In the undergraduate course about the wave, there stated for two harmonic waves propagating in opposite direction, then the resulting wave will be a standing wave. In math, it is like

$$y_1 = A\sin(kx + \omega t), y_2 = A\sin(kx - \omega t)$$

so $$ y = y_1+y_2 = A\sin(kx + \omega t) + A\sin(kx - \omega t) = 2A sin(kx)\cos(\omega t) $$

I am thinking what happen if we have the two plane waves propagating along two different direction (says making angle 60 degree, i.e. the two wave are making ). I know that if that's the case, we cannot write $kx$ but we need to consider the $k$ is a vector such that

$$y_1 = A\sin(\vec{k}\cdot\vec{r} + \omega t), y_2 = A\sin(\vec{k}\cdot\vec{r} - \omega t)$$

But if we look at the horizontal direction (i.e. x) and vertical direction (i.e. y), what can we tell about the resulting wave along x and along y? I am thinking from physical point of view, if we look at the horizontal direction, should the waves still added up to a standing wave because the x components of waves are propagating in opposite direction. But along the vertical direction, the y components of waves are propagating in the same direction so there is no standing wave. Is that correct? If so, how to prove that in math? The term $\vec{k}\cdot\vec{r}$ is very confusing!

In the undergraduate course about the wave, there stated for two harmonic waves propagating in opposite direction, then the resulting wave will be a standing wave. In math, it is like

$$y_1 = A\sin(kx + \omega t), y_2 = A\sin(kx - \omega t)$$

so $$ y = y_1+y_2 = A\sin(kx + \omega t) + A\sin(kx - \omega t) = 2A \sin(kx)\cos(\omega t) $$

I am thinking what happen if we have the two plane waves propagating along two different direction (says making angle 60 degree, i.e. the two wave are making ). I know that if that's the case, we cannot write $kx$ but we need to consider the $k$ is a vector such that

$$y_1 = A\sin(\vec{k}\cdot\vec{r} + \omega t), y_2 = A\sin(\vec{k}\cdot\vec{r} - \omega t)$$

But if we look at the horizontal direction (i.e. x) and vertical direction (i.e. y), what can we tell about the resulting wave along x and along y? I am thinking from physical point of view, if we look at the horizontal direction, should the waves still added up to a standing wave because the x components of waves are propagating in opposite direction. But along the vertical direction, the y components of waves are propagating in the same direction so there is no standing wave. Is that correct? If so, how to prove that in math? The term $\vec{k}\cdot\vec{r}$ is very confusing!