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Authors
Abstract(s)
A presente dissertação visa a sugestão de uma ab ordagem alternativa, e
recente, ao estudo da Acústica Submarina e consequente comparação com os princí-
pios clássicos. Alicerçada na geometria diferencial e na análise tensorial, e aplicada
à aproximação p or raios acústicos no limite das altas frequências da onda sonora, a
meto dologia que será apresentada parte de uma p ersp etiva relativista para explicar,
física e matematicamente, alguns dos fenómenos mais imp ortantes da acústica em
meios aquáticos em movimento.
Através de uma analogia evidente com a relatividade esp ecial e geral, passase a encarar o meio aquático, uma parte do o ceano p or exemplo, como uma variedade
ou um espaço pseudo-Riemanniano, no qual a métrica, ao incorp orar as variáveis
acústicas, define o espaço em si, os co eficientes de conexão p ermitem comparar
diferentes p ontos no espaço, o tensor de Riemann afere se existe curvatura ou não,
e p or fim os raios acústicos descrevem geo désicas numa esp écie de espaço-temp o
(aqui galileano), como na relatividade. A curvatura que os raios sonoros descrevem
em meios heterogéneos, derivada da constante alteração da velo cidade do som, é
de certa forma identificada com a curvatura que a tra jétória, das partículas ou até
mesmo da luz sofre quando o espaço-temp o é curvado p ela interação gravítica com
a matéria, na relatividade.
Por fim, ap ós to da a revisão teórica p or trás, tanto dos méto dos clássicos
como deste méto do alternativo, e o resultante contraste entre os dois, enaltecendo
as claras vantagens da geometria diferencial, não só p ela solução mais completa e
abrangente que dá para os problemas de acústica, incluindo os mais complicados
com uma certa arbitrariedade na corrente dos fluidos, mas tamb ém p elo valor p edagógico e académico que tem p ela aprendizagem de conceitos totalmente novos,
pretende prop or-se, para futuros trabalhos, a implementação da meto dologia computacionalmente.
The present dissertation aims to sugest an alternative and recent approach to the study of Underwater Acoustics and consequently the comparison with the classic principles. Based on differential geometry and tensor analysis, and applied to the ray theory aproximation of the sound waves in the limit of the high frequencies, the metho dology that will b e presented pro ceeds to explain some of the mos t imp ortant phenomena of acoustics in underwater moving media, under a relativistic p ersp ective. From an evident analogy with the sp ecial and general relativity, one starts to interpret the underwater medium, some part of the o cean for example, as a pseudo-Riemannian manifold, where the metric, filled with the acoustic variables that describ es the field, defines the space itself, the connection co eficients allow the connection b etween different p oints in the space, the Riemann curvature tensor provides the ultimate test on the existence of curvature or not, and finally the acoustic rays describ e geo desics in some typ e of spacetime (here galilean), like in the theory of relativity. The curvature that the rays describ e in non homogeneous media, due to the constant variation of the sound sp eed, is in some way identified with the curvature suffered by the tra jectory of particles, or even the light, caused by the gravitational interation with matter and energy that curves spacetime. In the end, after all the theoretical revision that stands b eside the b oth metho ds, the classical one and this new one, and the resulting contrast b etween them, extolling the notorious advantages of the differential geometry, not just by the completeness and generality that adds to a solution for the acoustic phenomena, including the most complex with arbitrary background flows, but also for the pedagogic and academic value in bringing new concepts, we propose, in future developments, the computational implementation of this methodology.
The present dissertation aims to sugest an alternative and recent approach to the study of Underwater Acoustics and consequently the comparison with the classic principles. Based on differential geometry and tensor analysis, and applied to the ray theory aproximation of the sound waves in the limit of the high frequencies, the metho dology that will b e presented pro ceeds to explain some of the mos t imp ortant phenomena of acoustics in underwater moving media, under a relativistic p ersp ective. From an evident analogy with the sp ecial and general relativity, one starts to interpret the underwater medium, some part of the o cean for example, as a pseudo-Riemannian manifold, where the metric, filled with the acoustic variables that describ es the field, defines the space itself, the connection co eficients allow the connection b etween different p oints in the space, the Riemann curvature tensor provides the ultimate test on the existence of curvature or not, and finally the acoustic rays describ e geo desics in some typ e of spacetime (here galilean), like in the theory of relativity. The curvature that the rays describ e in non homogeneous media, due to the constant variation of the sound sp eed, is in some way identified with the curvature suffered by the tra jectory of particles, or even the light, caused by the gravitational interation with matter and energy that curves spacetime. In the end, after all the theoretical revision that stands b eside the b oth metho ds, the classical one and this new one, and the resulting contrast b etween them, extolling the notorious advantages of the differential geometry, not just by the completeness and generality that adds to a solution for the acoustic phenomena, including the most complex with arbitrary background flows, but also for the pedagogic and academic value in bringing new concepts, we propose, in future developments, the computational implementation of this methodology.
Description
Keywords
Tensor Métrico Curvatura Espaço de Riemann Raios Acústicos Espaço-tempo Geodésica