Acoustics is the science of sound, including its production, transmission, and effects.

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Acoustics is the science of sound: its production, transmission and effects. Olson opens by placing the audible band inside the whole spectrum of elastic waves — from the infrasonic rumble below hearing to the ultrasonic range far above it — and by insisting that all of it obeys the same equations. The rest of the book is those equations, applied.

Where speech and music sit inside the audible band.

Sound is an alteration in pressure, particle displacement or particle velocity propagated in an elastic medium. The medium does not travel: each particle oscillates about its own position and passes the disturbance on, so what moves through the air is the compression, not the air.

From 17 metres at 20 cycles to 17 millimetres at 20 kc.

The fundamental equation of sound wave propagation in three dimensions. It relates the spatial variation of sound pressure (p) to its temporal variation, governed by the speed of sound (c).

Where ∇² is the Laplacian operator:
∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z²

This implies that the acceleration of the particle is proportional to the pressure gradient.

In a plane wave, the sound pressure (p) and particle velocity (u) are everywhere in phase. The ratio of pressure to velocity is the specific acoustic resistance (ρc) of the medium.

Pressure (Solid) and Velocity (Dashed) in Phase

Unlike plane waves, in a spherical wave expanding from a point source, the particle velocity ($u$) leads the pressure ($p$) by an angle that approaches 90° near the source, and 0° far from the source.

Notice the phase shift between P and U near the source.

Two plane waves of equal amplitude travelling in opposite directions interfere into a wave system characterised by the existence of nodes. Equations 1.48 and 1.51 show the result: pressure and particle velocity are both standing, but their maxima are a quarter wavelength apart — where the pressure has an antinode, the velocity has a node.

Pressure antinodes sit exactly on velocity nodes.

The sound energy per unit volume, in ergs per cubic centimetre. In a plane wave it follows equation 1.52 from the pressure alone. Olson then adds a curiosity that is really a measurement technique: a sound wave pushes on the wall it strikes, with a positive radiation pressure given by equation 1.53.

Energy density and radiation pressure against level.

The energy transmitted per unit time through a unit area normal to the direction of propagation. Equation 1.54 gives three equivalent forms, and the fact that they are equal is what makes acoustic measurement practical: measure the pressure alone and the intensity follows, because in a plane wave pressure and velocity are locked together by ρc.

The three forms of 1.54 agree exactly, at every level.

The bel is the logarithm of a power ratio; the decibel is a tenth of it. Equation 1.55 covers powers. When the quantities are the square roots of power — currents, voltages, and their acoustical analogues pressure, volume current, force and particle velocity — the factor becomes 20, equations 1.56 and 1.57. Hence the 10 and the 20 that everyone confuses.

Olson's Table 1.2, computed: a ratio of 2 is 3 dB of power and 6 dB of pressure.

The change in pitch due to relative motion of source and observer, equation 1.58. Olson then makes the point that most people get wrong: substituting v + w for v gives equation 1.59, which shows that wind produces no change in pitch unless there is some relative motion of source and observer.

Observed pitch for a 1000 Hz source: -- Hz.

Refraction is the change in direction produced by a change in the medium that affects the velocity: sound bends downward or upward depending on the relative temperatures at different heights, which is why a distant sound carries at night and dies in the afternoon. Diffraction is the bending around an obstacle, and it depends entirely on the size of the obstacle against the wavelength.

An obstacle only casts an acoustic shadow once it is bigger than the wavelength.

Swap the source and the receiver and the reading does not change. Equation 1.74 states it for two pressures: p′X″ = p″X′. It holds provided the system is invariable, linear, reversible and contains no internal source of energy — and it is the reason a microphone can be calibrated by using it as a loudspeaker.

The two curves are identical because the theorem says they must be.

For any acoustical system involving diffraction it is possible to build the same system on a different scale, provided the wavelength is altered in the same ratio as the linear dimensions. Build the room at one fifth size, test it at five times the frequency, and the performance is the same — the whole basis of scale-model acoustics.

Full size and model land on the same curve once frequency is scaled.

Apply the same derivation to a solid rod instead of a gas: Young’s modulus replaces the bulk behaviour of the air, and equation 1.82 comes out identical in form to the plane wave equation of 1.4. The velocity therefore follows equation 1.83 — and it is an order of magnitude faster than in air, which is why structure-borne sound arrives first.

Longitudinal velocity in solids against 344 m/s in air.

Twist the rod instead of compressing it and each transverse section stays in its own plane. For a circular, homogeneous bar the equations are analogous to the longitudinal case, but Poisson’s ratio enters and the velocity of equation 1.84 comes out slower — roughly six tenths of the longitudinal speed for ordinary metals.

Torsional waves travel at about 0.6 of the longitudinal velocity.

A long cylinder expanding and contracting radially. Between the plane wave, which does not decay, and the spherical wave, which decays as 1/r, sits the cylindrical wave: pressure and particle velocity fall off inversely as the square root of the distance, so the intensity falls off inversely as the distance — 3 dB per doubling instead of 6.

Plane, cylindrical and spherical decay compared.

There are almost an infinite number of different types of sound sources: the human voice, musical instruments, machinery noise, loudspeakers. In some, like musical instruments, it is almost impossible to analyse the action; but in most sound reproducers the action may be predicted with amazing accuracy. This chapter is that prediction — the directional pattern, the radiation efficiency and the output as a function of frequency.

Radiated power against frequency for the elementary sources.

A point source is a radiator whose dimensions are small compared to the wavelength. It radiates sound equally in all directions, yielding a spherical directional pattern.

A dipole consists of two point sources separated by a distance d, vibrating in opposite phase. It exhibits a bidirectional directivity pattern.

n equal sources in a straight line, spaced d apart and vibrating in phase. The array factor is the ratio of the summed field to n times one element, and its shape depends only on the spacing measured in wavelengths. Once d approaches a wavelength the grating lobes appear — the reason line arrays are built with the elements as close together as they will fit.

Grating lobes appear as soon as the spacing passes one wavelength.

A continuous line source of length L produces a highly directional pattern perpendicular to its axis. It is the foundation of line array theory.

By applying a progressive phase shift (δ) along the length of a line source, the main radiation lobe can be electronically "steered" without physically moving the array.

A tapered (apodized) line source applies a weighting function across its length, reducing side lobes at the expense of a wider main lobe. This is the foundation of modern line array shading.

Equation 2.28 covers the general case: the strength varies as a function of the distance x along the line. Taper the drive from the centre outwards and the side lobes collapse, at the price of a wider main lobe. It is the same trade every array shading scheme makes, stated in 1957.

Taper trades side lobe level for main lobe width.

Feed the same line with a progressive time delay between elements corresponding to the time of wave propagation over that distance in free space, and the maximum directivity swings from broadside to along the line itself. The pattern becomes symmetrical about the line as an axis — and this is exactly the end-fire subwoofer array of modern practice.

The lobe points along the line, not across it.

Drive adjacent elements in opposition and the array becomes a gradient source: the main lobe narrows far beyond what its physical length would suggest. Olson is careful about the cost — the efficiency collapses and the side lobes grow, so super directivity buys pattern control with output.

Order 0 is the plain line; each order narrows the lobe and costs output.

An arc source distributes elements along a circular arc of radius R subtending angle 2φ. This is the theoretical basis of J-shaped and curved line arrays.

A circular ring of sources of radius a produces a directivity pattern governed by the Bessel function J₀. This is the basis for ring-radiator transducers.

A circular baffled piston of radius R approximates a typical loudspeaker cone. Its directivity depends on the Bessel function of the first kind ($J_1$).

The piston of 2.13 with the velocity tapered from the centre to the rim instead of uniform. The first side lobe of the uniform piston, at −17 dB, is the price of its hard edge; taper the edge and the lobes go away, exactly as with the line source of 2.8.

Tapering the rim velocity removes the side lobes of the hard-edged piston.

