One of the fundamental steps in the process of bounded thin liquid films atomization by transverse ultrasonic vibrations is the hydrodynamic instability. It is assumed that liquid droplets are formed from trie crests of surface waves as a result of this instability. The case of ideal liquid films has been already treated . It has been established that the liquid film free surface behaviour is described by the Mathieu equation. This work presents a linear theory derived from the hydrodynamics in order to study the behaviour of the free surface of viscous bounded liquid films.