This article is about those electroacoustic transducers that convert a varying electrical signal into a varying acoustic pressure, usually in air but sometimes in water or other liquids and solids. These are the inverses of Microphones, and much of what is said in that article is applicable to them as well. The earliest such transducer was the magnetic telephone receiver, invented by Bell, which can also serve as a transmitter. Its commonest manifestation is as a loud speaker, or loudspeaker, or simply a *speaker*, which appeared as an appurtenance of a radio receiver. Let's use speaker as a common term for all such devices, of any size or application.

Rather than describe speakers in detail, we shall concentrate here on three fundamentals that explain their operation: acoustic radiation, motional impedance, and the theory of horns. All of these are intrinsically interesting and have wider application. The reader will need to know something about oscillators and mechanical impedance, which can be found in the article on microphones.

Any passive microphone can be inverted to make a speaker, but in most cases the electrical and acoustical characteristics are not suitable, so that speakers must be specifically designed for their applications. An active microphone, such as the carbon granule microphone, has no speaker analog. Similarly, there are some speakers that cannot be turned into a microphone, though most can.

Every speaker consists of two elements, the *motor element* that converts the electrical signal into a mechanical force, and the *acoustic radiator* that matches the mechanical output to the acoustic medium. The first is the interface between electrical and mechanical systems, the second between the mechanical system and the medium. Most microphones have only a rudimentary acoustical receiver, but it is an essential part of a speaker.

Inventive ingenuity has discovered many speakers, so we should first briefly note those ideas that have not been used to any extent. A hot wire of very small thermal inertia creates variations in pressure as the current through it is varied. Such pressure variations have been used to calibrate microphones, and a thermal earphone was even developed. Without a biasing current, double frequency acoustic output results (as double-frequency electrical ouput occurs in the hot-wire microphone). Electrical signals have been used to operate a delicate valve controlling compressed air. This *stentorphone* gave very loud sound, and could also be controlled by the needle of a phonograph. Sound is produced when electricity flows through certain frictional contacts. The sound quality is not high, but the curiosity value is. A "talking arc" can be created that radiates sound, but an arc is not a comfortable companion. Magnetostriction and electrostriction also create sound, but it is double frequency since both phenomena depend on the squares of the fields. A magnetostrictive receiver was used with Reis' telephone, but it could only reproduce tones, not speech. This should give some idea of the unusual types of speakers that have existed.

Practical speakers have capacitor, piezoelectric or magnetic motor elements. Capacitor speakers are the converse of the capacitor microphone. They must be biased, since the electrical force is independent of the direction of the electric field. Electrets make biasing much easier, but a capacitor microphone still must be supplied with a high voltage to make the force on the diaphragm sufficiently large. Electret microphones cannot be inverted to make a speaker, mainly because of the FET output amplifier, but also because they are not built for the large voltages required. I do not know of any commercially-available capacitor speakers, so we will not consider them further. Their principle of operation, however, is quite clear.

Piezoelectric speakers are very widely used, especially as acoustic transducers that radiate simple tones. The surface of the piezoelectric crystal usually forms the radiating diaphragm in the simplest transducers. The motion of the crystal can also be communicated mechanically to the diaphragm. Piezoelectric speakers have a high input impedance, and generally do not require high voltages.

Most speakers have magnetic motor elements, which are of three kinds. In the first kind, part of the iron constituting the magnetic circuit moves, so they can be called *moving-iron* speakers. This is the case in the Bell telephone receiver, which has an iron diaphragm a short distance from a U-shaped permanent magnet. Coils of many turns of fine wire are wound on pole pieces attached to the permanent magnet, so that the input current strengthens or weakens the field of the permanent magnet, changing the tractive force on the iron diaphragm. In early days, the steel permanent magnet was conveniently accommodated in the receiver handle. Better permanent magnet materials greatly reduced the size of the magnet, so that it could easily be placed in a handset, or even in an earphone. Careful development made this into a very satisfactory receiver, a perfect companion to the carbon granule transmitter, making possible a telephone system without electronic amplification. When used as earphones, these are the *high-impedance* type with kilohms of impedance that are so useful in radio experiments. This type of receiver has the benefit not only of high impedance, but also of high sensitivity. Only recently has it been replaced in telephone receivers by piezoelectric receivers that are much cheaper, but of poorer acoustic quality.