The same piston, now set in the end of an infinite pipe rather than an infinite baffle. The tube walls remove the baffle beyond the rim, so the low frequency pattern opens up toward a hemisphere and the impedance drops — the case Olson uses in 5.15 for the throat of a horn.

Piston in a pipe against the same piston in an infinite baffle.

Remove the baffle altogether and the rear of the piston radiates too, in opposition. At low frequencies the two cancel and the pattern becomes the figure of eight of the doublet in 2.3 — which is precisely why a loudspeaker without a baffle has no bass, the subject of 6.8.

Unbaffled, the piston degenerates into a doublet at low frequencies.

A square piston in an infinite baffle. Along each principal axis the pattern is the sinc function of a line source of the same width, so the square radiates a pattern that is not a surface of revolution: it is narrower across the diagonal than across the face.

Face and diagonal cuts of the same square source.

Make the sides unequal and the two axes separate completely: the long side controls the narrow plane and the short side the wide one. This is how every horn mouth and every flat panel is aimed — the aspect ratio is the coverage pattern.

Horizontal and vertical patterns of the same rectangle.

The directivity of a horn mouth depends on the ratio of its opening diameter to the wavelength. Larger mouths produce narrower beams at high frequencies.

Bend the radiating surface and the coverage angle is set by the arc, not by the wavelength. Where a flat source narrows as frequency rises, a curved one holds its pattern — the principle behind every constant directivity device, including the curved line array of 2.11.

Curved against flat: the arc holds the pattern as frequency rises.

A cone radiator (like a real loudspeaker cone) has a half-angle β. Its directivity narrows as the cone becomes shallower and as frequency increases.

The preceding chapters dealt with simple systems: point sources, homogeneous mediums and simple harmonic motion. But real sources of sound — strings, bars, membranes and plates — are liable to vibrate in more than one mode at a time, and each mode has its own frequency and its own pattern of nodes. This chapter is the catalogue of those modes.

Overtone series of the four classic vibrating systems.

The fundamental frequency of a stretched string depends on its length (l), tension (T), and mass per unit length (m). It is the basis of all stringed instruments.

Fundamental Frequency: 15.8 Hz

A bar clamped at one end (cantilever) vibrates transversely. The restoring force is due to stiffness, governed by Young's modulus (Q) and the radius of gyration (K).

Animation showing transverse cantilever vibration.

A stretched membrane (like a drumhead or condenser microphone diaphragm) vibrates in 2D modes. Its restoring force is tension.
Tap the membrane below to strike it!

A plate under no tension, stiff in its own right. Equation 3.14 gives the frequency of the free circular plate vibrating with one nodal circle; 3.15 gives the mode with two nodal diameters. Unlike the membrane of 3.4, the restoring force is the plate’s own elasticity, so Young’s modulus and Poisson’s ratio enter and the frequency scales with thickness over radius squared.

Fundamental -- Hz; the second mode -- Hz.

A free rod struck on the end rings at the frequency where its length is half a wavelength of the longitudinal wave of section 1.14. The overtones are true harmonics of the fundamental — f₂ = 2f₁, f₃ = 3f₁ — which is exactly what a xylophone bar is not, and what a struck steel rod is.

Fundamental -- Hz in steel; the overtones are exact harmonics.

Twist the same bar instead of compressing it and the modes are analogous, but the wave travels at the torsional velocity of equation 1.84. The overtones are again harmonics. Olson points out the practical consequence: comparing the longitudinal and torsional frequencies of the same bar gives you Poisson’s ratio directly.

Longitudinal -- Hz against torsional -- Hz, ratio --.

The gas column is analogous to the solid bar. An open pipe needs a displacement loop at both ends, so its fundamental is c/2l (equation 3.20) and all harmonics are present. Close one end and it becomes c/4l with only the odd harmonics. Olson adds the correction that makes it work in practice: the open end behaves as if the pipe were 0.82R longer.

Open -- Hz, closed -- Hz, end correction -- cm.

Engineers found that reducing a vibrating system to its analogous electrical network is a valuable tool in the analysis of vibrating systems — Olson says so in his own preface. The reason is economic: a century of circuit theory already exists, and if a mass behaves exactly as an inductance, all of it transfers over for free.

The same resonance, seen in four domains.

Before the analogy can be used it has to be defined precisely, and Olson spends a full section doing it: impedance, admittance, the distinction between an element and a device, and the four domains — electrical, mechanical rectilineal, mechanical rotational and acoustical. Sloppy definitions are what make analogies break down at the third step.

Impedance magnitude of a series resonant system.

An element is the abstraction: resistance, inductance and capacitance, as distinguished from the devices called resistor, inductor and capacitor. A resistor idealised to have only resistance is a circuit element. The whole method rests on this separation — the real cone is not a mass, but it can be represented by one.

Reactance of the three elements against frequency.

The dissipative element: whatever turns the energy into heat. Electrical resistance turns current into heat, mechanical resistance turns velocity into heat through friction, rotational resistance does it through torque and angular velocity (equation 4.3), and acoustical resistance does it through viscosity as a volume current passes through a narrow slit.

Resistance takes power out; it is the only element that does.

The inertia element: it stores kinetic energy and resists change. In the acoustical domain it is inertance, the mass of the air in a tube divided by the square of its area — equation 4.10 — which is exactly the quantity that tunes a bass reflex port in 6.12. Change the tube and you change the tuning.

Inertance -- g/cm⁴ — reactance rises with frequency.

The storage element: it stores potential energy and resists displacement. Electrical capacitance stores charge, rectilineal compliance stores it in a spring, and acoustical capacitance stores it in the compression of a volume of air — the same C_A = V/ρc² that becomes the sealed box of 6.10. Compliance and inertance together are every resonance in the book.

Acoustical capacitance -- cm⁵/dyne; with the tube of 4.5 it resonates at -- Hz.

Olson's masterpiece: unifying physics. A series resonant circuit behaves identically across domains. Toggle the domain below to see how terms change while the math remains exactly the same!

Chapter 4 drew the analogies; this chapter builds the parts. An electrical circuit is composed of electrical elements, and in the same way an acoustical system is composed of acoustical elements. Which element you get — resistance, inertance or capacitance — depends on how the medium is confined: a hole gives resistance, a tube gives inertance, a cavity gives capacitance.

The three elements, and the frequency where each one rules.

Acoustical resistance is obtained by forcing air through a small hole. The resistance is due to viscosity, which may be considered as friction between adjacent layers of air. Olson notes the practical catch immediately: the resistance of a single hole is ordinarily much too high, so the desired value is reached by using a sufficient number of holes in parallel.

Parallel holes are how a usable resistance is built.

A tube whose diameter is small compared with its length, and whose length is small compared with the wavelength. Viscosity does two things at once: it introduces acoustical resistance as dissipation, and it adds to the acoustical reactance. The resistance goes as the fourth power of the radius, so halving the bore multiplies the resistance by sixteen.

r_A = -- acoustical ohms; reactance crosses it at -- Hz.

A narrow slit behaves like the narrow tube, but the exponents differ: in equation 5.2 the acoustical resistance varies inversely as the cube of the thickness and the inertance inversely as the thickness. Therefore practically any ratio of inertance to acoustical resistance may be obtained — which is why a pile of washers with shims, or a spiral of tape, is a classic acoustic resistance.

r_A/M ratio changes as 1/t²: -- at this thickness.

Silk cloth provides a simple means of obtaining an acoustical resistance, and the magnitude is governed by the size and nature of the holes in the material. Olson measured it layer by layer, and the answer is a straight line: stack more layers and the resistance adds. It is still how a microphone gets its damping.

Olson's Fig. 5.1, measured layer by layer.

Inertance is the mass of the medium divided by the square of the area over which the driving pressure acts. For closed systems the resistance term drops out entirely, because there is no radiation, and the whole acoustical impedance is positive reactance — pure inertia.

Double the area and the inertance falls by four.

The most common acoustical capacitance is a cavity with rigid boundaries, its linear dimensions small compared with the wavelength. The air in it stores energy by compression, and the capacitance is simply the volume divided by ρc² — the sealed box of 6.10 and the compliance of 4.6 are the same element seen from different chapters.

A bigger cavity is a softer spring.

The reference case of the whole book: a piston set in an infinite baffle. Below ka = 1 the radiation resistance climbs as the square of frequency and the load is almost pure mass; above it, the resistance flattens at ρc per unit area and the piston finally radiates efficiently. Every loudspeaker in chapters 6 and 7 is read against this curve.

Resistance and reactance cross at ka ≈ 1, here -- Hz.

The same piston, now radiating into an infinite tube instead of a half space. The load is real and constant at every frequency — ρc per unit area, with no reactive part — because a plane wave in a tube never spreads. It is the ideal a horn tries to imitate.

The tube gives full load from the very bottom.

A rigid sphere oscillating back and forth, the acoustic model of an unbaffled loudspeaker or a swinging body. It behaves as a doublet: the radiation resistance falls off as the fourth power of frequency at the low end, twice as fast as the simple source, which is why an unbaffled radiator is so poor in the bass.

Doublet against simple source: 12 dB per octave against 6.

A long cylinder expanding and contracting radially — the source of the cylindrical waves of 1.16. Its radiation resistance rises more slowly than the sphere and its far field falls as 1/√r, so a line-shaped source loses only 3 dB per doubling of distance instead of 6.

The line source loads up more gradually than the point.

A long narrow strip in a baffle: the ribbon of a velocity microphone, and the driver of a slot. Its load is that of a line rather than a disc, so the transition frequency is set by the width alone and the strip stays reactive far higher than a piston of the same area.

Same area, very different loading.

Halfway between 5.8 and 5.16: the piston sits in the end of an infinite tube, so it has a baffle out to the tube wall and nothing beyond. In the region below ka = 1 the radiation resistance is half that of the piston in an infinite baffle — the comparison Olson draws in 5.15, and the reason a horn throat is calculated with this case.

Exactly a factor of two below the transition.

No baffle at all. An example of a vibrating piston in free space is a loudspeaker without a cabinet: front and rear radiation cancel, the load collapses at low frequency, and the pattern becomes the doublet of 2.16. Chapter 6 spends thirty pages on cabinets because of this one curve.

The cost of having no baffle, in one picture.

A circular hole in a wall of infinite extent: the port of 6.12, the vent of a microphone case, the leak in an enclosure. It carries an inertance equal to the air plug in the hole plus the end correction on both faces, which is why a port always behaves as if it were longer than it is.

Effective length -- cm, inertance -- g/cm⁴.

The open end of a pipe terminated in a large flange radiates as a piston in an infinite baffle, so the mouth impedance of 5.8 can be used directly. Without the flange the end correction is 0.6a instead of 0.85a, and that difference is enough to shift the tuning of an organ pipe or a bass reflex port by a noticeable amount.

The flange doubles the effective baffle and raises the load.

A horn is an acoustical transducer consisting of a tube of varying sectional area. Its principal virtue is the possibility of presenting practically any value of acoustical impedance to the sound generator — which is how a horn loudspeaker makes the driving force work against resistance instead of against the inertia of the diaphragm. Everything in the sections that follow comes from one choice: how the area grows along the axis.

Area against axial distance: cylindrical, parabolic, conical, exponential and hyperbolic.

Take a tube whose diameter is small compared with the wavelength, apply continuity to an element of volume SΔx, and equation 5.38 falls out: the wave equation for axial motion in a tube of varying section. The whole chapter is this equation solved for different S(x).

The middle term is the horn: it is the only difference from the plane wave equation of 1.4, and it depends on ∂(log S)/∂x alone. For the exponential horn that gradient is a constant, m — which is exactly why the exponential horn has a single, sharp cutoff frequency.

∂(log S)/∂x along the axis: constant only for the exponential.

The degenerate case: no flare at all, S = S₁ (equation 5.39). The flare term vanishes and 5.38 collapses back to the plane wave equation 5.40. The throat impedance is pure resistance at every frequency — there is no cutoff, but also no transformation.

r_A = -- acoustical ohms, flat from DC upward.

Area proportional to distance, S = S₁x (equation 5.46). The solution is no longer an exponential but a pair of Bessel functions, and the throat impedance of equations 5.51 and 5.52 approaches the resistive value slowly and without a definite cutoff.

Normalized throat impedance r_AS₁/ρc and x_AS₁/ρc.

The conical horn is the simplest expanding waveguide. Its acoustic impedance behaves similarly to a pulsating sphere, exhibiting poor low-frequency resistance compared to exponential profiles.

Acoustic Resistance (R) & Reactance (X)

The exponential horn is characterized by a "cutoff frequency" (f_c). Below f_c, the throat resistance is zero (pure reactance). Above f_c, it acts as a highly efficient acoustic transformer.

Notice the high-pass filter behavior at f_c.

Salmon's family, equation 5.68: S = S₁(cosh α + T sinh α)², with α = x/x₀ and the family parameter T < 1. One knob spans the whole range of flares — at T = 0 it is the exponential horn of 5.22, and as T grows the horn loads the diaphragm harder just above cutoff.

Slide T to 0 and this becomes 5.22 exactly.

Olson's Fig. 5.5, rebuilt live: the five infinite horns compared on equal terms — the same throat area of 1 cm², and the same 100 cm² at 100 cm from the throat for the four with flare, with T = 0.5 for the hyperbolic. This one figure is the argument for the exponential and hyperbolic families.

Cylindrical loads flat but never transforms; parabolic and conical climb slowly; exponential and hyperbolic snap to full load just above cutoff.

Cut the pipe to a finite length and the mouth reflects. Equation 5.78 transforms the mouth impedance Z_A2 back to the throat through the length l, and the result is the familiar series of resonances: the variations in resistance and reactance are quite large. The mouth is loaded as a piston in an infinite baffle (equation 5.12).

Every peak is a standing wave between throat and mouth.

The same transformation applied to the cone of 5.21, working between the apex distances x₁ and x₂ with the phase angles kθ₁ = tan⁻¹kx₁ and kθ₂ = tan⁻¹kx₂. The flare tames the resonances compared with the straight pipe, but a short cone still ripples badly at the low end.

Compare with 5.21: the infinite cone has no ripple at all.

Unlike the infinite horn, a finite horn has acoustic reflections at the mouth. This creates standing waves (ripples in throat impedance) which cause uneven frequency response.

The companion to 5.24, now for horns of finite length. The mouth reflection puts a ripple on the throat impedance whose spacing is set by the length; making the mouth larger, or the flare faster near the mouth, damps the ripple. These results may be applied to horns of a different flare by scaling, which is what makes the chart usable in design.

Ripple every -- Hz; a bigger mouth flattens it.

A short exponential section used as a transformer between two areas rather than as a radiator. It matches a small throat to a large pipe over a band, and below its own cutoff it stops transforming altogether — the acoustic equivalent of a transformer running out of core.

Above cutoff the connector transforms; below it, nothing passes.

Several exponential sections of different flare in series, each taking over where the previous one runs out. The low cutoff of the slowest section is preserved while the fast section near the throat keeps the high frequency load, which is the multiple flare horn Olson recommends in 7.3 for lower distortion.

Bandwidth of the slow flare, top end of the fast one.

A pipe closed at the far end reflects everything, so its throat impedance is pure reactance that swings between mass and compliance every quarter wavelength. It is the element behind every quarter-wave trap and every stub, and the reason a closed tube can be used to cancel one frequency exactly.

Zeros every c/2l; first quarter-wave resonance at -- Hz.

The general problem of the last third of the chapter: what happens when a wave travelling in a tube meets a change. Everything that follows — a change of area, three pipes in series, a change of medium — is this same reflection and transmission calculation with different constants.

The bookkeeping that runs to the end of the chapter.

