Acoustical Impedance and the Theory of Horns and of the Phonograph.

The introduction more than thirty years ago of the term 'impedance' by Mr. Oliver Heaviside has been productive of very great convenience in the theory of alternating currents of electricity. Unfortunately, engineers have not seemed to notice that the idea may be made as useful in mechanics and acoustics as in electricity. In fact, in such apparatus as the telephone one may combine the notions of electrical and mechanical impedance with great advantage. Whenever we have permanent vibrations of a single given frequency, which is here denoted, as usual, by n/27r, the notion of impedance is valuable in replacing all the quantities involved in the reactions of the system by a single complex number. If we follow the convenient practice of denoting an oscillating quantity by ei" and taking its real part (as introduced by Cauchy) all the derivatives of eft are obtained by multiplication by powers of in, or graphically by advancing the representative vector by the proper number of right angles. If we have any oscillating system into which a volume of air X periodically enters under an excess pressure p, I propose to define the impedance by the complex ratio Z = p/X. If we call dX/dt = I the current as in electricity, if we followed electrical analogy we should write Z = pI so that the definition as given above makes our impedance lead by a right angle the usual definition. I believe this to be more convenient for our purposes than the usual definition and it need cause no confusion. If we have a vibrating piston of area S as in the phonometer, we shall refer its motion to the volume St it carries with it and the force acting on it to the pressure, so that F = Sp. The differential equation of the motion is