No, a plane wave is not inherently incompatible with a spherical coordinate system. In fact, plane waves can be described and analyzed in different coordinate systems, including the spherical coordinate system.
A plane wave is a type of wave that propagates in a straight line with a constant phase across any plane perpendicular to the direction of propagation. Its wavefronts are planar surfaces, hence the name "plane" wave. The direction of propagation is typically specified using Cartesian coordinates (x, y, z) or spherical coordinates (r, θ, φ), depending on the coordinate system being used.
In a Cartesian coordinate system, a plane wave can be expressed as a function of position and time, such as:
ψ(x, y, z, t) = A * exp[i(kx - ωt)]
Here, k represents the wave vector, ω is the angular frequency, and A is the complex amplitude of the wave.
In a spherical coordinate system, the same plane wave can be expressed as a function of position and time using the spherical coordinates (r, θ, φ). The wave vector and angular frequency will have appropriate representations in the spherical coordinate system.
Therefore, while the plane wave equation is commonly expressed in Cartesian coordinates, it can be transformed and represented in other coordinate systems, including the spherical coordinate system, without any fundamental incompatibility.