The behavior of the amplitude and phase of the "intermediate wave", which we previously introduced as certain fractional solutions to the standard scalar Helmholtz equation, is addressed and presented. These waves effectively behave as intermediate cases between the canonical cases of the plane-wave and cylindrical wave propagation. We show that the amplitude and phase of such intermediate wave undergo interesting "evolutions" as the fractionalization parameter ν attains fractional values between zero and unity. Possible extension into the novel concept of intermediate guided-wave geometries is just speculated.