zero. However, the fact that $(\partial_t u_3)^2=0$ does allow you to conclude that $u_3$ is constant in time. And you also know that $u_3(t=0,x)=0$ so yes, $u_3=0$ and hence $u_1=u_2$ and thus thereis only one unique solution to the given wave equation.

 

[fluidistic]

In order to say that the integral will be zero for all LaTeX Code: t , you also need to show that LaTeX Code: u_3 satisfies the wave equation, so that you can apply the proof of LaTeX Code: E being constant.

Ok, what about this argument: "$$u_1$$ and $$u_2$$ are solutions to the WE, hence any linear combination of them also is a solution to the WE; thus $$u_3$$ also satisfies it."?
Thanks for the rest.

 

[gabbagabbahey]

"$$u_1$$ and $$u_2$$ are solutions to the WE, hence any linear combination of them also is a solution to the WE; thus $$u_3$$ also satisfies it."

Exactly.

 

[fluidistic]

Exactly.

Ok. A big thank you for everything for all the help you provided me and all the time taken to try to teach me a lot of things. I'm going to copy the full solution on a sheet of paper once again and try to digest it completely.

I have a similar problem although harder where the energy is not given and I have to deduce it; so you might see me again posting in the same forum!