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  ?

When we have 1D standing waves, we can write them as the sum of two propagating wave in opposite directions that give the formula $\sin kx\cos\omega t$.

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: $$\exp\left(i\left(k_x x + k_y y - \omega t\right)\right)+\exp\left(i\left(k_x x + k_y y + \omega t\right)\right)=\exp\left(i\left(k_x x + k_y y\right)\right)2\cos(\omega t).$$

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 is it not true that 2D stationary waves are the sum of two propagating waves in opposite directions?

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 :

$e^{i(k_x x + k_y y - wt)}+e^{i(k_x x + k_y y + wt)}=e^{i(k_x x + k_y y)}.2cos(wt)$

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 stationnary waves are the sum of two propagating waves in opposite directions ?

Thank you.

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 ?