Another receiver of the moving-iron variety was widely used in early radio. The coils were wound on a soft-iron armature placed in a magnetic field so that currents in one direction would cause the armature to move one way, and currents in the opposite direction would move it the other way. This movement was mechanically transmitted to a diaphragm radiator. These speakers worked, but had a very poor high-frequency response.

In a second kind of magnetic receiver, the force is due to eddy currents induced in the diaphragm, which is placed in a magnetic field. This kind of speaker was developed quite successfully by Hewlett, but has not come into general use.

The third kind, and now by far the most widely used, is the *moving-coil* speaker. A light coil that is connected directly to the diaphragm moves in a strong annular magnetic field. A current in the coil creates a proportional force driving the diaphragm. This is the inverse of the dynamic microphone, but the proportions are quite different in the two cases. Moving-coil speakers are highly satisfactory, and illustrate very clearly some basic principles, so most of this article will be devoted to their theory.

A good speaker should have an even frequency response, so that no frequencies are unduly emphasized or weakened; nonlinear distortion, the creation of new frequencies in the output, should be minimized; and the damping should be sufficient to give a good transient response. Efficiency in producing acoustic energy is only a minor consideration, since electrical power is easy to come by, and very little acoustical power is normally required. Even large auditoriums are satistifed with a few watts of acoustic power. The usual large-diaphragm speaker has an efficiency of around 1%, though a horn loudspeaker may have an efficiency of 30%. The usual amplifier powers bragged of are just electrical input power, a hundred times greater than the acoustical power actually produced. The frequency range desired for the reproduction of fine sound is 50 Hz to 15 kHz. It has so far proved difficult to design one speaker to cover the whole range, but subdividing the range and using two or three speakers is a very practical alternative. It is relatively easy to design one speaker for 50-1000 Hz, and another for 500 Hz - 15 kHz.

The acoustic pressure p in an isotropic homogeneous medium satisfies the scalar wave equation in three dimensions: part;^{2}p/∂t^{2} = c^{2} div grad p. This is the same equation that is satisfied by the amplitude in the Fresnel-Kirchhoff diffraction theory in optics, so there are some very close analogies between sound and light propagation. The speed of sound c in air at room temperature is about 343 m/s, and in water is 1500 m/s, values we shall use in examples. The pressure p is the overpressure, the difference from the static pressure p_{o}, which is about 1.013 x 10^{5} Pa. The theory based on the wave equation is valid for small ratios p/p_{o}; in acoustics, this ration is normally less than 0.01, so the theory is quite valid.

In optics, the wavelength is much smaller than the size of ordinary objects, resulting in rectilinear propagation that is well-described by light rays. Normal distances are very many wavelengths long, and phase or time delays short. Atomic sources are much smaller than the wavelength, simplifying their radiation properties. In acoustics the wavelengths corresponding to the audible spectrum, from 50 Hz to 15 kHz are from 6.9 m to 2.3 cm, the dimensions of familiar objects, and within which the sizes of electroacoustic transducers lie. Usual distances may only be a few wavelengths, and echoes and delays quite perceptible. Nevertheless, description by scalar waves is the same in both cases; only the scale is different.

The particle displacement **d** is a vector with rectangular components (ξ,η,ζ) in the Lagrange picture. The *condensation* s = - div **d** is the ratio of the change in density to the static density, s = Δρ/ρ_{o}. It is a dimensionless number. The equation of state then gives p = ρ_{o}c^{2}s. The static density ρ_{o} is about 1.293 kg/m^{3} for air, or 1000 kg/m^{3} for water. In the equations below, ρ will stand for ρ_{o}. For air, this means that p = 1.52 x 10^{5} s Pa, so that if p is small, s is smaller still. In water, it is even more true.

