Time-harmonic waves propagate along a cylindrical waveguide in which there is an obstacle. The problem is to calculate the reflection and transmission coefficients. Simple explicit approximations are found assuming that the waves are long compared to the diameter of the cross-section d. Simpler but useful approximations are found when the lateral dimensions of the obstacle are small compared to d. Results for spheres, discs, and spheroids are given.

We derive an approximate but fully explicit formula for the mean first-passage time (MFPT) to a small absorbing target of arbitrary shape in a general elongated domain in the plane. Our approximation combines conformal mapping, boundary homogenisation, and Fick–Jacobs equation to express the MFPT in terms of diffusivity and geometric parameters. A systematic comparison with a numerical solution of the original problem validates its accuracy when the starting point is not too close to the target. This is a practical tool for a rapid estimation of the MFPT for various applications in chemical physics and biology.

A metasurface comprising cavities in a soft medium has been proven to be highly efficient for control of water-borne sound waves. We formulate an analytical model to predict the acoustic performance of a soft elastic medium embedded with disk-shaped cavities and submerged in water. Each layer of cavities is approximated as an effective boundary that incorporates the effect of multiple scattering of waves and accounts for different damping mechanisms. The results from our analytical model are compared with numerical and experimental results from the literature.

We present an analytical framework to investigate the acoustic performance of an array of closely spaced spherical cavities embedded in a thin soft medium submerged in water. Each layer of cavities is approximated as a homogenised layer with effective properties. Strong monopole resonance of the cavities and multiple scattering of waves between cavities in proximity are taken into account. Analytical results for the metasurface with and without a rigid backing are compared with numerical simulations as well as with experimental results from literature.

A laminar stationary flow of viscous fluid in a cylindrical tube enhances the rate of diffusion of Brownian particles along the tube axis. This so-called Aris-Taylor dispersion is due to the fact that cumulative times, spent by a diffusing particle in layers of the fluid moving with different velocities, are random variables which depend on the realization of the particle stochastic trajectory in the radial direction. Conceptually similar increase of the diffusivity occurs when the particle randomly jumps between two states with different drift velocities. Here we develop a theory that contains both phenomena as special limiting cases. It is assumed (i) that the particle in the flow can reversibly bind to the tube wall, where it moves with a given drift velocity and diffusivity, and (ii) that the radial and longitudinal diffusivities of the particle in the flow may be different. We derive analytical expressions for the effective drift velocity and diffusivity of the particle, which show how these quantities depend on the geometric and kinetic parameters of the model.