Sound Wave Propagation through Periodic and Nonreciprocal Structures with Viscous Components Page: 17
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The first and second equality are the Snell's law for the longitudinal-to-longitudinal refrac-
tion, sin 01a/ sin 01b = Ca/Cb, and for the longitudinal-to-transverse refraction, sin 01a/ sin 0tb =
Ca/Ctb. The equality - sin Ola = sinlOta means that sin eta ~ 6a = kja < 1. Thus, due
a8a COe)ta a a
to low viscosity (see Eq. (2.13)) the transverse mode "propagates" almost perpendicular to
the boundary and does not suffer from refraction. Note that this property is additional evi-
dence that this mode is similar to the electromagnetic wave, which for any angle of incidence
"propagates" perpendicular to metal surface at a distance of the order of skin layer. .
Separating the imaginary part of Eq. (2.24), one obtains the following relation for
the angles eta and Yta:
Ea W . sin(Ota - Yta)
(2.26) sm(O'a - hta) = ,
Ca ba cos Yta
which assumes that the difference eta -Yta is cubic over a small parameter EaaW Ca (kaba)3
Finally, the set of equations 2.24 can be written in a compact form
Sin Ola Ca sin Ola Ca Wsa
(2.27) - , . - , e = Yta - sin 01(a) << 1.
sin eb Clin sm tb Ctb Ca
The relation between the angles 0la and 'Ya depends on the initial conditions of excitation
of sound and remains indefinite. This, however, does not affect further calculations since
dissipation of the longitudinal mode gives negligible contribution to the attenuation of sound.
The principal contribution comes from the transverse mode, which excites oscillations of fluid
parallel to the interface at any direction of propagation of sound wave in the layered structure
but the direction exactly parallel to the axis when eta = Yta = 0.2.3.3. Boundary Conditions
Within each layer the velocities of the longitudinal and transverse mode are given by
Eqs. (2.22) and (2.23) respectively. These equations contain four indefinite coefficients for
each layer, i.e., in total there are eight unknowns. They are obtained from the boundary
conditions at the interface x = a, see Fig. 2.1). When sound wave passes through the
interface (the oscillating total velocity in Eq. (2.21) satisfies the no-slip boundary condition,
which means that v = vi + vt is continuous at x = a. Also, the normal, Fx = o-xxnx, and the17
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Shymkiv, Dmytro. Sound Wave Propagation through Periodic and Nonreciprocal Structures with Viscous Components, dissertation, May 2024; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc2332612/m1/29/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .