Speaker Math and Physics

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Math & physics: The tools of the trade

By Roy Johnson, loudspeaker designer, Green Mountain Audio, Inc.

Speakers are transducers. They convert electrical signals into mechanical motion. The relationships between these two worlds are described by many simple mathematic expressions that lead to equally simple tests and measurements.

The simplicity comes from looking at speaker performance only on steady tones or noise. As a result, these equations and tests permit little insight to the performance of a speaker on music, which is why all speakers sound different.

The most common tests became the ones for tone balance and distortion, as they do yield useful measurements, ones also easily understood. This made them popular for marketing and permitted the rise of hundreds of speaker companies.

These tests look at a speaker's performance only on steady tones or steady noise (all tones at once). Because these tones have no beginning nor end, and no changes in between, they do not stimulate a speaker in a dynamic manner. A direct analogy is testing a sports car by driving it in a straight line at 50 miles per hour. We learn how loud it is, its gas mileage, and its sensitivity to wind. We learn nothing about its handling, acceleration, nor braking. Other tests are required to examine those dynamic situations and even then, potential owners still take test drives on their own roads.

Each musical sound and every sonic event is unique, as each is made from a unique sequence of pure tones. Each one of those pure tones happens in its own time, having a defined beginning, ending, and all sorts of changes to its loudness in between.

To get a fair idea of what those tones are, a computer is told to calculate a Fourier Transform on the microphone's signal. The result shows all the tones inside that note, perhaps from a trumpet. This would be the fundamental tone, such as the A440 a tuning fork can produce and then all of the trumpet's higher-pitched harmonics at 880Hz, 1760Hz, ... All of these tones reside 'inside' that trumpet's one note to make the note sound like it came from a trumpet. This is because each tone has its own 'trumpet' loudness, at each and every point in time.

To look at every moment in time, at every tone in the audible range and beyond, is not a simple task, requiring at least four dimensions: Frequency, Time, and +/- loudness (the loudness in both the +/- directions of cone stroke). Between these four variables, there is uncertainty (thank you, Herr Heisenberg), such as the more time we take to look at a frequency, then the more certain we are of its exact pitch (e.g., at 438.7Hz, not 440.0), and the less we know about the loudness variation of that 438.7Hz as time passed.

A Fourier Transform lets one 'see' the individual tones, indicating their frequency and how loud each is, but not very clearly what happens to each as time passes. It minimizes the element of 'time passing' to find each of those tones inside this one note, by averaging their loudness out across time, like a tape-deck meter too slow to show the peaks.

At left is the display of a 'Spectrum Analyzer' showing several different frequencies as spikes marching left to right, from low to high on the musical scale. The height of each spike shows its loudness. The width of each spike shows how precisely we know the central frequency, because it either drifts left or right (high or low) as time passes or we may not have the computer set to measure it very sharply.

Any little spikes near the main spikes are unwanted, additional frequencies, called sidebands, sounding like fuzz or 'mud'. The colors indicate changes in loudness as time passes. Note we cannot see the upper and lower halves of any tone, only the averaged loudness of each one's total 'swing' between + and - (their 'RMS' loudness, as on a tape-deck meter).

So, to make a better speaker, we must know what the speaker can do to each of those tones, all of them, as time goes along. The speaker can change their loudness (+/-, as above), and add sideband distortions, harmonics, and noise, and finally delay their arrivals and departures at your ear. Thus, we have look for what goes wrong when the amplifier tells the speaker's cone, dome or panel to begin moving, change direction, and then stop.

To look for what goes wrong, we have to know first what is likely to go wrong. This requires the element of 'time passing' be inserted back into all the simple equations describing 'a moving piston making sound'. For this, one has to examine how those equations were derived. What results are equations often too complex to be solved, but ones that make it quite clear what aspects of speaker design are important to its dynamic response, and furthermore, how specialized tests can be made for dynamic response, for both the `scope and the ear.

Unfortunately, there is no university path to loudspeaker design, which is a shame because speakers are now all around us. This article describes where many usual equations fall short and what university studies are required to understand the fundamentals behind them, so you can then modify them for what you need.

Ask what equations can do for us

Equations were invented to categorize relationships, and if an equation fits the observations, it can then be used to predict the behavior of other, similar relationships.

An equation can be developed out of other equations or simply from examining experimental results, and we owe much to Galileo for this. He made the observation that a stone falling for twice as long traveled four times the distance. If it fell for three times longer, it traveled nine times farther. He then created simple equations for the observed distances and times, which Sir Isaac Newton would rely upon for his work on gravity. Renee DesCartes later invented the Cartesian system that allow the graphing of equations and measurements.

Newton invented the concepts of Force and Energy, which physicists use to look at relationships. He also invented calculus, which lets us examine relationships in unexpected ways. All of this allowed him and everyone who followed to recognize many other relationships between 'things', a very important aspect of any mathematical analysis. For example, Einstein's equations for relativity came from using simple algebra to relate the speed of an observer and the passage of time to the constant speed of light. Although the result was also expressed in simple algebra, his genius lay in recognizing the implications of the result.

Looking for flaws in the logic

The equations commonly used for speaker design were simplified by leaving out the element of time passing. Finding how the element of time fits back into those equations requires particular studies in physics, math, materials science, and psycho-acoustics. Those studies automatically force one to also ask the right questions.

Specialists

When the first speakers were invented, it was already clear how to move the air -- just push and pull on it. This was how a Victrola's needle moved the diaphragm back and forth on the horn attached. In 1920, a speaker-inventor's challenge was to first make one that even worked, then refine it for better tone balance, clarity, dispersion, and power handling. Eighty years of refinement have led to the production of billions of raw woofers and tweeters per year if we include those used in phones, computers, TVs, headphones, and answering machines.