The wave equation is linear, so its solutions obey the principle of superposition. We now assume that any solution can be expressed as the sum of solutions depending on time through e^{jωt}, where the angular frequency ω = 2πf. Time derivatives now are replaced by multiplication by jω, so the wave equation becomes -ω^{2}p = c^{2} div grad p, or (div grad + k^{2})p = 0, where k = ω/c is the propagation constant (Helmholtz's equation). In rectangular coordinates, the solutions of this equation are the familiar plane waves e^{j(ωt - k·r)}, where k and r are vectors, and k is in the direction of propagation.

What we need here are solutions in spherical coordinates, and of these we look at those that are spherically symmetrical, not depending on the angles. Then Helmholtz's equation becomes (1/r^{2})(d/dr)[r^{2}(dp/dr)] + k^{2}p = 0. The equation is satisfied by p = e^{±kr}/r, which correspond to incoming and outgoing spherical waves. The pressure for an outgoing spherical wave will be p = (A/r)e^{j(ωt - kr)}. The particle displacement and velocity will then also be radial. If u is the particle velocity, then the acceleration jωu = -(1/ρ)(dp/dr), so that u = -(1/jωρ)(dp/dr). For our outgoing wave, -dp/dr = p(1/r + jk), so u = (p/jρc)(1/kr + j). The relation between p and u is so important that we call it the *wave impedance* z. Its dimensions are kg/s-m^{2}. From the last equation, z = ρc/(1 - j/kr). When kr is large, z becomes the usual plane-wave impedance, ρc = 444 kg/s-m^{2} for air.

Now we relate our wave to a source, so that we can determine the constant A. Let us assume that the source is a small sphere whose radius oscillates around a value a, which we'll call a "point source." If ka is small, then the velocity in the neighborhood of the sphere will be u = (p/jρc)(1/kr), or, at the surface of the sphere p = (jkaρc)U, where U is the amplitude of vibration of the sphere. The radiated wave is p = (A/r)e^{-kr}, or, on the surface of the sphere, p = A/a, approximately. This gives A = jkρca^{2}U. Since 4πa^{2} is the surface area of the sphere, the combination 4πa^{2}U = Q is a *source strength* imagined as the rate of change of a volume of fluid. Indeed, the dimensions are m^{3}/s. Instead of an oscillating sphere, we could have an oscillating amount of fluid introduced at that point, with the same radiation. Therefore, the pressure due to a small source Q at a point P a distance r from the source is p = (jρck/4πr)Qe^{j(ωt - kr)}, as illustrated in the figure. There is no electromagnetic analogy to this monopole point source, since the conservation of electric charge does not allow an oscillating point charge.

The intensity of the wave at any point is I = p^{2}/2ρc, where p is the peak amplitude of the pressure. The magnitude of the wave impedance is ρc cos θ, where tan θ = 1/kr. Therefore, I = (pu/2) cos θ, a relation analogous to the electrical power, P = (VI/2) cos θ, where V and I are peak values. The total power radiated is the integral of I over a sphere of large radius. For our small source, this is just multiplication by 4πr^{2}, so we find P = (ρck^{2}/8π)Q^{2} W.

Now consider a source of strength Q a distance d from a similar source of strength -Q. If d is small compared to the wavelength, this is called an *acoustic doublet* or dipole of strength Qd m^{4}/s, analogous to an oscillating electric dipole. Just as in the electric case, we can find the pressure by superposing the pressures of the two small sources, with the result p = (jρck/4πr)(Qd cos θ)(jk + 1/r). This is the superposition of an "induction" field depending on 1/r^{2} and a "radiation" field depending on 1/r. θ is the angle between the radius vector to the point of observation and the axis of the dipole. The radiation has forward and backward lobes of opposite phase, with no radiation perpendicular to the axis of the doublet. The radiated power is now P = (ρck^{2}/8π)Q^{2}[(kd)^{2}/3]. The factor in square brackets is always less than unity when our approximations are valid, so a doublet radiates less power than either simple source.

A vibrating diaphragm is analogous to an acoustic doublet, radiating to front and back in opposite phases. Suppose we wish to radiate waves of 100 Hz, or wavelength 3.43 m. The factor (kd)^{2}/3 = 0.0112 if we assume that the distance between the simple sources is 0.1 m. This means that only about 1% of the power is radiated that would be radiated by either the front or back of the diaphragm alone. Acoustic doublets are very inefficient radiators.

We can radiate much more power if we place the diaphragm in a *baffle* that separates the radiation from the two sides. To be effective, the baffle should be at least a wavelength in size; for 100 Hz, this is about 11 ft, so the difficulty in radiating such frequencies is obvious. Many ingenious speaker cabinets have been designed for the purpose of inverting the phase of the low frequencies and radiating them in aid of the front surface, while absorbing the high frequencies, for which the front surface will be effective.

If we place the point source of strength Q in an infinite baffle, the pressure p = (jρck/4πr)Qe^{j(ωt - kr)} does not change, since the particle velocity is parallel to the baffle. However, only Q' = Q/2 is the source strength effective to the right of the baffle, so p = (jρck/2πr)Q'e^{j(ωt - kr)} is the pressure in terms of the source strength, which is twice what the pressure would be if it radiated into 4π steradians, of course. The baffle has no effect on the pressure field, unless we agree that it doubles the source strength.

This result can be used to calculate the pressure due to a distribution of source strength over the baffle, of amount Q' = UdS, just as we calculate the diffraction pattern of a plane source of light. Each area dS contributes a pressure dp = (jρck/2πr)UdSe^{j(omega;t - kr)}, where r is the distance from dS to the observation point P. Let us calculate the radiation field of a piston of radius a that moves with velocity Ue^{jωt}. This is exactly the same problem as the diffraction from a circular aperture, so the result is the same, p = (jρck/2πr)(πa^{2}U)[2J_{1}(ka sin θ)/(ka sin θ)]e^{j(ωt - kr)}, at distances large compared to a. The total source strength is Q = πa^{2}U. The first zero of the angular factor is ka sin θ' = 3.83, where θ' is the half-angle of the main lobe of radiation. sin θ' = 3.83/ka = 1.22λ/d, where d = 2a is the diameter of the piston. This should be quite familiar from optics as the Airy diffraction pattern. There are side lobes of smaller intensity.

Since 2J_{1}(x)/x = 1 - x^{2}/8 + ..., and so approaches unity when x is small, the pressure on the axis θ = 0 is the same as that due to a simple source of the same strength Q. The pressure decreases off axis due to interference between the waves coming from different parts of the piston. The total power radiated is P = (ρcQ^{2}/2πa^{2})[1 - 2J_{1}(2ka)/2ka]. A piston whose diameter is small compared to the wavelength will radiate rather inefficiently in all directions, while a piston whose diameter is large compared to the wavelength will radiate mainly a strong concentrated beam.

We will need the reaction force of the medium on the piston in considering speaker dynamics. This can be found by integrating the force on an element of area dS due to all other elements of area of the piston dS', over the area of the piston. The result is that the reaction force f' = -ρcQe^{jωt}[R(2ka) + jX(2ka)], where R(x) and X(x) are the piston impedance functions, defined by power series. For small values of x, R(x) = x^{2}/8, and X(x) = 4x/3π. For large values of x, R(x) = 1 and X(x) = 4/πx. These functions are tabulated in Reference 1. Then, the ratio of the reaction force -f' to the velocity u of the piston will be z' = ρcπa^{2}[R(2ka) + jX(2ka)] = r' + jx'.

A very interesting aspect of speakers is the effect of the movement of the diaphragm on the electrical characteristics of the speaker. These effects are easy to describe for the moving-coil speaker. First, let's look at the mechanical system of the coil, speaker cone and its suspension. This is an oscillator of mass m, stiffness s, and damping r. The ratio of the force driving the oscillator f to its velocity of motion u is its mechanical impedance z = r + j(ωm - s/ω), so that f = zu. The radiation reaction is -f' = z'u = (r' + jx')u, so that with both forces acting on the oscillator, f + f' = zu, or f = (z + z')u = z"u, where z" = z + z' is the total mechanical impedance of the mechanical system, and f is the applied force.

If the coil has a total length of wire of L in a magnetic field B, then the force driving the oscillator is f = BLI, where I is the current in the coil. Therefore, in the steady state, u = BLI/z". All the forces, currents and velocities in these expressions vary as e^{jωt}, and may be expressed as peak or rms values. When the coil moves with velocity u, an emf E' = BLu is generated, that will oppose the emf driving the current that causes the motion. Therefore, the net voltage applied to the coil will be E - E'. If the impedance of the coil is Z = R + jωL, then E - E' = (R + jωL)I, or E = (BL)(BLI/z") + (R + jωL)I = [(BL)^{2}/z" + R + jωL]I = Z"I. The impedance of the coil is seen to be its impedance when the coil does not move, or is *blocked*, plus the *motional impedance* Z' = (BL)^{2}/z". The coupling factor BL is often denoted by φ

The dimensions of φ are either N/A or V-s/m, while the dimensions of z" are N-s/m, so the dimensions of Z" are (V-s/m)(N/A)(m/N-s) = (V/A) = Ω, which is consistent. The motional *admittance* Y' = 1/Z' = z"/φ^{2} = (r + r')/φ^{2} + j(ωm - s/ω + x')/φ^{2} has an easy interpretation as a conductance G' = (r + r')/φ^{2} = G + G_{r}, a capacitance C' = (m + x'/ω)/φ^{2} and an inductance L' = φ^{2}/s in parallel. This parallel combination is in series with R and L. The equivalent circuit of the coil is shown in the diagram. The conductance has been split in two parts; one, G reflects coil damping, while the other, G_{r} is the effect of radiation. The power in this conductance is the radiated acoustic power.

This equivalent circuit is really rather exciting, since it contains all the electromechanical effects that are operating, and permits the evaluation of the radiated power. It is divided into two parts, the blocked impedance that can be measured with the coil immobilized (in which case C' becomes infinite, or effectively a short circuit), and the motional impedance. There will be a resonance at f = 1/2π√(L'C') at which the motional impedance will consist of the conductances only. At lower frequencies, the motional impedance will be inductive as the cone is stiffness-controlled, while at higher frequencies it will be capacitive, and series-resonant with the blocked inductance. In fact, this behavior is what makes the coil impedance roughly equal to its DC resistance over a rather wide range. This makes the voltage across the radiation conductance relatively constant, so the power radiated is more independent of frequency.

The motional impedance can also be represented as a series circuit, but in this case the "constants" are not as constant. While it gives the same numerical results, it does not seem to me to be as heuristically valuable as the parallel circuit.

The moving-iron telephone receiver can be represented the same way, but its behavior is more complicated. First of all, the blocked inductance is rather large, since the coils have many turns and are wound on an iron core. This causes the resistance to increase with frequency because of core losses. Nevertheless, the fundamental behavior is very much the same. Early receivers had a resonance right in the middle of the operating range, at 800-900 Hz, which helped to reinforce these important frequencies at the expense of distortion.

Our model has only one resonant frequency, for simplicity, but more complicated circuits can be drawn when there are several resonant frequencies, which will give the same information.

