The Effect of Intermediate Layer on the Stability of Helmholtzian Wave

Abstract

According to the recent investigation by H. Solberg and C. L. Godske the “Scherungswirkung” due to the difference of velocity across the discontinuous surface is considered to be the only factor inducing the instability of wave motion. Therefore this factor is very important in the wave theory of cyclone. The present paper aims at the modification of the above conclusion by the presence of intermediate layer. Consider the three layers of different velocity distribution, the upper and the lower stratum having the uniform velocity U₂ and U₁ respectively, both of infinite extent, and the transient layer of the width b' varying linearly from U₁ to U₂ in velocity.
The period equation is derivable as, on the assumption of no density difference, The discriminant of this equation is given by which, in case of b'→O, reduces to the usual characteristic equation of Helmholtzian wave: Here β is the frequency, Δ=(U₂-U₁)/b' and α=2π/L, L being the wave length. It is readily seen that the instability occurs in the following domain: 4.9 b'<L, while in the domain: 4.9 b'>L, the wave motion is stable. The above result is very interesting in the light of usual theory on the Helmholtzian wave in which the wave motion of any wave length is unstable when the distribution of density is uniform.