Let we use the Figure 3.2 (from one of Brekhovskikh books) - where shown the (n-1) layers between the half-spaces, starting as from below at (while could be and at any other height) and having the incidental acoustical wave from above under the angle

As usual in acoustics for many texts here is the speed of the sound in each medium, is the layer's ordinal number; - is using for the angle of incidence of the sound wave in each layer , - is the wave number, - is the wave length

This is the often used and well accepted assumption about interface surface. For this particular problem and its statement it brings simplification into the both scales treatment.

The HSP-VAT upper scale acoustic equation in the phase one simplifies for the general 2D situation (y,z), here we took advantage of more vivid compliance with the Cartesian 3D figure that we used to apply, to the following one

when the interface is completely rigid. The Lower scale acoustics equation in the phase one is as usually

or in the harmonic form 2DL

This equation, while a 2D incident field will impact the superlattice at or at

when used the statement set-up as in the Brekhovskikh's book, then it becomes in the Upper scale

or when strictly vertical incident field applied then

These above two averaged acoustics equations are the exact copies of the simple linear EM Upper Scale wave propagation equations from -

The difference is that for the EM fields equations used

These equations have been studied without the last (source) term for the globular morphologies by me and V.Kushch in 94-96, and we get many results (most are unpublushed), as, for example:

1) Has been developed the Simulation Averaging Techniques (SAT) for the analytical solution for each term on the Upper Scale and numerical calculation of those terms on the Upper Scale.

2) Has been developed the mathematics of analytical averaging of the fields;

3) Has been solved "exactly" (as allowed the computer power at that time) - few two scales conductivity and laminar flow problems for the two phase heterogeneous and porous media with globular morphology fillers;

4) Has been developed the methodology for verification of the WSAM theorem by analytical modeling with then simulated problems using the DMM-DNM; etc.

See some outcomes also in the section on Ht Electrodynamics -

We would give here few outstanding features from the Two Scale solution for the conventional one scale known and studied by anyone involved in the field problem of acoustic wave incidental upon the multilayer medium consisting of two or more phases of alternating layers- the acoustic wave propagation in superlattices.

Actually, we have found more than one method of solving this famous problem as the two scale Heterogeneous Acoustics problem. One of the methods is completely analytical - and this is the second of generally known analytical methods for the two scale VAT problems for superlattices.

Another one was developed for the steady-state thermal (diffusion, electrical) conductivity problem for the superlattice in-cross distribution of fields -

The Upper Non-Local scale solution of this problem would be nice to illustrate with the few figures related to the Upper scale functions.

The plots of function in the presented on the following two figures are pretty much common when the wave length gets in the range comparable to the width of the separate layer in the pack.

Few outstanding features that presented in these function plots have been used for the Upper scale acoustics equations solution for layered (superlattice) morphologies.