It is instructive to plot the resistance R and the reactance X of the speaker as a function of frequency, with R the abscissa and X the ordinate. The blocked impedance, R + jωL plots as a vertical straight line, with X proportional to ω. The motional impedance plots as a circle of radius R' = 1/(G + G_{r}), with the horizontal axis as X = 0, and R = R' at the resonant frequency, the origin corresponding to ω = 0 and ω = ∞. The circle is described in the clockwise direction with increasing frequency. If x = R'(ωC' - 1/ωL'), then (R/R') = 1/(1 + x^{2}) and (X/R') = -x/(1 + x^{2}), so that the equation of the circle is (R/R')^{2} + (X/R')^{2} = R/R'. An *approximate* plot for a speaker is very much like a straight line with a circle tangent to it. With a little effort, an accurate plot can be made.

The impedance function R(2ka) for the piston speaker in an infinite baffle shows how little a small piston can get a "grip" on the air in front of it, and produce a reasonable overpressure for a given amplitude of oscillation. The air simply slides away to the sides without raising the pressure. In other words, there is a large impedance mismatch between the piston and the air. This can be overcome by confining the air so that it cannot move away, by means of a *horn*. Cupping the hands around the mouth to intensify the voice is one way of doing this, as is the use of a conical horn by cheerleaders. Trumpets increase the radiation from vibrating lips, modified by the resonance of the air column. Although a horn hinders spreading of the sound a little, its main effect is to increase the pressure at the sound source, making a better match, allowing more radiation for the same amplitude of vibration. Using a horn with a speaker greatly increases the pressure at the diaphragm, and so the radiated sound intensity, as well as improving the frequency response.

The shape of the horn is a matter of some importance. For a conical horn to be effective at low frequencies, its rate of widening at the throat should be no greater than that of an exponential horn at the same point. Since this is small, the conical horn will be much longer than even the exponential horn. For this reason, conical horns usually diverge far too rapidly to be effective. The usual cheerleader's horn is easy to make, but gives only small aid to the voice. A properly designed exponential speaking trumpet can be much more effective. A horn can be used in reverse to strengthen a received sound, as in an earhorn. Speaking trumpets and earhorns have been developed for verbal communication over remarkable distances; this communication is rather public, however. The exponential horn is the preferred shape because it is shorter than all other horns of the same efficiency at low frequencies.

An *exponential horn* is sketched at the right. Its cross-sectional area A is given by A = A_{0}e^{mx}, or its radius by r = r_{0}e^{(m/2)x}. The radius increases by a factor e in a distance 2/m, which is also the distance at which a line of the initial slope intersects the x-axis. The air in a distance dx where the area is A moves to a distance (1 + dξ/dx)dx to where the area is A' = A + (dA/dx)ξ, approximately. ξ is the particle displacement of the air. The change in volume of this air is ΔV = (A + dA/dx)(1 + dξ/dx)dx - Adx = (dA/dx)ξ + [A(dξ/dx) + (dA/dx)(dξ/dx)]dx. Although dx is infinitesimal, ξ need not be. However, we presume that the nonlinear term is small compared to the rest, so that the change of volume is approximately ΔV = [d(Aξ)/dx]dx. This holds for any form of horn where the increase in area in gradual, incidentally, not just for an exponential horn.

We can now relate the change in volume to the change in pressure, through p = ρc^{2}s = ρc^{2}(-ΔV/V). ΔV is given above, and V = Adx. Therefore, p = -(ρc^{2}/A)[d(Aξ)/dx]. The force on a slice of air of thickness dx is -A(dp/dx)dx, and its mass is ρAdx, so Newton's Law gives us (ρAdx)(d^{2}ξ/dt^{2}) = ρc^{2}A (d/dx)[(1/A)d(Aξ)/dx]dx. Removing identical factors gives d^{2}ξ/dt^{2} = c^{2}(d/dx)[(1/A)d(Aξ)/dx]. Since ξ is a function of both x and t, all these derivatives should be partials, of course. If A is a constant, as in a cylindrical pipe, then this reduces to the ordinary wave equation, with its well-known solutions.