Sound travels from pipe S₁ into pipe S₂ of different cross section. The transmitted fraction depends only on the ratio of the areas, and it is symmetrical: a two to one expansion loses exactly as much as a two to one contraction. Mismatch is mismatch, in either direction.

Transmitted power --% — and the same for the reciprocal ratio.

A section of different area inserted between two equal pipes. Now the length matters as well as the ratio: at multiples of half a wavelength the section is transparent, and in between it reflects. This is the low pass muffler and the expansion chamber, described three decades before it was called that.

Transmission loss returns to zero every -- Hz.

The same calculation with ρc instead of area: sound crossing from one medium into another. The numbers are brutal — air into water transmits about one tenth of one per cent of the incident power — and that single fact is why underwater sound needs its own chapter and why a wall works at all.

--: -- transmitted, -- dB loss.

A wall between two rooms is three transmissions in series: air to wall, through the wall, wall back to air. Because the mismatch dominates, the transmission loss climbs with the mass per unit area and with frequency — the mass law, and the reason a heavy limp wall beats a stiff light one.

Double the mass, gain 6 dB; double the frequency, gain 6 more.

Line the walls and the tube stops being lossless. The attenuation per unit length rises with frequency, because at high frequencies the wave grazes the lining many times over the same distance — the dissipative silencer, and the damping that removes the resonances of the labyrinth in 6.14.

A lined duct is a low pass filter you can walk through.

The chapter closes by assembling the three elements into the simplest complete system: one inertance, one capacitance, one resistance. Its response is the resonance curve that every later chapter reuses — the loudspeaker of 6.2, the microphone of 8.2, the port of 6.12 — and the only thing that changes is what the letters stand for.

Peak -- dB, bandwidth -- Hz — the curve behind the whole book.

The direct radiator loudspeaker is the opposite bargain to Chapter 7's horn: the cone radiates straight into the air with no acoustic transformer, so efficiency is low — a few per cent against thirty — but the response is smooth, the size is small and the cost is low. Olson devotes his longest chapter to it because that bargain is the one almost every loudspeaker actually makes.

The same driver, radiating directly and through a horn.

The whole machine in one impedance: suspension, moving mass and air load in series (equation 6.2), reflected into the voice coil circuit as a motional impedance (6.3). Efficiency is then the share of that motional resistance which is actually radiation (6.4), and in the mass controlled region it reduces to 6.6.

Mass controlled efficiency: --% — flat, and small.

Several small mechanisms on one baffle instead of one large cone. The radiating area adds, so the low frequency output rises with the number of units, while each cone stays small enough to keep its directivity wide at the top of the band. The catch is interference: once the spacing approaches a wavelength the units comb against each other off axis.

More units buy low-frequency output, and lose it again where the spacing combs.

One voice coil drives a large cone through a compliance and a small cone directly. At low frequencies both move together; above the compliance break the large cone is decoupled and only the small cone radiates. One motor, two effective radiators, and a directivity pattern that stays open at the top.

The coupling compliance hands the top end over to the small cone.

Two voice coils on the same cone, fed through a dividing network. The low frequency coil can be heavy and long for force, the high frequency coil light and short for speed, and each sees only the band it is good at — without the mass penalty of covering the whole range with one winding.

A mechanical two-way with a single cone.

The logical end of the previous two: separate coils and separate cones in one mechanism and one magnet structure. It is the coaxial loudspeaker, and it buys the performance of a two-way system in the space and cost of one unit.

Coaxial: both radiators on the same axis, so the response does not change with angle.

It is usually desirable to attenuate the response above a certain limit — in radio, for instance, to kill the 10 kc adjacent-channel beat note. Electrical filters are costly; a mechanical filter built into the cone and its coupling costs nothing per unit. A mass and compliance between coil and cone form a low-pass filter, and a tuned branch forms a band rejection.

A mass and a compliance in the cone assembly do the work of a filter.

Without a baffle, the rear wave cancels the front wave at low frequencies (acoustic short circuit). A finite baffle pushes this cancellation to lower frequencies.

The open-back cabinet is a folded baffle, and it behaves like one: a marked increase of about 6 dB where cabinet and mechanism resonate together, and below that the pressure response falls off at 18 dB per octave. Making the cabinet deeper does not fix the peak — it makes it worse.

Deeper cabinet, more accentuated resonance — Olson's Fig. 6.27.

Close the back and the trapped air becomes a spring in parallel with the suspension. Equation 6.16 gives that spring: C_MB = V / (ρc²S_c²). The smaller the box, the stiffer the air, and the higher the system resonance climbs above the free-air resonance of the mechanism — the sealed box trade in one line.

System resonance -- Hz, box compliance ratio α = --.

A small driving loudspeaker works into a sealed chamber that loads a large radiating loudspeaker. Because the driver is small its resonance with the cabinet can be placed below that of the radiating unit, and the two-stage system extends the response in the low frequency range beyond what either would give alone.

Olson's Fig. 6.32: the same cabinet and radiator, an octave lower.

Olson's name for the bass reflex. A port in the closed box turns the rear radiation, which would have cancelled the front, into radiation that adds to it: the constants can be chosen so the phase angle between the velocity of the cone and of the port is very small. The system increases the radiation mechanical resistance at the low frequencies.

Port tuning f_B = -- Hz. Below it the port output turns and cancels: −24 dB/octave.

Replace the port with a second, undriven cone. Its area can be made the same as the active cone, so the particle velocity is relatively low and the losses are low — where a small port has to move air fast enough to whistle. The tuning is now set by the drone's own mass against the box compliance.

Drone tuning -- Hz, against a 32 sq. in. and a 10 sq. in. port — Olson's Fig. 6.36.

A lined tube behind the cone. When the tube is half a wavelength long the mouth output is additive and the response rises over about an octave; an anti-resonance occurs when the tube is one-quarter wavelength long. Olson's design rule is to make the cone's fundamental resonance coincide with that quarter-wave anti-resonance — and the cabinet resonance peak disappears.

Anti-resonance at -- Hz; put the cone resonance there. Transmission is very low above 150 Hz.

Horn load the front of the cone for the middle band, radiate directly at the low end where the horn would need to be enormous. The horn gives efficiency where the mouth is large enough to work, and the direct radiator covers below it — the practical compromise behind most large cabinets of the period.

Efficiency where the horn can work, extension where it cannot.

The inverted-magnet field structure delivers the same performance as the conventional design with a cone of the same diameter and weight and a magnet of the same weight, while occupying considerably less depth. The constraint that shapes a loudspeaker is often not acoustics but the cabinet it has to fit inside.

Same response, roughly half the depth.

Make the amplification deliberately higher than necessary and feed the output back so as to throw the excess gain away — and include the loudspeaker inside that loop. Nonlinear distortion and response irregularities of both the amplifier and the loudspeaker are reduced by the same factor as the gain.

20 dB of loop gain divides the distortion by ten.

The cabinet is not only a baffle, it is a diffracting object. Sharp edges close to the loudspeaker produce reflections that arrive late and add or subtract at the listener, putting ripples of several decibels into an otherwise smooth response. Rounded edges and an offset mounting spread those reflections out.

Every edge is a secondary source arriving late.

Where the loudspeaker sits in the panel matters as much as the panel size. Centred, every edge is the same distance away and all the diffraction ripples land on the same frequencies, reinforcing each other. Offset, the path lengths differ and the ripples spread out and partly cancel.

Offsetting the mechanism trades one big ripple for several small ones.

In a television receiver the loudspeaker is wherever the picture tube is not: below the screen, at the side, or firing down. Olson measured them all. Firing into the floor or into the back of the cabinet costs the top end, because the high frequencies are directional and never reach the listener.

The high frequencies are the ones that pay for a bad location.

The phonograph puts the loudspeaker under a lid that is usually closed while playing, and beside a pickup that will happily reproduce the loudspeaker through the cabinet as acoustic feedback. Both the response and the stability of the machine depend on where the mechanism is mounted.

The closed lid adds a peak and takes away the top.

