# How can the amplitude of a wave determine if it is plane, spherical or cylindrical?

it's known that typical expressions for plane, spherical and cylindrical waves are (for instance in terms of electric field uniformly propagating along r axis, in frequency domain):

• Plane Wave: $$E(r) = E_0 e^{-jkr}$$

• Spherical Wave: $$E(r) = \frac{E_0}{r} e^{-jkr}$$

• Cylindrical Wave: $$E(r) = \frac{E_0}{\sqrt r} e^{-jkr}$$

Which is the formal difference between these expressions?

• The plane wave is not attenuated with the distance r

• The spherical wave is attenuated with distance as $$\frac{1}{r}$$

• The cylindrical wave is attenuated with distance as $$\frac{1}{\sqrt r}$$

So, the difference is in how amplitude depends on the distance between the source and the observation point.

But, "plane, spherical, cylindrical" refers to a wavefront's property. A wavefront is a set of points where the wave has the same phase. What does it has to do with it the wave amplitude? I can't see the physical link bewtween the physical meaning of wavefront and the amplitude dependence on the distance.

• The definition of $r$ (at least in relation to other coordinate variables) and direction of $\mathbf{E}_0$ is different for each of these three cases. Choosing a different coordinate system without altering the expression implicitly changes what the expression means. Feb 7, 2021 at 20:16