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williams j dimensional analysis principle similitude 2022

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Textbook in PDF format The purpose of physics is the understanding of the world around us, not the solving of differential equations. Such equations are often a means to an end, but not the end in themselves. And if that end can be reached by simpler means; especially more physically transparent and intuitive means, all to the good. In this volume, we will look at the technique of dimensional analysis. And how the foundations of dimensional analysis were laid in the first years of the last century; particularly with the work of Lord Rayleigh (John William Strutt, 3rd Baron Rayleigh, 1842–1919). Interestingly, Lord Rayleigh’s paper in Nature (1915), which demonstrated the power of the technique for handling complex problems without the need for long, difficult experiments, and the solution of coupled differential equations, appeared at about the same time as other applications of dimensional analysis. When presenting his model of the hydrogen atom in 1913, Niels Bohr used a dimensional argument in his article. In addition, in 1911, Albert Einstein looked at the dimensionality of the heat capacity of solids. Evidently, dimensional analysis was very much the Zeitgeist in physics at the beginning of the 20th Century. One of the advantages of treating complex problems by dimensional analysis, rather than by a straightforward solution of a set of equations is that dimensional analysis allows one to incorporate physical insight into the resolution of the problem. This idea goes back to the very first use of dimensional analysis. In 1915, Lord Rayleigh commented that it was a shame that more physicists did not look at problems from the point of view of the underlying dimensions, rather than simply constructing sets of differential equations, and then attempting to solve those equations, or even undertaking a long series of complex experiments. ‘I have often been impressed by the scanty attention paid even by original workers in physics to the great principle of similitude. It happens not infrequently that results in the form of ‘laws’ are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes’ consideration. However, useful verification [of a hypothesis by experiment] may be, whether to solve doubts or to exercise students, this seems to be an inversion of the natural order. One reason for the neglect of the principle may be that, at any rate in its applications to particular cases, it does not much interest mathematicians.’ Apart from the splendid Victorian prose, this is quite an indictment from one of the then greatest living physicists. But Lord Rayleigh was making a general comment upon the attitudes of both experimentalists and theoretical physicists, that both should spend more time examining the dimensions involved in the phenomena they investigated. Dimensional analysis, or the principle of similitude, to use Lord Rayleigh’s term, usually gets short shrift in textbooks. What one finds is mostly an admonition about checking the dimensional homogeneity of equations as a way of finding errors. This is good advice, but perhaps lacking in profundity. The remarkable power of dimensional analysis to obtain approximate results quickly and easily is not often fully exploited. By the time most students come across dimensional analysis, they seem to have been thoroughly inculcated in the belief that physics problems must entail the solving of differential equations, and any attempt to side step this rite of passage is a form of cheating. But solving physical problems by dimensional analysis is perfectly respectable; Lord Rayleigh’s law involving the inverse fourth power of wavelength for electromagnetic scattering by objects, small compared with the wavelength of the incident radiation, was first obtained by dimensional arguments — and that analysis explains why the sky is blue

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