In a small radio the cabinet is smaller than the wavelengths it is asked to reproduce, so the acoustic problem is almost entirely the low frequency end. Suitable compensation may be employed in the audio amplifier — Olson is describing the loudness contour of every table radio ever built.

Boosting what the cabinet cannot radiate has a limit: cone excursion.

The combination instrument — radio, phonograph and television in one cabinet — has to serve all three sources with one loudspeaker system, and the cabinet is shared with a turntable that is sensitive to vibration. Separate compartments and elastic mounting are what make it work.

Give the loudspeaker its own box inside the furniture.

Sixty-seven small mechanisms mounted on a section of a spherical surface, all in phase at the centre of curvature. The result is a gain of more than 20 dB in signal level at the focus compared with locations removed from it — high level over a limited zone without flooding everything around it.

67 units, one focal point: the acoustic spotlight of 1957.

Steady state response says nothing about what the cone does when the signal starts and stops. A resonant system with little damping keeps ringing after the drive is removed; the damping is set by the total mechanical resistance against the reactance — and most of that resistance comes back through the voice coil from the amplifier.

Decay to 10% after -- cycles of hangover.

The general trend is more output without a corresponding increase in the size of the loudspeaker, so the amplitude of the cone keeps rising. Beyond a point the suspension leaves the region where Hooke’s law holds, the voice coil leaves the uniform part of the gap, and the air in front of the cone stops being linear — three separate distortions that all grow with excursion.

At the low end the suspension dominates; at the top, the air.

The cone is a compromise made of paper: stiff enough to move as a piston at the bottom, light enough to keep going at the top, and lossy enough that when it finally breaks up it does so gently. The suspension has to be compliant for a low resonance and linear over the excursion, and the voice coil light for efficiency but big enough to take the heat.

Break-up cannot be avoided, only damped.

A cone becomes directional exactly where its diameter approaches a wavelength, so the top of the band collapses into a narrow beam. A distributor — vanes or a small diffusing element in front of the cone — spreads that beam back out, trading a little axial level for coverage that holds up off axis.

The distributor buys back the off-axis top end.

The magnet is where the efficiency comes from: the force on the coil is Bl times the current, so the whole design pushes flux density in the gap as high as the iron allows. Soft iron saturates around 20,000 gausses; beyond that Olson uses Permendur pole tips to reach 23,000.

Sensitivity in decibels against gap flux: the reason magnets are expensive.

No cone and no coil: a diaphragm between charged plates, driven over its whole surface at once instead of at the centre. With an effective spacing of about a thousandth of an inch the sensitivity is comparable to conventional dynamic loudspeakers, but that small spacing limits the amplitude and confines the operation to the frequency range above 7500 cycles.

Push-pull construction and wider spacing extend it downward.

The total acoustic output is the pressure squared integrated over the whole sphere (equations 6.65 and 6.66), not the pressure measured on axis. Two loudspeakers with the same axial reading can emit very different total power, which is exactly what the directivity factor of the next section measures.

Total radiated power -- acoustic watts for the same axial pressure.

The directivity factor Q is the ratio of the power that would be radiated if the axial pressure held over the whole sphere to the power actually radiated (equation 6.67). Expressed in decibels it is the directivity index, equation 6.68 — the number that says how much of the amplifier's work is aimed where it is wanted.

DI at 1 kHz: -- dB — the cone beams as soon as it is a wavelength wide.

High power amplifiers are costly, so Olson's answer is to spend the watts on acoustics instead of electronics. The principal virtue of a horn is that it can present practically any value of acoustical resistance to the generator, which is what makes maximum overall efficiency possible — and with a suitable combination of horns, a directional pattern that is independent of frequency.

The trade is not free: reproducing intense sound through a horn introduces frequencies that were never in the input, because the elements of the vibrating system are not perfectly linear. Chapter 7 is the accounting of that trade — efficiency, distortion and power handling.

Typical efficiency of a horn system against a direct radiator of the same driver.

Initial efficiency is the ratio of sound power output to electrical power input where the mechanical reactance is negligible and all mechanical resistance comes from radiation (equation 7.2). Substituting the throat load 7.4 gives the design equation 7.5: efficiency is set by the flux-turns product against the area ratio and the voice coil resistance.

Initial efficiency: --% at a compression ratio of 10:1.

A sound wave of large amplitude cannot travel through air without changing shape: an increase in pressure produces less volume change than an equal decrease. Equation 7.20 gives the second harmonic generated at the mouth of an exponential horn — it grows with throat pressure and with frequency, and falls as the flare constant increases.

Second harmonic distortion rises linearly with frequency: the horn is its own limit.

No single horn and diaphragm covers the whole band at high efficiency: the low frequency end wants a large mouth and a low flare rate, the high frequency end wants a small throat and a light diaphragm. Olson's answer is the multichannel system — two, three or four horns, each driven over the band where its own efficiency is highest, combined through dividing networks.

Each channel is used only over its high efficiency region.

Here the diaphragm stops being the source of energy and becomes a valve. A steady stream of compressed air is fed to the throat and the audio signal modulates a valve that throttles it, so the acoustic output is drawn from the air supply rather than from the amplifier. The electrical input only has to move the valve, which is why the system reaches acoustical outputs far beyond any conventional driver.

The penalty is distortion and noise: the valve is a nonlinear element and the airstream itself is a broadband noise source. Olson places it where nothing else will do — very high power warning and signalling service, not high quality reproduction.

Output climbs with supply pressure while the electrical input stays constant.

The last horn driver in the chapter has no moving parts at all. A corona discharge in the throat heats the air; the intensity of the corona follows the audio signal, so the air expands and contracts and a sound wave is produced directly in the throat of the horn.

Because the generator is of the constant amplitude type, for constant electrical input the amplitude is independent of frequency — so coupled to a horn the output rises in proportion to frequency, and the practical limit on amplitude confines the ionophone to the top of the audio band.

Rising response of a constant amplitude generator on a horn: +6 dB per octave.

A microphone is an electroacoustic transducer actuated by power in an acoustical system and delivering power to an electrical system. Olson sorts every microphone ever built by what actuates the diaphragm: the pressure of the wave, its gradient, or a combination of both. Everything else — carbon, crystal, condenser, ribbon — is only how the motion becomes a voltage.

The three actuating principles, and the three basic patterns.

The diaphragm is closed at the back, so it responds to the pressure of the wave regardless of direction: the omnidirectional microphone. It stays omnidirectional only while it is small compared with the wavelength — above that its own body diffracts the sound and the response rises on axis, which is the pressure build-up every measurement microphone has to be corrected for.

Pressure build-up starts near -- Hz for this diameter.

A ribbon microphone operates on pressure gradient, giving it a natural Figure-8 (bidirectional) polar pattern, following cos(θ).

Combine a pressure element and a gradient element in one microphone and the patterns add on the front and cancel on the back. The ratio between the two decides everything: all pressure gives the circle, all gradient gives the figure of eight, and equal parts give the cardioid.

-- — front to back ratio -- dB.

Take the gradient of the gradient. A second order microphone responds to cos²α and a third to cos³α, so the pattern narrows sharply without the microphone getting any bigger. The price is the same one the super directivity source pays in 2.10: the sensitivity collapses at low frequency, badly.

Order --: random efficiency --, low end falls -- dB/octave.

Directivity from size instead of from gradients: the line microphone, the parabolic reflector and the lens. Because they work by path difference they are as directional as they are long in wavelengths — very sharp at high frequency and useless at low, the opposite failing to the gradient family.

Same line, very different pattern at each frequency.

Strapped against the throat, it picks up the vibration of the larynx directly instead of the sound in the air. Intelligibility suffers because the radiating action of the mouth is missing, but in an aircraft cockpit or next to a running engine that is a fair trade for immunity to noise.

It hears the larynx, not the room.

Three ways of putting the microphone where the talker is. The lapel and lavalier sit on the chest, so they need a rising response to make up for the loss of the direct mouth radiation; the boom keeps the microphone on axis at a fixed distance and needs no correction at all.

