The equation describes a vectorial field, where for any (t,x,y,z) there is a vector. Each component of the vector is a complex number.

The meaning of $\mathbf k$ in the exponent is the direction of the maximum change of each component of the field, for a given time $t_0$. So, if we take the gradient of one of the components:

$\nabla \psi_x = i\mathbf k\psi_x$

On the other hand, planes normal to $\mathbf k$ for a given $\mathbf r_0$, all $\mathbf r$ such that $\mathbf k.(\mathbf r - \mathbf r_0) = 0$ have a constant value for the vector component in a given time. It is a level surface.

The field can be understood as a travelling wave because an observer travelling with a velocity $v = \frac{\omega}{|\mathbf k|}$ will see a static field.

The equation describes a vectorial field, where for any ($t,x,y,z$) there is a vector. Each component of the vector is a complex number.

The meaning of $\mathbf k$ in the exponent is the direction of the maximum change of each component of the field, for a given time $t_0$. So, if we take the gradient of one of the components:

$\nabla \psi_x = i\mathbf k\psi_x$

On the other hand, planes normal to $\mathbf k$ for a given $\mathbf r_0$, all $\mathbf r$ such that $\mathbf k \cdot(\mathbf r - \mathbf r_0) = 0$ have a constant value for the vector component in a given time. It is a level surface.

The field can be understood as a travelling wave because an observer travelling with a velocity $v = \frac{\omega}{|\mathbf k|}$ will see a static field.