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Euler first theory of resonance

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Abstract

We examine a publication by Euler, De novo genere oscillationum, written in 1739 and published in 1750, in which he derived for the first time, the differential equation of the (undamped) simple harmonic oscillator under harmonic excitation, namely the motion of an object acted on by two forces, one proportional to the distance traveled, the other varying sinusoidally with time. He then developed a general solution, using two different methods of integration, making extensive use of direct and inverse sine and cosine functions. After much manipulation of the resulting equations, he proceeded to an analysis of the periodicity of the solutions by varying the relation between two parameters, \(a\) and \(b\), eventually identifying the phenomenon of resonance in the case where \(2b=a\). This is shown to be nothing more than the equality between the driving frequency and the natural frequency of the oscillator, which, indeed, characterizes the phenomenon of resonance. Graphical representations of the behavior of the oscillator for different relations between these parameters are given. Despite having been a brilliant discovery, Euler’s publication was not influential and has been neglected by scholars and by specialized publications alike.

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Fig. 1

(Source: https://commons.wikimedia.org/w/index.php?curid=37807147, Retrieved May 20, 2021)

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Notes

  1. This solution assumes that at the initial time \((t=0)\), the initial position and the initial velocity are both zero.

  2. In §23 of E126, Euler describes his method as “… Indeed, my method is firstly handled as follows …”, “…The value of \(s\) will be found from a rule postulated by me as follows …”. As such, this method can be considered as an ad-hoc method, meaning that this solution was designed for a specific problem or task, non-generalizable, and not intended to be adapted to other purposes.

  3. The velocity of the body in its motion is \(\frac{{\rm d}x}{{\rm d}t}\), and if we suppose that this velocity is equal [to the velocity] a heavy body acquires when it falls from the height [\(x\)], it is necessary to take [\({\left(\frac{{\rm d}x}{{\rm d}t}\right)}^{2}=x$$]\;\text{or}\;[$${\rm d}t=\frac{{\rm d}x}{\sqrt{x}}\)]; from the latter, the relation between the time \(t\) and the distance \(x\) is known [after integration].

  4. The values for \(s\) and \(\sqrt{v}\) in this table were obtained by Euler from \(s=\frac{{-a}^{2}}{4g}\mathit{sin}\frac{\boldsymbol{t}}{a}+\frac{a\boldsymbol{t}}{4g}\mathit{cos}\frac{\boldsymbol{t}}{a}\) and \(\sqrt{v}=\frac{\boldsymbol{t}\sqrt{a}}{4g}\mathit{sin}\frac{\boldsymbol{t}}{a}\), which are derived in §35 of E126 by series expansion of Eqs. 8 and 9.

  5. Perhaps not for Euler, but for a reader today to discern the phenomenon of resonance from such a list is not an easy task. Although some historians have gone deep into the subject of graphical representation in eighteenth-century science, here we have adopted the graphical representation to better convey Euler’s results and conclusions to a modern reader. As put by Laura Tilling in an extract taken from “Early Experimental Graphs” (Tilling 2009) “…The graphical presentation of experimental data in the physical sciences has several advantages which today are too familiar to require very detailed enumeration. Its greatest strength lies in the clarity and succinctness with which it displays the information contained in tabulated results: for the experimenter a graph provides a rough and immediate check on the accuracy and suitability of the methods he is using, and for the reader of a scientific report it may convey in a few seconds information that could only be gleaned from a table of measurements by hours of close study. There are occasions where only the analysis of experimental graphs will provide the information we require, but usually the actual analysis of results is carried out nowadays by computational methods. The use of graphs is therefore not so much a necessary part of scientific procedure as an extremely useful one, and one that is often taken very much for granted”.

    One might ask why graphs are included in a historical account of Euler's paper, since Euler himself did not include them in the original paper and in fact was moving in his general approach to analysis to eliminate diagrams and visual representations. The answer is that one may view Euler's paper as part of the heritage of mathematics with substantial points of interest for a modern reader. Some historians of mathematics have argued that there is historical validity in looking at past mathematics or physics in a modern way “… It would furthermore be regrettable, in our opinion, if celebration of textual esoterica, and zero tolerance for moderate readability adaptations, made the reading of historical mathematical and scientific texts a fortified protectorate for specialized scholars only. Let us not give up on trying to reach a wider audience, including modern scientists, teachers, students, and the educated public in general. By striving to make translations as accessible as possible, one can hope to stimulate recognition of and wonder about the past among a wider readership than just fellow professional historians of science”. Extracted from “On Translating Mathematics” by Blåsjö and Hogendijk (2018).

References

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Correspondence to Sylvio R. Bistafa.

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Communicated by Craig Fraser.

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Bistafa, S.R. Euler first theory of resonance. Arch. Hist. Exact Sci. 76, 207–221 (2022). https://doi.org/10.1007/s00407-021-00280-5

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