Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 45
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He deduced the wavelength from thelength of the tube, and multiplied it by the sound frequency to obtain the velocity. Therewas a difficulty in that the measured sounding frequency depended slightly on the intensityof the blowing. Also, the theoretical stationarity condition completely failed if air wasreplaced by water, as Wertheim demonstrated with a special water organ of his owninvention.
Helmholtz suggested that the boundary conditions assumed by previous theorists were oversimplified. In particular, he pointed to the need for a more realistictreatment of the motion of the fluid near the opening of the tube.4Helmholtz's first publication in the field of acoustics, in 1 856, concerned anotherproblem, belonging to the physiological register. Organists had long known that whentwo successive harmonics of the same tone are played together loudly, the base tone isheard. Acousticians verified and generalized this result, but disagreed on the exact combination rule. According to Wilhelm Weber and Georg Simon Ohm, the combination ofthe frequencies mf and nf, where m and n are two integers without a common divisor,yielded the frequency f According to Gustaf Hallstriim, it yielded frequencies of the form(pn - qm) f, where p and q are two other integers.
In order to decide the issue experimentally, Helmholtz invented a clever monochromatic source by placing a tuning fork in acavity resonator whose proper frequencies were mutually incommensurable. Playingtogether pure mf and nf sounds, he heard the combined frequency (m - n)f, and also(m + n) f after he had convinced himself that it should theoretically exist.
His theory wasthat combination tones occurred when the mechanical response of the ear was no longerlinear and involved a term proportional to the square of the sound amplitude. 5The nature of combination tones bore on a central issue of contemporary acoustics,namely, the relevance of Fourier analysis to the perception of sounds, on which Ohm andThomas Seebeck famously disagreed. Helmholtz's main goal was to put an end to thecontroversy and to base the science of acoustics on non-controversial facts. However, thelack of experimental facilities at Bonn, where he had been recently appointed, promptedhim to work on the more mathematical aspects.
Among the acoustic systems in urgentneed of a better theory were resonant cavities, which played a central role in the production and detection of monochromatic sounds, and organ pipes, which Helmholtz used toproduce strong, sustained tones in his acoustic experiments.
63Helmholtz [1 852/53]. On the tripartite structure, cf. Hehnholtz [1 863a] p. 7.4Hehnholtz [1852153] pp. 250 (Doppler), 242-6 (Wertheim); Wertheim [1 848].5Helmholtz [1856] pp. 497-540. The quadratic terms include the cross-product cos 2=ft cos (2·mift + </J),which is the superposition of(1 /2) cos [27T(m + n)ft + </>] and (1/2) cos [Z1r(m - n)ft - </>].60n Bonn, cf.
Helmholtz to Du Bois-Reymond, 5 Mar. 1858, in Kirsten et al. [1986]. On the aims ofHehnholtz's work on combination tones, cf. Turner [1977], Vogel [1993].1484. 1 .2WORLDS OF FLOWOrgan pipesIn 1 859, Helmholtz published in Crelle's mathematical journal a major memoir 'On themotion of air in open-ended organ pipes', with an appendix on spherical resonators.
In theintroduction, he recalled that the 'most important mathematical physicists', includingDaniel Bernoulli, Euler, Lagrange, and Poisson, had dealt with organ pipes. All of themhad simplified the conditions at the opening of the pipe, generally assuming a compressionnode there, and complete rest for the air outside the tube. This procedure only gave a firstapproximation of the true motion, and neglected the damping of the vibrations throughsound emission. Moreover, it disagreed with Wertheim's frequency measurements, and itcontradicted recent experimental determinations of the locations of the nodes.7In order to remedy these defects, Helmholtz grappled with the daunting problem of themotion of the air near the opening of the tube.
He managed to determine the empiricallyinteresting parameters of the sounding pipe-position of the nodes, frequency and intensity of the emitted sound, and phase relations-without simplifying this motion. The secretof this mathematical feast was a multiple application of Green's theorem, which was wellknown to the Germans since the publication of Green'sEssay in Crelle's journal. With thistheorem and a number of analytical tricks, Helmholtz not only solved a particularproblem of acoustic importance, but he also inaugurated a general strategy for determining relations between controllable aspects of wave propagation when the explicit solutionof the wave equation is inaccessible.
Gustav Kirchhoff's diffraction theory is a directdescendent of Helmholtz's paper on organ pipes; modern scattering theory or wave-guidetheory are more remote ones.8Hehnholtz compared his theoretical formulas for node location and sounding frequencywith measurements made by Wertheim and by Friedrich Zamminer. The agreement wasreasonably good for wide tubes, but poor for narrow tubes for which it should have beenbest (since the theory presupposed a wavelength much larger than the opening). When bepublished his memoir, in 1 859, Helmholtz believed that the discrepancy could be explainedby the known difference between the sounding frequency of blown pipes and theirresonance frequency.9As was well known, friction broadens the response of a resonator to periodic excitations.
