*Article* **Periodic Waves and Ligaments on the Surface of a Viscous Exponentially Stratified Fluid in a Uniform Gravity Field**

**Yuli D. Chashechkin 1,\* and Artem A. Ochirov <sup>2</sup>**


**\*** Correspondence: chakin@ipmnet.ru; Tel.: +7-(495)-434-01-92

**Abstract:** The theory of singular perturbations in a unified formulation is used, for the first time, to study the propagation of two-dimensional periodic perturbations, including capillary and gravitational surface waves and accompanying ligaments in the 10−<sup>4</sup> < ω < 10<sup>3</sup> s−<sup>1</sup> frequency range, in a viscous continuously stratified fluid. Dispersion relations for flow constituents are given, as well as expressions for phase and group velocities for surface waves and ligaments in physically observable variables. When the wave-length reaches values of the order of the stratification scale, the liquid behaves as homogeneous. As the wave frequency approaches the buoyancy frequency, the energy transfer rate decreases: the group velocity of surface waves tends to zero, while the phase velocity tends to infinity. In limiting cases, the expressions obtained are transformed into known wave dispersion expressions for an ideal stratified or actually homogeneous fluid.

**Keywords:** waves; ligaments; gradient flow; singular solution; stratification; viscous fluid
