Exploring Wave Equation Solutions for E-fields

[dyn]

Hi.
I'm following the "derivation" in some lecture notes which shows that the Electric and Magnetic fields are perpendicular to each other and to the direction of propagation. There are 2 points I don't understand

A solution to the wave equation for E-fields is given as E = E₀ exp i(ωt-kz).
It then states that if the propagation is along z only then ∂/∂x and ∂/∂y of any property is zero. Why is this so ?

Using Gauss's law this then leads to ∂E_z/∂z = 0 which implies E_z is a constant which is set to zero. But we also have ∂E_x/∂x = 0 and ∂E_y/∂y = 0 so why aren't E_x and E_y equal to a constant which can be set to zero ? I realize this would give no wave.
Thanks

 

[Dale]

It then states that if the propagation is along z only then ∂/∂x and ∂/∂y of any property is zero. Why is this so ?

That is what defines a plane wave. They vary only along the direction of propagation, and they are uniform in the plane normal to that direction. Not all waves behave this way, but those that do are called plane waves.