By Harry W. Reddick
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Additional info for Differential Equations (first edition)
A remote sensing method of estimating the temperature profile of the atmosphere is based on a collection of microwave signals transmitted from a satellite. The physical basis for the method relies on the fact that the microwave radiation is absorbed by molecules. The excited molecules then re-emit radiation, at a rate dependent upon the temperature, according to a known physical law (Planck's law). The idea of the method is to infer the temperature profile from measurements of the emergent intensity of radiation at the base of the atmosphere.
0 This exercise points to an equally interesting inverse problem associated with the draining vessel. Namely, suppose that the water level y can be measured, but the cross-sectional area A(y) is inaccessible. 3 (a sealed institutional coffee urn with unknown internal geometry is an appropriate visualization). 6). 1!.. 7) dt (note that in practice there is no region where 'fjf = 0 and hence A(y) 1s m principle identifiable). Although (3. 7) uniquely determines A(y), we see that practical determination of A(y) will be at the mercy of the instabilities inherent in the differentiation of the measured water level y(t).
When sufficiently heated, a material incandesces and radiates energy in the form of heat and light. The power dissipated as incandescent radiation is distributed over a range of wavelengths and a spectrometer may be used to analyse the power spectrum, that is, the distribution of power over various wavelengths. 15 is a simplified picture of a spectrometer based on an interferometer ~ fixed mirror sample beam splitter t/~ Q ,... 15 In the interferometer, a beam leaves the sample and strikes a beam-splitter (a pane of half-silvered glass) which sends part of the beam vertically and allows part to pass through.