## 1 Introduction  
Marangoni-Benard convection still generates much interest in fluid and nonlinear dynamics due to its complexity. When the fluid layer is heated from below, convective instabilities can be driven by surface or buoyancy forces [1]. The role of surface-tension gradients in inducing convective instability through Marangoni stresses at the air-liquid interface in a thin layer initially at rest, heated from below, was characterized in the seminal works of [PERSON] and [PERSON] [2] and [PERSON] [3] and the role of surface deformation and surface tension gradients in the onset of patterned convection and oscillatory instability is reviewed in [4]. When the layer is heated from above, however, only overstability can be excited at sufficiently high Marangoni or Rayleigh (buoyancy) numbers. The most common chemical engineering context for a cold liquid heated from above is a condenser, which comprises a hot vapor that condenses over a colder liquid chilled from the solid support below. There are many different configurations for condensers, but all of them would condense faster if interfacial waves are excited on the interface. Thus, a stability theory that shows under what conditions self excitation occurs could better inform the design and operation of condensers. The scenario also describes an evaporator that is operated by contacting hot gas over the cold liquid, as long as there is no fluid motion imposed. Of course,imposing fluid motion adds additional complication but has been found to accelerate performance in novel contactors (evaporators, condensers and distillers) [5]. Classical experiments by [PERSON] and coworkers [6] demonstrate a series of wave instabilities excited at high Marangoni numbers, although they could not discern whether the excited waves were surface or internal waves. It was later posited that the waves were surface manifestations of soliton solutions to a dissipative variation of the Korteweg-de Vries equation. Experiments have shown that solitary waves, excited and sustained by Marangoni stresses, undergo interactions and collisions that return the waves to the pre-collision celerities and shapes, experiencing at most a phase shift, but with either sense possible [7]. [PERSON] and [PERSON] [8] demonstrated via multiple scale perturbation methods the definitive derivation of the dissipative nonlinear evolution equation (termed the KdV-KSV equation) in a sufficiently thin layer that buoyancy effects are neglected. Their study assumed a stress-free lower boundary. The KdV-KSV theory predicts a critical Marangoni number \(M_{crit}=12\), irrespective of capillary, Prandtl and gravity numbers. The interpretation of this result is simply that surface-tension gradients, if sufficiently strong, overcome viscous dissipation in the surface layer to start the fluid oscillations. These oscillations must then combine two modes--the gravity waves modified by Marangoni stresses that are normal to the surface (termed transverse waves) and elastic-like waves that are tangential to the surface (termed longitudinal waves). A study of solutocapillary Marangoni-induced interfacial waves is given in [9]. The purpose of this paper is to test the regime of validity of the KdV-KSV theory on two points--(i) the assumption of the multiple scale theory that there is a long wave cut-off for a wave-packet of unstable waves just above critical stability; (ii) the affect of the no-slip lower boundary on the coefficients of the linear terms of the KdV-KSV equation--by computing the full linear stability theory (LST) of the Benard-Marangoni problem when heated from above. Since, in the basic state, the fluid is at rest and only the static pressure among the field variables has a vertical dependency, it is possible to formulate the analytic solution to the linearized equations and subsequently the secular equation for excitation of non-trivial solutions for the streamfunction, temperature, and surface displacement.