No, the background electromagnetic field does not need to be non-zero for electromagnetic (EM) waves to propagate. In fact, EM waves can exist and propagate even in the absence of a background field.
The wave equation you mentioned, derived from Maxwell's equations using the Fourier series, describes the behavior of electromagnetic waves. It states that the second derivative of the electric or magnetic field with respect to both time and space is proportional to the Laplacian of that field. Mathematically, it can be written as:
∇²E = με ∂²E/∂t²
where ∇² is the Laplacian operator, E represents the electric field, μ is the permeability of the medium, ε is the permittivity of the medium, and ∂²E/∂t² denotes the second derivative of the electric field with respect to time.
This wave equation describes how the electric field propagates through space and time. It does not depend on the initial values of the electric and magnetic fields being non-zero at all points. The equation tells us how the field evolves over time based on the initial conditions, boundary conditions, and the properties of the medium through which the wave is propagating.
When the initial values of the electric and magnetic fields are zero at all points, it means that there is no pre-existing field present. However, disturbances or variations in the medium can still generate EM waves. For example, an oscillating charge or current in a conductor can create an EM wave that propagates through space, even if the background field is initially zero.
In summary, the existence and propagation of electromagnetic waves do not require a non-zero background electromagnetic field. The wave equation describes how the fields evolve and propagate based on the initial conditions and the properties of the medium.