Consider a 2D wave equation
$$\partial_x^2 f(x,y,t)+\partial_y^2 f(x,y,t)=\partial_t^2 f(x,y,t)\tag1$$
on $\mathbb R^2$. The solution $f$ is to be bounded at $|\vec r|\to\infty$ (where $\vec r=(x\;y)^\intercal$), and some contour $\Gamma\subset\mathbb R^2$ is a boundary where $f$ also must vanish.
A plane wave $w(\vec r)=\exp(i(\omega t-\vec k\vec r))$ is scattered by $\Gamma$, and we need to find the steady-state solution
$$f(x,y,t)=w(x,y)+g(x,y)\exp(i\omega t).\tag2$$
Suppose we have guessed some function $p(x,y,t)$, which is bounded, vanishes at $\Gamma$ and satisfies $(1)$. It could be used as the solution if not one problem: it may not be the steady state we are looking for. Namely, it could have e.g. a component which cancels $w(x,y,t)$ exactly, or it could have an arbitrary number of additional incident wave components with wavevector rotated away from $\vec k$ along with corresponding scattered fields.
My question is thus: how can we check whether a given function is really the steady state of the wave $w$ scattered by $\Gamma$, and doesn't include anything extra?