The lavalier is built with the correction already in it.

A heated wire cooled by the particle velocity of the wave: its resistance follows the air motion, so it responds to velocity rather than pressure and has no moving parts at all. It is slow, nonlinear and rectifying — it responds to the square of the velocity — which is why it stayed a laboratory instrument.

Thermal inertia sets the top limit.

A microphone with a small radio transmitter, so the talker can move without a cable. Olson describes it in 1957 as a laboratory curiosity for special applications; it is now the default in every theatre and every broadcast studio in the world.

The transducer is unchanged; the link sets the band.

The number that says how much a pattern helps: the ratio of the response to random incidence sound to the response on axis. The figure of eight and the cardioid both give one third, so both reach the same distance in a reverberant room — a result that still surprises people.

Cardioid and figure of eight are equally good in a live room.

Wind is not sound: it is turbulence at the diaphragm, and it is worst exactly where the microphone is most sensitive to gradients. A screen works by keeping the turbulent air away from the diaphragm while letting the wave through, which is why it has to be large and open rather than dense.

A screen buys about 18 dB where it matters.

A microphone distorts for the same reasons a loudspeaker does, only at the other end of the chain: the suspension leaves its linear range, the air in the case behaves nonlinearly, and at very high levels the wave itself is already distorted before it arrives. The limit is stated as the level for one per cent.

At -- dB the distortion is --%.

The steady state response hides what a resonant diaphragm does with a sudden sound. A microphone with a high Q rings after the transient has passed, and the ringing lands squarely in the band where speech carries its consonants — which is why damping matters more than a flat curve on paper.

Rings for -- ms after the sound stops.

The floor of the whole chain: thermal noise in the microphone resistance, amplifier noise, and the acoustic noise of the room itself. Olson computes the equivalent noise level in decibels of sound pressure so it can be compared directly with the sound being picked up — the self-noise figure on every modern data sheet.

Thermal noise -- nV, equivalent to -- dB SPL.

The chapter ends with the case rather than the transducer. Every microphone is an obstacle in the field it is measuring, and its shape decides how much it disturbs that field: a sphere, a cylinder and a flat plate of the same size give three different pressure build-ups on axis, which is why measurement microphones are small and round.

The microphone is part of the sound field it measures.

Chapters 6 to 8 covered the loudspeaker and the microphone. This one collects every other electroacoustic transducer Olson could find a use for: telephone receivers, phonographs, vibration pickups, tape, film, hearing aids, sirens, seismic detectors and stethoscopes. The physics never changes — only what is coupled to what.

Each machine is designed to a band, not to perfection.

A small diaphragm in a cavity clamped against the ear. The receiver does not have to radiate into a room, only into the few cubic centimetres of the ear canal, so it can be tiny and still be loud. Its band is deliberately narrow: everything outside 300 to 3400 cycles is spent bandwidth.

The ear cavity is what makes the receiver efficient.

Thirty pages on the record player, because in 1957 it was the highest quality reproduction most people owned. The pickup is a vibration transducer reading a groove, and the two problems Olson dwells on are tracking — the stylus must follow a curve whose radius shrinks with frequency and amplitude — and the wear that follows when it does not.

The inner groove moves slower, so it loses the top first.

A mass on a spring inside a case. Below its resonance the mass follows the case and the pickup reads acceleration; above it the mass stays still in space and the pickup reads displacement. Choosing which side of the resonance to work on is the whole design decision.

Flat to -- Hz as an accelerometer with this resonance.

A telephone with no battery and no amplifier: the talker’s voice drives a microphone that is efficient enough to power the receiver at the far end directly. It works because both ends are tightly coupled to ears and mouths rather than to rooms, and it is still on every ship because it cannot lose power.

Narrower band, but it never runs out of power.

A horn loudspeaker with a microphone and amplifier strapped to it. It is chapter 7 applied to the smallest possible system, and its design is all compromise: efficiency for battery life, a rising response for intelligibility, and a mouth small enough to carry.

Everything outside the speech band is thrown away on purpose.

The upper limit of a tape machine is set by the head gap against the tape speed: when the recorded wavelength equals the gap, the head averages a full cycle and the output falls to zero. Doubling the speed doubles every wavelength and buys an octave — which is exactly why studios ran tape so fast.

Gap null at -- Hz; −3 dB near -- Hz.

Systems for transferring between tape formats and speeds without losing what the first machine captured. Every conversion is a second pass through the same gap loss and the same noise floor, so Olson’s rule is to convert as few times as possible — the analogue generation loss that defined studio practice for forty years.

Copies of copies, measured.

The optical sound track: the audio is a photographic image beside the picture, read by a lamp and a photocell. Its band is limited by the width of the scanning slit against the film speed, exactly as tape is limited by the gap, and its noise floor is the grain of the emulsion itself.

Why magnetic film replaced optical for the studio.

Magnetic stripe applied to film stock, which keeps the quality of tape while staying in sync with the picture. Synchronism is the whole problem: the sound and the image have to be locked frame by frame, and the machinery to do that is more elaborate than the transducer at either end.

Sound quality of tape at the frame rate of film.

Circuits whose gain depends on their own signal level. Below the threshold the transfer is one to one; above it, every decibel of input becomes a fraction of a decibel of output. A ratio of infinity is a limiter, a ratio below one is an expander, and the whole of broadcast loudness practice sits in between.

A -- dB input comes out at -- dB: -- dB of reduction.

Reverberation added on purpose, when the studio is too dead or the take was too close. Olson’s means are a reverberation chamber, a spring and a plate; the requirement is the same in all three — a decay that is smooth, dense and long enough, without a discrete echo you can count.

Falls 60 dB in -- s, that is -- dB per second.

The most demanding size constraint in the book: a microphone, an amplifier and a receiver small enough to be worn. The response is not flat but shaped to the loss it is correcting, and the practical limit is acoustic feedback — the microphone and receiver are centimetres apart.

Peak gain -- dB near 4 kHz for this loss.

The loudest acoustic source of its era and the only one in the book that is not a transducer at all: a rotor chops a stream of compressed air into pulses. The frequency is set by geometry alone — ports times revolutions — and the power comes from the air supply, so it reaches levels no diaphragm can survive.

Fundamental -- Hz; the harmonics carry most of the power.

The vibration pickup of 9.4 taken to its extreme: a mass so large and a spring so soft that its resonance is below one cycle per second. Everything above that frequency leaves the mass standing still in space while the earth moves around it, which is exactly what a seismometer needs.

Below its own resonance a pickup is deaf; put it low enough and it hears everything.

A chest piece, two tubes and two ear tips — an acoustical system of pure elements from chapter 5. Olson analyses it as such: the chamber is a capacitance, the tubes are inertances, and the response is the resonance they form, which is why a stethoscope emphasises the band where heart sounds live.

Two chest pieces, two different bands on purpose.

A transducer designed to transmit nothing. The attenuation follows the mass law of 5.36 at high frequency and is limited at low frequency by leakage around the seal and by conduction through the bone — which is why no ear defender exceeds about 50 dB no matter how heavy it is.

The bone conduction limit is the ceiling for every design.

Active noise cancellation, in 1957. A microphone senses the noise, the amplifier inverts it and a loudspeaker radiates the opposite. Olson is clear about the limit: it only works where the wavelength is long compared with the region being quieted, so it belongs at low frequency and close to the ear.

Effective at the bottom, useless at the top: still true today.

The electrical counterpart: reduce the noise of the medium rather than the noise of the room. Compress before recording and expand on playback, and the noise of the medium is pushed down by the difference — the companding principle behind every noise reduction system that followed.

The medium noise moves down; the signal does not.

A chapter of instruments rather than devices. Olson opens with the point that gives the book its engineering status: a design is only as good as the measurement that confirms it, and acoustical measurements are hard because the room, the instrument and the observer are all part of the system being measured.

