Some problems in acoustics of emulsions

On the spectrum of an integro-differential equation arising in viscoelasticity theory

“…The interest in the study of spectral properties of such problems is caused by numerous applications of integro-differential equations to mechanics and physics, in particular, in the visoelasticity theory and the theory of heat transfer. For example, the dynamics of a one-dimensional viscoelastic medium with long-time memory is described by the following integro-differential equation:where ρ is the density of the viscoelastic medium, u(x, t) is displacement of the point with abscissa x at time t at the equilibrium state, α and β depend on properties of the viscoelastic medium, g(t) is the convolution kernel, the prime denotes the derivative with respect to x. Equations of the form (1) also arise as a result of homogenization of the acoustic equations for periodic combined media of two viscous fluids [4,5] or of porous or viscoelastic material and a viscous liquid occupying pores [6]-[8].This paper is devoted to the spectral analysis of Equation (1) with the homogeneous initial and boundary conditions u(0, t) = u(l, t) = 0, t > 0, u(x, 0) =u(x, 0) = 0, x ∈ (0, l). …”

“…where ρ is the density of the viscoelastic medium, u(x, t) is displacement of the point with abscissa x at time t at the equilibrium state, α and β depend on properties of the viscoelastic medium, g(t) is the convolution kernel, the prime denotes the derivative with respect to x. Equations of the form (1) also arise as a result of homogenization of the acoustic equations for periodic combined media of two viscous fluids [4,5] or of porous or viscoelastic material and a viscous liquid occupying pores [6]- [8].…”