Explicit formulas for the reflection and transmission coefficients of one-component waves through a stack of an arbitrary number of layers

• The reflection and transmission of one-component waves through a stack of arbitrary number of layers are considered.
• The explicit formulas for the reflection and transmission coefficients of one-component waves are obtained.
• From these formulas, the explicit expressions of the reflection and transmission coefficients for an FGM layer are derived.
• Approximate formulas are established for a stack of thin layers and for a thin FGM layer.
• It is shown that they are good approximations.

The transmission and reflection of one-component elastic, acoustic, optical waves on a stack of arbitrary number of different homogeneous layers have been intensively studied in the literature. However, all obtained formulas for the reflection and transmission coefficients are in implicit form. In this paper, we provide the explicit formulas for them. From these formulas we immediately arrive at the explicit formulas for the reflection and transmission coefficients of one-component waves through an FGM layer. Based on the obtained exact formulas, approximate formulas for the reflection and transmission coefficients are established for a stack of thin layers and for a thin FGM layer. It is numerically shown that they are good approximations. Since the obtained formulas are totally explicit, they are useful in evaluating, not only numerically but also analytically, the transmission and reflection coefficients of one-component waves.