Sound Wave Propagation through Periodic and Nonreciprocal Structures with Viscous Components Page: 2
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1.2. Nonreciprocal Acoustic Structures
Reciprocity is a fundamental principle of the wave propagation. In optics it is known
as a Lorentz reciprocity theorem. Rayleigh [73] was the first who formulated a reciprocity
theorem for acoustic waves.
(1.1) pA(rB) = pB(rA)
This equation states the equality of pressure when positions of emitter and receiver are
exchanged. It is valid for a point source and emphasizes a symmetry between two points.
From 1950s it was known that an external magnetic field can break reciprocity [47, 40].
Nowadays [24] there are various methods of breaking the reciprocity by either introducing
a quantity which breaks time-reversal (T) symmetry of the system or using a nonlinear
medium [51, 68]. A combination of linear metamaterial with appropriate band gaps and
nonlinear medium, which allows second-harmonic generation, is proposed to be used as a
sound rectification device [51, 50]. If an acoustic system involves fluid in motion, Doppler
effect is a source of nonreciprocity [66]. Time modulation of a periodic in space structure
can lead to the nonreciprocity. Dispersion of a simple mass-spring system with time periodic
spring constants under specific conditions [59] is different for forward (k) and backward (-k)
directions.
Recently, it was shown [86, 31, 79] that viscosity (intrinsic property of fluids) leads to
the nonreciprocity of sound waves in systems with broken parity (P) symmetry. Viscosity,
which is a source of dissipation, leads to irreversibility of acoustic wave propagation. Despite
this, there exists a prevalent opinion that the transmission of sound remains reciprocal evenwithin a dissipative medium [24, 60]. The Navier-Stokes equation, which is a governing
equation of fluid dynamics, is non time-reversible due to the terms proportional to the
viscosity coefficients. This equation is written for the field of velocities v(r), which does not
possess reciprocity in general case. Even in inviscid fluid, where velocity is proportional to
the gradient of pressure, velocity is not reciprocal. Pressure reciprocity can not be extended
to the reciprocity of its gradients since pressure is not required to be either even or odd2
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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/m1/14/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .