PHYSICS: A. G. WEBSTER 
275 
ACOUSTICAL IMPEDANCE, AND THE THEORY OF HORNS AND 
OF THE PHONOGRAPH 
By Arthur Gordon Webster 
Department of Physics, Clark University 
Communicated, May 8, 1919* 
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 fre- 
quency, which is here denoted, as usual, by w/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 e^"^ and taking its real part (as introduced by 
Cauchy) all the derivatives of e^"^ 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 = / 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 beheve 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 S^ it carries with it and the force acting on it to the 
pressure, so that F = Sp. The differential equation of the motion is 
we have 
Z^=^ {f -mn^ -\-iKn) / S\ ' (2> 
where m is the mass, k the damping, / the stiffness. The real part of S'^Z^. 
f — mn^, is the uncompensated stiffness, which is positive in a system tuned 
too high, when the displacement lags behind the force, by an angle between 
zero and one right angle, negative when the system is tuned too low, when the 
* This article was read in December 1914 at the meeting of the American Physical Society 
at Philadelphia, and has been held back because of the continual development of the experi- 
mental apparatus described in a previous paper in these Proceedings. 
