WAVE MOTION 11 (1989) 419-426 419 NORTH-HOLLAND THE EFFECT OF NEWTONIAN COOLING ON THE REFLECTION OF VERTICALLY PROPAGATING ACOUSTIC WAVES IN AN ISOTHERMAL ATMOSPHERE Hadi Y. ALKAHBY and Michael YANOWITCH Department of Mathematics and Computer Science, Adelphi University, Garden City, N Y 11530, U.S.A. Received 8 April 1988, revised 12 September 1988 Upward travelling small amplitude acoustic waves in an isothermal atmosphere will be reflected downward if the gas is viscous or thermally conducting. The presence of small viscosity creates a layer which acts like an absorbing and reflecting barrier for waves generated below it. Above it the motion decays while below it the effect of viscosity is negligible. In the case of thermal conduction the effect is different since the oscillatory motion in the upper region approaches an isothermal one because of the exponential increase with height of the thermal diffusivity.Thus the local sound speed in the upper region approaches the Newtonian sound speed, and the difference in the sound speed in the two regions results in partial reflection and transmission. The addition of Newtonian cooling affects mainly the lower region, in which it produces wave attenuation and alters the wave length. If the Newtonian cooling is large, the temperature in the lower region evens out and the wave motion in the lower region approaches an isothermal one. This eliminates the attenuation since the isothermal regime is dissipationless. The effect on wave reflection is most pronounced when large Newtonian cooling is combined with thermal conduction. In this case the motion approaches an isothermal one in both regions, as a consequence of which the reflection is eliminated altogether. The case of a horizontal magnetic field is also considered since its effect can be easily combined with that of viscosity. 1. Introduction In this paper we study the effects of N~,wtonian cooling on the vertical propagation of acoustic waves in an isothermal atmosphere. Newtonian cooling, which refers to radiative heat exchange proportional to the temperature perturbation, is significant enough to be taken into account in many wave problems. Lindzen [4] considered vertically propagating waves in an isothermal atmosphere with Newtonian cooling increasing exponentially with height, a somewhat unrealistic model which mimics the results of viscous dissipation. In [5] Lindzen dealt with internal gravity waves in realistic atmospheres and found the effect of Newtonian cooling to be 'not profound'. It is also customary to include Newtonian cooling in hydromagnetic wave problems connected with the solar atmosphere (see e.g. [2], [11]). We will consider one-dimensional motion only, in which the effect of Newtonian cooling is combined with that of viscosity and a horizontal magnetic field, and with thermal conduction. It will be shown that Newtonian cooling, when it is large, can have a surprisingly interesting effect on the reflection and attenuation of waves. Of course, these results apply only to the simple model considered, which, because of the restriction to one-dimensional acoustic waves, rules out the internal gravity waves which are present in the three- dimensional problem. It was shown in [9] and [10] that the presence of even small viscosity or thermal conduction in an isothermal atmosphere can profoundly alter the propagation of waves because of the exponential increase with height of the respective diffusivities. If viscosity alone is present, a layer is created which dissipates 0165-2125/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)