Rays, beams and quasimodes on thin shell structures

We present a theory of flexural wave propagation on elastic shells having nontrivial geometry and develop an analogy to geometric optics. The transport of momentum within the shell itself is anisotropic due to the curvature, and as such complex classical effects such as birefringence are generically found. We determine the equations of reflection and refraction of such waves at boundaries between different local geometries, showing that waves are totally internally reflected, especially at boundaries between regions of positive and negative Gaussian curvature. We verify these effects by using finite element simulations and discuss the ramifications of these effects for the statistical mechanics of thin curved materials.

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“…(3) and (4) of the form ðx; tÞ ¼ e iqÁxÀi!t ðq; !Þ, ðx; tÞ ¼ e iqÁxÀi!t ðq; !Þ, obtaining a dispersion relation !ðqÞ as the solution to [29,30] …”

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confidence: 99%

Reflection and Refraction of Flexural Waves at Geometric Boundaries

Evans

Levine

2013

Phys. Rev. Lett.

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“…Consider such a wave propagating in the direction specified by a unit vector u at a point on the shell, with harmonic time dependence e Ϫit . The local wave number k satisfies 18,19…”

Section: B Thin Shell Ray Equationsmentioning

confidence: 99%

“…͑3͒ define the ray trajectories. 19 However, to a first approximation we can ignore the influence of the fluid-loading term, which is small, and instead consider the simpler eikonal equation…”

Section: B Thin Shell Ray Equationsmentioning

confidence: 99%

“…A differential equation for A follows from Eqs. ͑8͒ and ͑9͒ by taking variations with respect to a parameter orthogonal to s. The resulting equation is called the variational ray equation or the equation of geodesic variation ͑when cϵ0͒, and is 19 c d ds…”

Section: B Thin Shell Ray Equationsmentioning

confidence: 99%

“…Norris and Rebinsky decomposed Pierce's general dispersion relation into that associated only with each distinct wave type and expressed the ray and transport equations in relatively simple forms. 18,19 In addition, they derived general results for the interaction of an acoustic field with a smooth thin shell submerged in a fluid. 17 The ray-type representations developed by these authors are valid for arbitrary smooth shells of variable curvature.…”

Section: Introductionmentioning

confidence: 99%

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Ray tracing over smooth elastic shells of arbitrary shape

Yang

Norris

Couchman

1996

The Journal of the Acoustical Society of America

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An efficient numerical scheme based on ray theory is developed for the analysis of elastic waves traveling over a fluid-loaded smooth elastic shell of arbitrary shape. The shell's surface is first discretized into a number of small patches. The local geometry of each patch is then approximated in a parametric form using bi-cubic spline functions. A local curvilinear coordinate frame is defined on each patch. The ray trajectories and ray-tube areas are obtained by solving a set of ordinary differential equations, the ray and transport equations, within each patch. Several numerical tests of the accuracy and efficiency of the scheme were carried out on spherical and ellipsoidal elastic shells. The numerical results for the spherical shell agree well with analytical solutions. The ray trajectories and the ray-tube areas over an ellipsoidal shell with three distinct semiaxes clearly illustrate the inhomogeneous and anisotropic effects due to the variable curvature on the shell's surface. It is also observed numerically that the magnitude of the ray-tube area along a ray is directly correlated with the stability of its trajectory.

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Quasimodes in Integrable Systems and Semi-Classical Limit

Baldo

Raciti

2016

Essays in Mathematics and Its Applications

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Rays, beams and quasimodes on thin shell structures

Cited by 14 publications

References 15 publications

Reflection and Refraction of Flexural Waves at Geometric Boundaries

Reflection and Refraction of Flexural Waves at Geometric Boundaries

Ray tracing over smooth elastic shells of arbitrary shape

Quasimodes in Integrable Systems and Semi-Classical Limit

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