الفهرس 
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Abstract
Over the past decades a large amount of work has beer one on thestudy ofprogressive gravity waves on the freE urface because of their effects on ships, breakwaters, ocks, and beaches. Capillary progressive water wavee sessing having surface tension as a restoring force have of considerable importance because of their effect the reflection and scattering of light and microwave dar waves from the wind-swept surface of the sea. However, waves which occur in nature are never, in ~ct, free surface waves, since they are always beneath a VUid of some finite density. Despite this fact, very \ rttle work has been done on finite amplitude interfacial nd internal waves. Here, in this thesis, attention will be I onfined to obtain a periodic solution of the finite Imlitude interfacial progressive capillary and gravity lives. Thi’;; present thesis consists of four chapters :- chapter 1:- a review of the previous works in the field r the waves in fluids, such as: surface gravity rogress-i ve waves, cap ill arywaves, capillary-gravity rogressive waves, progressive interfacial () capillary- avity waves are presented. It also contains a complete ’- Jrivation of the equations governing the nonlinear capillary-gravity waves attheinterface two incompressible and inviscid fluids, which are by two fixed infinite horizontal planes in two Then, a third order expansion using perturbation technique are obtained. the solution of the third order equations nonlinear interfacial gravity progressive waves using scales method of the time variation will be profile and velocity of propagation are third order approximation. The effect of upper fluid to the amplitude of the higher harmonic in e wave profile and the velocity of propagation will also 3 :- the solution of the third order equations nonlinear interfacial capillary-gravity waves wave profile and the velocity of propagation third order approximation uniformaly valid for all Although this expansion is valid for a wide range of breaks down at two critical wave-numbersthe of two case semi-infinite fluids hese at wave-numbers : I 1 - p 2 3cha ter 4 :- The two-dimensional progressive interfacial pillary-gravity waves in two incompressible and inviscid uids are dealt with near the first singularity. The wave ofile and the velocity of propagation to the third order II. are I