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Suppose we obtain a solution for each of $\mathbf B$, $\mathbf E$ of maxwell equations in the vacuum ($\rho=0$). Clearly, for any constant vector $\mathbf k, \mathbf m$, $\mathbf {B+k}$ and $\mathbf{E+m}$ also satisfy the same set of differential equations. Presumably, we can call $\mathbf k, \mathbf m$ "constants of integration".

My question is, though, is it okay to choose those constants randomly as I like? It is really difficult for me to "choose" appropriate values of $\mathbf k, \mathbf m$.

I might be able to determine $\mathbf k, \mathbf m$ if $\mathbf E$ is assumed to vanish at $\infty$. However, in the case that $\mathbf E$ is a sinusoidal electromagnetic wave, it certainly does not vanish as infinity.

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    $\begingroup$ Are you really allowed to add a constant vector to E and still satisfy all of the equations? Presumably if the oscillations are reduced to zero you would be left with a uniform E field in all of space and that would require an infinite source at infinity. $\endgroup$ – ggcg Sep 13 '19 at 14:22
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Usually, you would formulate it in terms of Sommerfeld radiation condition:

$$ \lim_{r\to\infty} r\left( \frac{\partial u}{\partial r}-iku \right)=0 $$ Here $u$ would be some wave quantity, $r$ is the radial coordinate, $i=\sqrt{-1}$, and $k$ is the wavenumber.

See the following reference for a detailed discussion of its origin and the role in mathematical physics:

Excerpt from this paper's abstract:

In 1912 Sommerfeld introduced his radiation condition to ensure the uniqueness of the solution of certain exterior boundary value problems in mathematical physics. In physical applications these problems generally describe wave propagation where an incident time-harmonic wave is scattered by an object, and the resulting diffracted or scattered waves need to be calculated. When formulated mathematically, these problems usually take the form of an exterior Dirichlet or Neumann problem for the Helmholtz partial differential equation. The Sommerfeld condition is applied at infinity and, when added to the statement of the boundary value problem, singles out only the solution which represents “outgoing” (rather than “incoming” or “standing”) waves in the physical applications.

It is also important to note the uniqueness theorem (as it is usually called in the engineering graduate texts on electromagnetics). See, for example,

In this chapter, the proof first requires the medium to be lossy to prove the uniqueness (thus fields decay), and then uses the lossless case as a limiting case when loss approaches zero.

How deeply and how mathematical you want to dive into this topic – depends totally on your needs.

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We are almost always satisfying Maxwell's equations (or any set of differential equations) with respect to some boundary conditions. Usually we assume that a vector field goes to zero at infinity, which means it is uniquely specified by its divergence and curl. (See the Helmholtz decomposition.) If it doesn't go to zero at infinity, then can specify some other boundary conditions that account for that constant.

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  • $\begingroup$ So, what is the boundary condition for electromagnetic waves? $\endgroup$ – Ma Joad Sep 13 '19 at 22:10

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