TY - JOUR

T1 - Application of the transfer matrix approximation for wave propagation in a metafluid representing an acoustic black hole duct termination

AU - Guasch, Oriol

AU - Sánchez-Martín, Patricia

AU - Ghilardi, Davide

N1 - Funding Information:
The first author would like to acknowledge l’Obra Social de la Caixa and the Universitat Ramon Llull for their support under grant 2018-URL-IR2nQ-031 . The authors would also like to acknowledge Prof. Ramon Codina for his advice and comments.
Funding Information:
The first author would like to acknowledge l'Obra Social de la Caixa and the Universitat Ramon Llull for their support under grant 2018-URL-IR2nQ-031. The authors would also like to acknowledge Prof. Ramon Codina for his advice and comments.
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2020/1

Y1 - 2020/1

N2 - The transfer matrix method has been proposed to analyze the acoustic black hole effect in duct terminations. The latter is achieved by placing a retarding waveguide structure inside the duct, which consists in a set of rings whose inner radii decrease to zero following a power law. The rings are separated by thin air cavities. If the number of ring-cavity ensembles is large enough, wave propagation inside the waveguide can be treated as a continuous problem. A governing differential equation can be derived for the acoustic black hole which intrinsically relies on assumptions from transfer matrix theory. However, no formal demonstration exists showing that the transfer matrix solution is consistent and formally tends to the solution of the continuous problem. Proving such consistency is the main goal of the paper and an original option has been adopted to this purpose. First, we prove the differential equation for the acoustic black hole is identical to the wave equation for a metafluid with a power-law varying density. Transfer matrices are then applied to solve wave propagation in a discretization of the metafluid into layers of constant density. It is shown that when the number of layers tends to infinity and their thicknesses to zero, the transfer matrix solution satisfies the metafluid equation and therefore the acoustic black hole one. The transfer matrices for the metafluid discrete layers take a particularly simple form, which is a great advantage. This work allows one to interpret the retarding waveguide structure as a particular realization of the metafluid.

AB - The transfer matrix method has been proposed to analyze the acoustic black hole effect in duct terminations. The latter is achieved by placing a retarding waveguide structure inside the duct, which consists in a set of rings whose inner radii decrease to zero following a power law. The rings are separated by thin air cavities. If the number of ring-cavity ensembles is large enough, wave propagation inside the waveguide can be treated as a continuous problem. A governing differential equation can be derived for the acoustic black hole which intrinsically relies on assumptions from transfer matrix theory. However, no formal demonstration exists showing that the transfer matrix solution is consistent and formally tends to the solution of the continuous problem. Proving such consistency is the main goal of the paper and an original option has been adopted to this purpose. First, we prove the differential equation for the acoustic black hole is identical to the wave equation for a metafluid with a power-law varying density. Transfer matrices are then applied to solve wave propagation in a discretization of the metafluid into layers of constant density. It is shown that when the number of layers tends to infinity and their thicknesses to zero, the transfer matrix solution satisfies the metafluid equation and therefore the acoustic black hole one. The transfer matrices for the metafluid discrete layers take a particularly simple form, which is a great advantage. This work allows one to interpret the retarding waveguide structure as a particular realization of the metafluid.

KW - Acoustic black hole

KW - Metafluid

KW - Metamaterial

KW - Reflection coefficient

KW - Transfer matrix method

KW - Waveguide power-law radius

UR - http://www.scopus.com/inward/record.url?scp=85074629056&partnerID=8YFLogxK

U2 - 10.1016/j.apm.2019.09.039

DO - 10.1016/j.apm.2019.09.039

M3 - Article

AN - SCOPUS:85074629056

SN - 0307-904X

VL - 77

SP - 1881

EP - 1893

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

ER -