A SINGULAR EXACT SOLUTION FOR NONLINEAR SHALLOW-WATER ON A ROTATING EARTH

In this paper, we present an exact solution for nonlinear shallow water on a rotating planet. It is a kind of solitary waves with always negative wave height and a celerity smaller than linear shallow water propagation speed square-root gh. In fact, it propagates with a speed equal to (1 + a/h) square-root gh(1 + a/h) where a is the negative wave height. The lowest point of the water surface is a singular point where the first order derivative has a discontinuity of the first kind. The horizontal scale of the wave has actually no connection with the water depth.

Methane dehydro-aromatization over a Mo/phosphoric rare earth-containing penta-sil type zeolite in the absence of oxygen

The dehydro-aromatization of methane over a Mo-modified penta-sil type high-silica zeolite containing phosphoric and rare earth oxide (abbreviated as Mo/HZRP-1) was investigated. As a modification of HZSM-5, HZRP-1 is also a good support for the preparation of Mo-based zeolite catalysts, and is active for methane dehydro-aromatization. Mo/HZRP-1 catalysts are more active at high Mo loadings compared with Mo/HZSM-5 catalysts. Al-27 MAS NMR spectra of Mo/HZRP-1 reveal that there are two kinds of framework Al in HZRP-1. It is suggested that only the tetrahedral coordinated Al atoms in the form of Al-O-Si species in the zeolite, in the proton forms, are responsible for the formation of aromatics.