In an organ pipe there is friction due to the viscosity of the air. The width of theresonance should increase with this friction, and therefore with the narrowness ofthe pipe.Helmholtz probably had in mind this effect of viscosity when he faced the failure of histheory for narrow tubes.
His improved theory of 1 863, which we will consider shortly,established that viscosity implied both a broadening and a shifting of the resonancefrequency of organ pipes.4.2 Vortex motionHelmholtz studied the general effects of internal friction on fluid motion in the sameperiod, 1 8 58/59, probably because he had in mind an application to organ pipes. In any7Helmho1tz [1859] pp. 303-7.'cr. Darrigo1 [1998] pp.
7-10.9He1mho1tz [1859] pp. 314-15; Wertheim [1851]; Zamminer [1856].149VORTICEScase, he knew that the viscosity of fluids could cause considerable deviations of experimentfrom theory. Being unaware of previous mathematical studies of this problem by Poisson,Navier, Saint-Venant, and Stokes, he proceeded to 'define the influence [of friction] and tofind methods for its measurement'.
In his opinion, the most difficult aspect of this problemwas to gain an 'intuition of the forms of motion that friction brings into the fluid'. Suchwas the motivation of his memoir of 1 858 on vortex motion. 1 04.2.1Fundamental theoremsAs Helmholtz knew from Lagrange, when an incompressible, non-viscous fluid is set intomotion by forces that derive from a potential or by the motion of immersed solid bodies, apotential exists for the fluid velocity.
Frictional forces never derive from a potential (forthey are not conservative), and are therefore able to induce states of motion for which apotential does not exist. Helmholtz's first step was to study motions of this kind, independently of the forces that caused them.As Helmholtz explained without knowing of Stokes's earlier demonstration, the mostgeneral infinitesimal motion of the volume element of a continuous medium can bedecomposed into a translation, three dilations along mutually-orthogonal axes, and arotation around a fourth axis.
In the case of a fluid, the infinitesimal rotation has theangular velocity oo/2, with00 = \7 X V.(4. 1 )Hence the mathematical condition for the existence of a velocity potential, 'V x v = 0 , canbe interpreted as the absence of local rotation in the instantaneous motion of the fluid.Conversely, the absence of a velocity potential signals the existence of vortex motion in thefluid.
1 1Helmholtz next examined how the vortices evolved i n time. For this purpose he wroteEuler's equation asII8v- + (v 'V)v = - - 'VP - - 'V V,ar·PP(4.2)where P is the pressure, p is the constant density, and V is the potential of external forces(for instance, gravitational forces). The continuity equation reads\7 · V = 0.(4.3)Applying the operation 'V x to Euler's equation, Helmholtz obtained the further equation000er + (v .
'V)oo = (oo . 'V)v,(4.4)10Helmholtz [1 858] p. 102. That this publication antedated that on organ pipes by a few months does notexclude the reverse chronology adopted here for their genesis.1 1Helmholtz [1858] pp. 104-8. In a letter he wrote to Moigno (quoted in Les mondes 17 (1868), pp. 577-8),Helmholtz named Kirchhoff [1882] (memoir on vibrating plates) as his source for the decomposition, FranzNeumann as Kirchhoff's source, and Cauchy as Neumann's probable source. Kirchhoff only used the principaldilations ([1882] pp. 246-7).
Cauchy [1841] p. 321 introduced the 'rotation moyenne'. As was said in Chapter 3,Stokes [1849a] introduced the decomposition to prepare his derivation of the Navier-Stokes equation.WORLDS OF FLOW150already known to d'Alembert and Euler. Helmholtz's brilliant innovation lay in thefollowing kinematic interpretation.1 2The left-hand side of eqn (4.4) represents the rate of variation of the vector w for a givenparticle of the fluid moving with velocity v. Helmholtz first considered a particle for whichthe rotation w/2 vanishes at a given instant. Then eqn (4.4) implies that the rotation of thisparticle remains zero at any later time.
In the general case, it may be rewritten ass dw= v(r + sw) dt - v(r) dt,(4.5)where dw is the variation of w on a given fluid particle during the time dt, and s is anarbitrary infinitesimal quantity of second order. In order to interpret this relation, Helmholtz considered two fluid particles located at the points r and r + sw at time t. At timet + dt, the first particle has moved by v(r)dt and the second by v(r + sw) dt. According toeqn (4.5), the two new locations are separated by sw + sdw, which is parallel to the newrotation vector.13For a more intuitive grasp of this result, He!mholtz defined 'vortex lines' that areeverywhere tangent to the rotation axis of the fluid particles through which they pass,and 'vortex filaments' that contain all the vortex lines crossing a given surface element ofthe fluid. As a first consequence of eqn (4.5), two particles of the fluid that belong to thesame vortex line at a given instant still do so at any later time.
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