P-modes of a polytropic convection zone with an overlying hot envelope

Aims. To investigate the behavior of non-radial stellar p-modes with high horizontal wave numbers l, a plane layer approximation is sufficient. In the k - omega diagram, the ridges of the p-modes are strongly influenced by the structure of the atmospheric layers. We present a one-layer model the wave equation of which can be solved in closed form. The layer consists of a polytropic convection zone smoothly joined by an envelope with exponentially increasing temperature. We investigate the behavior of p-modes. As the model is convectively unstable there are no g-modes. This shortcoming is not significant as we discuss only p-modes. Methods. The adiabatic wave equation is reduced to Whittaker's equation. As the dispersion relation of the p-modes is a fourth order algebraic equation in omega, the omega(k)- relation can be given in closed form. Results. We discuss the form of the ridges of the diagnostic diagram. It is shown that the modes concentrate at the position of the temperature minimum in the high-frequency limit. A comparision of the ridges with the ridges of a convection zone with an isothermal atmosphere is performed. In the diagnostic diagram, below the f-mode, there is a continuous spectrum. The correspondig waves behave as gravity waves in the range of the exponential temperature increase. It is discussed whether there are resonances in the continuos spectrum. Solutions of the wave equation of vertically propagating waves are presented.