Suppose we have two waves $u_1$ and $u_2$ satisfying

$$ \begin{cases} \partial_{tt}u_i = \Delta u_i\\ u_i(0,x) = f_i(x) \\ \partial_t u_i(0,x) = g_i(x) \end{cases} $$ in $\mathbb{R}^+ \times \mathbb{R}^n$. Then $u = u_1 + u_2$ will be a solution of $$ \begin{cases} \partial_{tt}u = \Delta u\\ u(0,x) = f_1(x) + f_2(x)\\ \partial_t u(0,x) = g_1(x) + g_2(x) \end{cases} $$ i.e. the solution depends linearly on the initial data.

However, if we consider the energy of solutions $$ e(u;t) = \int \left(u_t(t,x)\right)^2 + \left|\nabla u(t,x) \right|^2 \,dx $$ it is pretty clear that this quantity doesn't depend linearly on the initial data. For example, if $f_1 = -f_2 =: f$ and $g_i \equiv 0$ (for simplicity), then $u \equiv 0$, so obviously its energy is $0$ as well. On the other hand, the energy of $u_1$ and $u_2$ are both $$ e(u_i;t) = \int \left|\nabla{f(x)}\right|^2\,dx > 0 $$ (by conservation of energy).

So, my question is: what kind of (if any) relationships do hold, or what kind of conditions are necc./suff. for interesting/useful conclusions to be drawn? For example, (as the beginning of an idea), if the supports of $f_1$ and $f_2$ are disjoint, then it seems we can (at least) conclude $$ e(u_1 + u_2;0) = \int \left| \nabla f_1 + \nabla f_2\right|^2\,dx \\ = \int \left| \nabla f_1\right|^2 + \left|\nabla f_2\right|^2\,dx = e(u_1;0) + e(u_2;0) $$ (it would be nice if this were satisfied for $t>0$ as well, but I don't know if I can conclude that just yet.) Maybe the statement is something like "if, at time $t$, the supports of the $u_i(t,\cdot)$'s are disjoint, then the total energy is the sum of the individual energies".

Anyway, this is of course a broad question, but thanks in advance for any thoughts you may have.


it would be nice if this were satisfied for $t>0$ as well

It is satisfied for $t>0$ (provided the initial supports were disjoint), because the energy is conserved by the wave equation. $$ e(u_1+u_2;t)= e(u_1+u_2;0) = e(u_1;0)+e( u_2;0) = e(u_1;t)+e( u_2;t) \tag{1}$$

Thus, additivity of energy for all times is equivalent to $e(u_1+u_2;0) = e(u_1;0)+e( u_2;0) $. In terms of the initial data, the latter says precisely that $$\int g_1g_2 + \int \nabla f_1\cdot \nabla f_2=0 \tag{2}$$

Thus, (1) holds if and only if (2) holds.

One can imagine situations in which (2) naturally holds withough the supports being disjoint: for example, the support of one wave is contained in a "flat part" of the other; and initial velocities are zero.

  • $\begingroup$ Great, thanks for the answer; though I wonder if you have any sense whether (2) is a necessary condition as well. Or further, if (1) holds at some $t>0$ does it imply either or both of (2) (with $\nabla f_i$ replaced by $\nabla u_i(t,\cdot)$ etc.)? Or even further, whether it could imply $\nabla f_1 \cdot \nabla f_2 = 0$ (and not just their integral). Thanks! $\endgroup$ – BaronVT Nov 19 '13 at 5:18
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    $\begingroup$ @BaronVT Edited with iff condition. No, (1) does not imply a pointwise statement $\nabla f_1\cdot \nabla f_2=0$, since (1) holds whenever the integral condition (2) holds. There is nothing deep going on here, in my opinion. $\endgroup$ – user103402 Nov 19 '13 at 5:53
  • $\begingroup$ Alright, thanks again for the discussion. $\endgroup$ – BaronVT Nov 19 '13 at 15:52

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