When we have 1D standing waves, we can write them as the sum of two propagating wave in opposite directions that give the formula $\sin(kx)\cos(wt)$.
When I try to do this for 2D waves (I mean 2D by the fact there are $k_x$ and $k_y$) I don't have the right expression :
$$\exp\left(i\left(k_x x + k_y y - wt\right)\right)+\exp\left(i\left(k_x x + k_y y + wt\right)\right)=\exp\left(i\left(k_x x + k_y y\right)\right)2\cos(wt)$$
If I take the imaginary part I will have $\sin(k_x x +k_y y)$ and not $\sin(k_x x)\sin(k_y y)$.
Am I wrong somewhere or we can't say that 2D stationary waves are the sum of two propagating waves in opposite directions ?