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The electrodynamical response of the edge of a compressible Quantum Hallsystem affects tunneling into the edge. Using the composite Fermi liquidtheory, we derive an effective action for the edge modes interacting withtunneling charge. This action generalizes the chiral Luttinger liquid theory ofthe Quantum Hall edge to compressible systems in which transport ischaracterized by a finite Hall angle. In addition to the standard terms, theaction contains a dissipative term. The tunneling exponent is calculated as afunction of the filling fraction for several models, including short-range andlong-range Coulomb interaction. We find that tunneling exponents are robust andto a large extent insensitive to the particular model.  We discuss recent tunneling measurements in the overgrown cleaved edgesystems, and demonstrate that the profile of charge density near the edge isvery sensitive to the parameters of the system. In general, the density isnon-monotonic, and can deviate from the bulk value by up to 30%. Implicationsfor the correspondence to the chiral Luttinger edge theories are discussed.Comment: 26 pages, 8 figure

Effective action of a compressible quantum Hall state edge: Application to tunneling