By Park K.
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Extra resources for A K -automorphism with non-isomorphic ZxZ2 actions
The porous medium is supposed to be transversely isotropic. Rotations around the axis x3 do not modify the acoustical properties of the layer. The case of transversely isotropic porous media is considered in Chapter 10 in the context of the full Biot theory. If the frame is motionless, the material can be replaced by a transversely isotropic ﬂuid. In the transversely isotropic ﬂuid, the bulk modulus is a scalar quantity. In the system x1 , x2 , x3 the density is a diagonal tensor with two different components ρ3 = ρN , ρ1 = ρ2 = ρP .
Hence, the reﬂection coefﬁcient describes a circle in the complex plane if k is real. If k is complex, the reﬂection coefﬁcient describes a spiral. It should be noted that the propagation of electromagnetic plane waves in a waveguide can be described with impedances and reﬂection coefﬁcients in a similar way to the use of those concepts in describing sound propagation. The Smith chart, which provides a graphical representation for the propagation of electromagnetic waves, can also be used to describe the acoustic plane-wave propagation.
2 Inhomogeneous plane waves in isotropic ﬂuids In Chapter 2, waves propagating parallel to a coordinate axis have been considered. The same plane waves can propagate in any direction deﬁned by a unit vector n if the ﬂuid is isotropic. Let n1 , n2 and n3 be the three components of n. 2) J. F. Allard and N. 3), k is perpendicular to the equiphase and equiamplitude planes. 4) to deﬁne k1 , k2 and k3 . 4). 11) From the above equations it is seen that the equiamplitude planes are parallel to the x2 = 0 plane, and the equiphase planes are parallel to the x1 = 0 plane.
A K -automorphism with non-isomorphic ZxZ2 actions by Park K.