Acoustic wave propagation in anisotropic media with applications to piezoelectric materials

ABSTRACT

In this paper, we analyze acoustic wave propagation in anisotropic fluids and solids. By formulating the acoustic system as an evolution equation over a Hilbert space, we obtain global in time solutions when the associated material parameters are bounded and measurable. In particular, we prove well-posedness of a Cauchy problem for wave propagation in piezoelectric crystals. We then provide a stability analysis of these solutions not assuming positive definiteness of the stress–strain tensor or the piezoelectric stress tensor. Finally we prove continuous dependence on initial data, allowing the piezoelectric tensor to depend on space and time, provided solutions belong to an appropriate function space.

Keywords:

• Anisotropy
• acoustics
• piezoelectricity
• stability
• Cauchy problem

2010 Mathematics Subject Classifications:

• 35L90
• 35L53
• 74E10
• 74B05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Alternatively (and equivalently) we can assume the analog of (H3) for the velocity $c\left(x\right)$.

2 Note that by assumptions on $\rho \left(x\right)$ it is easy to check that the space $\mathcal{H}$ is indeed a Hilbert space with respect to the “measure” $\rho \left(x\right)\mathrm{d}x$.