On the Green’s function for the biharmonic equation in an infinite wedge

Abstract:

The Green’s function for the biharmonic equation in an infinite angular wedge is considered. The main result is that if the angle a is less than ${a_1} \cong 0.812\pi$, then the Green’s function does not remain positive; in fact it oscillates an infinite number of times near zero and near $\infty$. The method uses a number of transformations of the problem including the Fourier transform. The inversion of the Fourier transform is accomplished by means of the calculus of residues and depends on the zeros of a certain transcendental function. The distribution of these zeros in the complex plane gives rise to the determination of the angle ${a_1}$. A general expression for the asymptotic behavior of the solution near zero and near infinity is obtained. This result has the physical interpretation that if a thin elastic plate is deflected downward at a point, the resulting shape taken by the plate will have ripples which protrude above the initial plane of the plate.