If A varies exponentially, as A_{0}e^{mx}, then this equation becomes d^{2}ξ/dt^{2} = c^{2}[d^{2}ξ/dx^{2} + mdξ/dx]. This equation has solutions of the form ξ = Ae^{j(ωt - βx)}, where β is a complex propagation constant β = β' - jβ". If this solution is substituted in the equation, we find that β = ±√(k^{2} - m^{2}/4) + jm/2, where k = ω/c. The imaginary part contributes a factor e^{-mx/2} to the solution, which is not an attenuation, but a decrease in the amplitude of the wave as the horn opens out. The energy is conserved. The real part contributes travelling waves in both directions, with a *cutoff* when k = m/2. The horn is a *high-pass filter* that rejects all frequencies below f = mc/4π.

This relation is more usefully expressed as m = 4πf/c, showing that if the horn is to pass low frequencies, m must be sufficiently small. For example, if f = 50 Hz, then m must be smaller than 200&pi/343 = 1.82 m^{-1}. The outer end, or mouth, of the horn should be about λ/4 in diameter, or 1.72 m in this case. If the diameter of the driving end, or throat, is 2 cm, then the length of the horn will be x = ln(A_{1}/A_{0})/m = 4.89 m. This is a sizeable piece of furniture for the sitting room; in fact, it would be rather dominating. This explains why horn loudspeakers are not common in home entertainment centres, and the much less efficient, but smaller, diaphragm speakers are found instead. It is possible to "fold" a horn to reduce its length, but this is not completely satisfactory.

Let's now relate the acoustic variables pressure and velocity for a wave in the exponential horn. First, we will consider an outgoing wave, ξ = Ae^{j(ωt - βx)} The pressure is related to ξ through p = -(ρc^{2}/A)[d(Aξ)/dx] = -(ρc^{2})[dξ/dx + mξ]. For our harmonic wave, p = -(ρc^{2})(-jβ + m)ξ, where p and ξ are now phasors. The particle velocity v = jωξ, so p/v = (ρc^{2})(j&beta - m)/jω= (ρc^{2})(β + jm) = (ρc^{2}/ω)[√(k^{2} - m^{2}/4) + jm/2] = ρc[√(1 - m^{2}/4k^{2}) + jm/2k]. This is the *wave impedance*, or the *specific acoustic impedance*, z, of the horn. If m is small or k large, there is a good impedance match with free air, whose impedance is ρc, so the horn will radiate effectively. Note that this impedance does not depend on the area, or location along the horn.

The mechanical impedance is f/v, or pA/v, so to find the mechanical impedance we must multiply the wave impedance by the area: z" = Az. If the throat of the horn is driven by a piston source, A is the throat area, and z" has real and imaginary parts that are used just like the R_{r} and X_{r} that we found above for radiation from a piston speaker in an infinite baffle. R_{r} for the horn speaker rises very rapidly above cutoff and is constant over a wide frequency range, in contrast to the piston speaker.

A high pressure at the throat implies a large v, and so a large displacement, especially at low frequencies. In order to limit the coil displacement, the piston is made of larger area A'. For the same volume velocity v'A' = vA, so the larger area means smaller displacement in the ratio of the areas. The practical arrangement is shown in the diagram. The piston works into a volume V_{2} connected to the throat. The volume V_{1} behind the piston adds a stiffness ρc^{2}A'^{2}/V_{1} to the stiffness of the support, while V_{2} adds a stiffness in parallel with the throat of the horn. At high frequencies, this stiffness short-circuits the radiation into the horn. The various components must be carefully proportioned for the best effect.

Experiments on inexpensive small moving-coil loudspeakers will be included when I can get around to preparing them.

L. E. Kinsler and A. R. Frey, *Fundamentals of Acoustics*, 2nd ed. (New York: John Wiley & Sons, 1962). Piston impedance functions are on p. 506. See Chapters 7 and 10.

A. L. Albert, *Electrical Communication*, 2nd ed. (New York: John Wiley & Sons, 1940). Chapter VI. Good treatment of moving-iron telephone receiver.

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Composed by J. B. Calvert

Created 12 September 2003

Last revised