The same loudspeaker, measured three ways.

The reciprocity method, and the reason 1.12 is in the book at all. Using a reversible transducer as both source and receiver, the sensitivity can be computed from electrical measurements and geometry alone — no calibrated reference microphone is needed anywhere in the chain. It is an absolute calibration from first principles.

Reciprocity constant J = -- at this geometry.

The response curve, the directional pattern, the efficiency, the distortion and the transient response, each with its own apparatus. Olson insists on the free field room for the response and the reverberation room for the power output, because the two questions are different: what reaches one listener, and what enters the room.

The gap between the two curves is the directivity index of 6.32.

A receiver is measured on a coupler: a cavity of standard volume with a microphone at the bottom, standing in for the ear. The whole art is in the coupler, because a receiver that measures well on a 6 cm³ cavity and badly on a real ear has been optimised for the wrong load.

The ear canal resonance the coupler does not have.

Test records with known signals: gliding tones for response, bands for distortion, and constant frequency for speed. Because the pickup and the record are measured together, the reference has to be manufactured to a tighter tolerance than the thing under test — the recurring problem of every acoustic standard.

What you measure is always the pair, never the pickup alone.

Speed variation heard as frequency modulation of the reproduced tone: slow variation is wow, fast is flutter. It is measured by recovering the modulation from a constant tone, and the ear is most sensitive to it around four cycles per second — which is precisely the rate a turntable produces.

-- at -- Hz, --% peak: --

The standing wave tube: drive a tube with a loudspeaker, terminate it with the sample, and traverse a probe microphone along it. The ratio of the maximum to the minimum pressure gives the magnitude of the reflection coefficient, and the position of the minimum gives its phase. From those two the impedance follows exactly.

S = -- → |R| = --, absorption coefficient --.

Drive the structure with a known force and measure the velocity it produces: the ratio is the mechanical impedance, and its shape identifies which element dominates. A rising line is mass, a falling line is compliance, and a flat line is resistance — the diagnosis of chapter 4 read straight off the meter.

Three slopes, three elements.

The fraction of a sample that is air rather than material, measured by compressing a known volume and reading the pressure change. Porosity, flow resistance and structure factor are the three numbers that decide how an absorbing material behaves, and this is the first of them.

Too dense reflects, too open passes: absorption peaks in between.

Push a steady flow of air through the sample and measure the pressure drop: the ratio is the d-c acoustical resistance, the second of the three numbers. It predicts the absorption of a porous layer better than any other single measurement, and it is the one a factory can check in seconds.

There is an optimum flow resistance, and it is not the highest.

Excite the room, cut the source, and record the decay. The slope of the straight part gives the time; the trouble is that the decay has to run 60 dB before it reaches the noise floor, which in a real room it usually does not — hence measuring 20 or 30 dB of it and extrapolating.

Usable decay -- dB → extrapolation from --.

The reverberation room method: measure the time empty, put the sample in, measure again, and the difference in absorption is the sample. It gives the coefficient for random incidence, which is what a room actually presents — unlike the tube of 10.7, which gives it only for normal incidence.

Added absorption -- sabins → α = -- for a 10 m² sample.

A microphone, an amplifier and a weighting network, because the ear does not measure noise the way a voltmeter does. The A, B and C curves are the inverted loudness contours of 12.6 at three levels, so the reading corresponds to what the noise sounds like rather than to how much energy it contains.

The A curve throws away the bass because the ear does.

The wave analyser: a narrow band filter swept across the spectrum, reading the amplitude of each component in turn. Everything modern spectrum analysis does was defined here — the trade between filter bandwidth, sweep speed and the ability to separate two close components.

Two components at 1000 and 1150 Hz: resolved, or not.

Two rooms and a wall between them: measure the level on each side and the difference, corrected for the absorption of the receiving room, gives the transmission loss. It is the practical form of 5.36, and the reason building codes are written in decibels rather than in materials.

At 500 Hz this partition gives -- dB; doubling the mass adds 6.

Measuring a surface where it stands, instead of cutting a sample and taking it to a laboratory. The methods are all comparisons — against a known reflector, or against the same room before treatment — and they are less accurate than the reverberation room but they measure the actual installation.

The installed value is never quite the laboratory value.

The only measurement in the book whose instrument is a person. Lists of syllables are read and scored, and the percentage correctly received is the articulation. Everything else in the chapter measures a machine; this measures whether the machine did its job, and it is still the final court of appeal for a speech system.

At -- dB signal to noise, syllable articulation is about --%.

A hearing aid is measured on a coupler like a telephone receiver, but three numbers matter more than the curve: the acoustic gain, the maximum output before it saturates, and the battery drain. Olson notes the obvious constraint — every one of them is fixed by the size the wearer will accept.

Smaller means less gain and a shorter battery, always.

An instrument for the magnet rather than the sound: it traces the hysteresis loop of a magnetic sample on an oscilloscope automatically. It is in an acoustics book because the field structure of 6.29 is where a loudspeaker’s efficiency comes from, and the loop is how the alloy is judged.

A hard magnet keeps its field; a soft one follows the current.

A stylus dragged across a surface with a pickup on it: the surface finish becomes an electrical signal, and its spectrum describes the roughness. Olson includes it because the same pickup that reads a record groove reads any other profile — a transducer does not care what made the bumps.

The phonograph pickup, employed as a metrology instrument.

The chapter closes on the machine rather than the sound it makes. Measuring the vibration of a machine, its supports, the floor, the walls and the ceiling is how noise is traced back to its source — and Olson is explicit that reducing it at the source beats treating it anywhere else.

Constant acceleration is rising velocity and steeply rising displacement.

The advent of sound reproducing systems changed the problems of architectural acoustics. Before them the concern was the optimum reverberation time and the geometry for the best artistic effect and the maximum intelligibility of speech; after them, the room has to serve a loudspeaker as well as a talker, and the two want different things.

Optimum reverberation time against room volume, speech and music.

Sabine's equation for T60 is the cornerstone of architectural acoustics.

The other half of the chapter: getting the sound into the system rather than out of it. Microphone placement in a live room is a fight between direct and reverberant field, and the distance at which they are equal — the critical distance — decides whether a pickup sounds close or distant.

Direct and reverberant level cross at -- m.

The last long chapter turns from machines to the people at both ends of them. Speech is the source, hearing is the receiver, and music is what the whole apparatus exists for. Olson insists the engineer needs their characteristics as precisely as those of any transducer — a reproducing system is only as good as its match to the ear.

Where speech, music and the ear actually live.

Olson draws the ear as an acoustical network, which is the only way this book could treat it: the canal is an inertance and the drum a compliance, the ossicles are a mechanical transformer matching air to fluid, and the cochlea is the analyser. The middle ear exists because of 5.35 — without its transformer, almost nothing would cross from air into fluid.

The canal resonance is why the ear is most sensitive near 3 kHz.

The other end: vocal cords as the generator, and the throat, mouth and nose as a variable acoustical network that shapes their harmonic-rich buzz. The resonances of that network are the formants, and moving them is how one set of cords produces every vowel.

Two vowels, one set of vocal cords: only the formants move.

If the tract is a filter, the buzz can come from anywhere. Olson describes the vocoder and its relatives: an electrical generator driving an adjustable network that reproduces the formants. It is speech synthesis a decade before computers, and the artificial larynx built on it is still in use.

Close enough to be understood, which is the whole test.

The sound spectrograph: frequency up, time across, intensity as darkness. It made speech something you could look at, and Olson reproduces the patterns because they show at a glance what no waveform does — that a consonant is a transition and a vowel is a steady state.

Three sounds of speech, seen as spectra.

Fletcher-Munson equal-loudness contours show the non-linear frequency response of human hearing.

Loudness level in phons is the level of the 1000 cycle tone that sounds equally loud; loudness in sones is how loud it actually seems. The relation is the one that matters in practice: ten phons more is twice as loud, so a doubling of subjective loudness costs ten times the acoustic power.

