Waves in space-dependent and time-dependent materials: a systematic comparison
arxiv(2023)
摘要
Waves in space-dependent and time-dependent materials obey similar wave
equations, with interchanged time- and space-coordinates. However, since the
causality conditions are the same in both types of material (i.e., without
interchangement of coordinates), the solutions are dissimilar.
We present a systematic treatment of wave propagation and scattering in 1D
space-dependent and time-dependent materials. After a review of reflection and
transmission coefficients, we discuss Green's functions and simple wave field
representations for both types of material. Next we discuss propagation
invariants, i.e., quantities that are independent of the space coordinate in a
space-dependent material (such as the net power-flux density) or of the time
coordinate in a time-dependent material (such as the net field-momentum
density). A discussion of reciprocity theorems leads to the well-known
source-receiver reciprocity relation for the Green's function of a
space-dependent material and a new source-receiver reciprocity relation for the
Green's function of a time-dependent material. A discussion of general wave
field representations leads to the well-known expression for Green's function
retrieval from the correlation of passive measurements in a space-dependent
material and a new expression for Green's function retrieval in a
time-dependent material.
After an introduction of a matrix-vector wave equation, we discuss propagator
matrices for both types of material. Since the initial condition for a
propagator matrix in a time-dependent material follows from the boundary
condition for a propagator matrix in a space-dependent material by
interchanging the time- and space-coordinates, the propagator matrices for both
types of material are interrelated in the same way. This also applies to
representations and reciprocity theorems involving propagator matrices.
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