We develop theory and algorithms for the design of coatings which either el
iminate or enhance reflection of waves from surfaces. For steady-state harm
onic waves with continuous frequency spectrum that covers an arbitrarily pr
escribed frequency band, coatings are designed that essentially eliminate r
eflections of all frequencies within the band. Although we focus on acousti
c waves in elastic media, the methods developed here can be adapted to elec
tromagnetism or other phenomena governed by variants of the linear wave equ
ation.
To create a nonreflective coating which is to operate in a frequency band [
Omega(0),Omega(1)], we select n frequencies, uniformly distributed in the b
and, and design an n-layer coating of given thickness that completely elimi
nates reflections of waves at these frequencies. We show that if n is large
, then the reflectivity of the coating designed by our method is small for
all frequencies in the band. More precisely, the reflectivity at an arbitra
ry frequency omega is an element of (Omega(0),Omega(1)) is O (1/n) if Omega
(0) = 0, while it is O (alpha(n)) if Omega(0) > 0, where 0 < alpha < 1. Fur
thermore, extensive numerical studies show that when this discrete n-layer
coating is smoothed out by spline interpolation, the reflectivities remain
small not only for frequencies in the original band but also for all larger
frequencies.
We also describe a procedure for designing coatings that maximizes reflecti
vity (or, equivalently, minimizes transmissivity). We show that through a p
roper layering technique, it is possible to obtain transmissivity of O(alph
a(n)), 0 < alpha < 1, in an n-layer design.