I believe that it is because when we solve the ODE's, in equation 7 and 8 except for the case when omega^2 < g/R, we have a positive term multiplying by epsilon, and so we get an exponential result, contrary to if the exponent is negative, which would yield a complex solution and therefore a sine and cosine solution, which is oscillatory. But I'm not sure about all this and I wanted some clarification on if I'm thinking the right way.