-- phons = -- sones; +10 phons doubles it.

Pitch follows frequency but is not the same thing: at low frequencies a tone flattens as it gets louder, and at high frequencies it sharpens. The mel scale measures what is actually heard, and it diverges from the hertz scale badly above 1000 cycles — one more reason the ear is not a spectrum analyser.

Above 1 kHz the two scales part company for good.

One sound raises the threshold for another. The masking spreads upward in frequency far more than downward, and the spread grows with level — which is why a loud low tone hides everything above it, and why noise reduction and, much later, perceptual coding both start here.

At -- Hz the threshold is raised to -- dB one octave above.

The ear generates harmonics and combination tones of its own. Feed it two loud tones and it produces sum and difference frequencies that are not in the air at all — which means a distortion measurement can never be compared directly with what a listener reports at high level.

Aural distortion against level: the ear adds its own.

Change the phase of the harmonics and the waveform changes completely while the sound barely changes at all. Olson concludes that the ear is largely insensitive to phase for steady tones, which is what allows a loudspeaker with severe phase shift to sound perfectly acceptable.

Two very different waves that sound the same.

Vibrato is frequency modulation of a musical tone at about six or seven cycles per second, and tremolo is the amplitude version. Both are perceived as richness rather than as a defect, which is exactly the same modulation that becomes wow and flutter in 10.6 when it is not wanted.

Vibrato spreads the line into sidebands.

How small a change can be heard at all: about one decibel in level and about three tenths of one per cent in frequency, over most of the useful range. These two numbers set the tolerance for everything else in the book — there is no point holding a response to a tenth of a decibel.

At -- dB: -- dB and --% are the smallest audible changes.

What makes the same note on a violin and a clarinet different. Timbre is the harmonic structure plus the way it changes over the life of the note, and Olson is careful to say that the second part matters as much as the first: a steady tone stripped of its attack is very hard to identify.

The clarinet keeps the odd harmonics and drops the even ones.

A tone must last long enough to be heard as a tone. Below about a tenth of a second the ear cannot settle on a pitch and the sound becomes a click, and the shorter it is the louder it must be to be heard at all — the ear integrates energy over roughly 200 milliseconds.

Shorter than 200 ms and the threshold starts climbing.

How the ear follows a sound that starts and stops. Loudness grows fast and decays slowly, and that asymmetry is why a room with a long reverberation time masks the sounds that follow — and why the transient response of 6.25 matters more than its steady state curve suggests.

The perceived sound outlasts the physical one.

Two ears, two cues. Below about 1500 cycles the difference in arrival time between the ears locates the source; above it, the head shadows one ear and the difference in level takes over. Stereophony works because both cues can be manufactured with two loudspeakers.

At --°: -- μs between the ears, about -- dB of shadow.

When the same sound arrives twice, the ear takes its direction from the first arrival and hears the second as part of it, provided the delay is short enough. Beyond about fifty milliseconds the second arrival separates into an echo — the single most useful fact in sound reinforcement.

-- ms: --

Olson reproduces the survey data: hearing loss with age is real, it starts at the top of the band, and it is worse in men than in women. It is the reason a system judged excellent by its designer may be judged dull by a younger listener and shrill by nobody at all.

At -- years, typical loss is -- dB at 4 kHz.

The chart every engineer of the period had on the wall: the frequency and level territory occupied by speech, by the orchestra and by the ear itself. The orchestra needs about 100 dB of range and ten octaves; speech needs 40 dB and four. That difference decides how much system you have to build.

The territory a reproducing system has to cover.

Olson’s listening tests on band limitation: cut the top and the bottom of reproduced music and ask people what they think. The result is the design brief for every system in the book — the losses are not symmetrical, and cutting the top is judged far more damaging than cutting the bottom.

Both cuts lose energy; only one of them loses the listener.

The companion experiment: how much distortion can be present before it is noticed, and how that threshold depends on the bandwidth of the system. The finding is uncomfortable and important — a wider band system reveals distortion that a narrow one hides, so improving one thing exposes the other.

The better the system, the less distortion it can hide.

Absolute numbers rather than relative curves: the actual sound pressures produced by speech, by musical instruments and by noise, measured at defined distances. This is the section a designer opens to find out how much output a system genuinely needs.

Real levels, at a stated distance.

Every stage adds noise, and the ear judges the result against the quietest passage rather than the loudest. Olson works the requirement backwards from the room: the system noise has to sit below the room noise, or the extra dynamic range is wasted.

Below the room noise, more dynamic range buys nothing.

The masking of 12.9 applied to the living room. The noise of the room sets the level at which the quiet passages disappear, so the useful dynamic range of a reproducing system is the distance between the room noise and the loudest level the room will tolerate — usually far less than the medium can carry.

The room decides how much of the medium you can use.

Two tones through a nonlinear transducer produce sum and difference frequencies, and the difference tone is the dangerous one: it falls below both originals, where nothing masks it. This is why intermodulation is a more revealing test than harmonic distortion.

Difference -- Hz, sum -- Hz, cubic terms at -- Hz.

Subjective tests of reproduced sound against measured distortion, graded in three steps: not perceptible, perceptible but not objectionable, objectionable. Olson’s numbers are still roughly where modern listening tests land, which says something about both the ear and the method.

Where listeners actually draw the lines.

What the systems of the day actually delivered: the telephone, the radio receiver, the phonograph, the sound motion picture and the best laboratory system, each with its band drawn against the range of the orchestra. The gap between the last two is the size of the engineering problem this book set out to solve.

Four media, four bands, one orchestra to carry.

Given a choice, listeners preferred a restricted band over a full one — a result that startled everyone. Olson traced it to distortion and noise: with the systems of the day, opening the band let in more faults than music, so the preference was rational.

Clean up the system and the preference moves to full range.

The control experiment that settles it. With a live source behind a curtain and an acoustic filter in front of it, listeners preferred the full range every time. The band restriction was never the preference — it was the distortion that came with the band.

The definitive answer to 12.29.

The same method applied to stereophony: listeners preferred two channels to one, and the preference held even when the single channel was given more bandwidth. Spatial information turned out to be worth more than frequency range — which is why the industry went stereo before it went wideband.

Space beats bandwidth, in the listening room.

The hardest test in the book: an orchestra behind a curtain, alternating with its own reproduction, and listeners asked which is which. Olson reports where the differences were heard — the extremes of the band, the loudest passages and the sense of space — a list that reads like the agenda for the next fifty years of audio.

What was left to fix, measured in 1957.

The fundamental ranges of the instruments of the orchestra and of the singing voice, laid out end to end. The double bassoon reaches down to about 30 cycles and the piccolo up past 4000, but every one of them has harmonics far above its own fundamental — which is why a 5 kHz system sounds wrong on all of them.

Fundamentals only; the harmonics go far higher.

Equal temperament divides the octave into twelve equal ratios of the twelfth root of two, which makes every key equally usable and every interval equally slightly wrong. Olson gives the arithmetic and the comparison with just intonation, where the intervals are exact small ratios but only in one key.

With A₄ = -- Hz: C₅ = -- Hz; the tempered fifth is -- cents flat of just.

Instruments where the sound is generated or shaped electrically: the electric organ, the electric guitar and their relatives. Olson treats them as transducer problems, and points out the freedom they gain — the generator and the radiator are now independent, so the tone no longer depends on what will physically resonate.

Drawbars are additive synthesis, built in 1935.

The RCA synthesizer: frequency, amplitude, timbre, duration and envelope of every note punched on a paper roll and generated electronically. Olson describes it as producing any sound within the range of hearing, which is exactly the claim every synthesizer has made since — and this one was first.

Two classic waveforms, specified rather than played.

The book closes its long chapter on hearing with a machine that listens: speech in, printed text out. Olson is honest that it worked only for a limited vocabulary from a known talker — and he sets out exactly why the general problem is hard, which is the same reason it stayed hard for another fifty years.

The same word from two talkers: this is why it is hard.