Of course, that production has pushed down the price of finished speakers, originally equal to a house payment to now much less than a day's labor. The large market now also supports many specialty manufacturers dedicated to building what they believe are the world's best woofers and tweeters, in smaller quantities. This is good, because the vast range of modern materials now requires their full-time specialists to explore.

Their work on raw drivers has freed the speaker designer to become a specialist, focusing on improving the design of the enclosure, choosing among the best woofers and tweeters, and blending them via the speaker's crossover circuit. During that process, the appeal of new hi-tech parts is strong, but too often any improvement in performance they may offer is overshadowed by the overall design of the speaker system, because that design never had its performance goals clearly delineated.

Setting the right goals

A speaker designer must first understand what it takes to both deliver and to blend the sounds from a woofer and tweeter at every instant on the dynamics of music, over there, by your ear. Physics shows that projecting the best sound requires first answering these questions:

• Where are the listeners?

• How loud for them?

• Over what range of bass to treble?

• In what acoustic environment?

Most of those answers are known: We can control a room's acoustics. We know how much low bass and high treble will be required and how loud they must be. What is left to guide a speaker's design is the distance to the listener -- vitally important, as we all know sounds change over distance.

The right path from the right math

Sounds become weaker with distance. A speaker makes only a given amount of pressure and when we move away from the speaker, we receive less sound pressure simply because it has had time (and space) to spread everywhere else.

The dynamic contrasts of sound are also reduced when we move farther away, which affects the impact and clarity. This reduction in dynamics with distance is not explained by the usual equations and measurements

The trouble begins with the typical equation for 'acoustic radiation resistance.' That equation allows us to calculate how much pressure a cone or dome will make in front of it at each tone from low to high. When solved, it shows a woofer will not deliver low bass as loud as it will the voice range (think what happens when you put a speaker outdoors -- no bass at all). It shows a tweeter, made small for the wide dispersion of treble, cannot produce any voice range or bass.

Into that simple equation, one plugs in only the size of the cone or dome, how far it strokes, then assumes it is mounted flush in an infinite wall, and given only steady test tones. This produces the chart at the right, showing graphically at what what frequency the lower-range tones will begin to roll off in their loudness, their pressure, before we even build that woofer or tweeter, let alone measure that with a microphone. However, because the element of time passing was left out of this equation, it cannot describe what happens when the woofer's cone or tweeter's dome first begins to stroke, what happens as the sound pressure first spreads into space (with no infinite wall around the speaker cone), nor when the cone is told to suddenly change direction or to stop. Therefore, this equation cannot describe what happens to the dynamics of the music that needs to be broadcast into a real space.

To fix this equation, one must apply advanced math skills now taught only in upper-level physics courses. The book Elements of Acoustics shows the derivation of the usual equation using simple calculus. There, one will see how that standard approach forces the element of time to be left out at the very beginning. Advanced calculus and physics studies reveal how to alter that equation to include the element of time passing.

What we hear is not what we measure

Speakers send us direct sounds, and also bounce sounds off the walls and other surfaces. What we hear from the reflections are not what measurement microphones hear, especially for the earliest reflections, those which arrive immediately after the direct sound from the speaker. This includes the reflections from the floor, the wall behind the speakers, the large video screen in between, off the side walls near the speakers, and off the face of the speaker cabinet itself.

It turns out that we respond to the early reflections differently in each tone range. In the lows, we hear them as 'more bass.' In the voice range, we hear something wrong with the overall tone balance and that the sound is very much 'in one's face.' Early reflections of high tones are irritating and fatiguing. These responses are in part described by the Haas effect. Unfortunately, the general interpretation of Dr. Haas' psycho-acoustic research makes little distinction about what we hear in each tone range. One must revisit his work and that of Daniel Queen to learn more about this.

When the reflections arrive a little later, such as those from the room's sidewalls that are many feet from the speakers, that allows us just enough time to hear the concert-hall echoes on the recorded CD before those sidewalls' contributions. The extra few thousandths of a second time delay for these reflections turns out to be just enough to allow the mind time to filter off the familiar sound of that room to enjoy what is in the recording. Control rooms in studios are now designed specifically to absorb the early reflections and then widely diffuse the later-arriving reflections. This is known as "Live-End, Dead-End design," originated by Chips Davis.

Controlling what can be controlled

Knowing that any voice-range and treble reflections quite near the speakers are bad for sound, a speaker designer can request an owner place the speakers away from a wall, video screen, home entertainment center, and any rack full of stereo gear. Hopefully, that advice is followed.

A speaker designer has control of only what reflects off the face of the speaker's enclosure and what travels around it to then rebound from other room surfaces. Unfortunately, neither situation is seldom considered by most designers, even on the 'curved-front-panel' speaker designs, because cabinet-face reflections are very difficult to measure and extremely difficult to calculate. Most designers simply say 'we can't hear them.' What escapes around the sides of the speakers is also difficult to measure in the same manner that we hear the results, and is also difficult to calculate.

For cabinet-face reflections, the speaker's front can be covered with an absorptive wool felt. But this creates tone-balance problems because most midrange drivers and tweeters produce their 'proper tone balance' only with those reflections. Re-deriving the radiation-resistance equation for a woofer or tweeter without the caveat of either one being mounted flush into an infinite wall reveals how that woofer or tweeter should measure without the cabinet-face reflections.

Below the voice range, any acoustic-absorption material on the face of the speaker enclosure begins to lose its effectiveness. Thus, in that range, the physical size and the proportions of the cabinet face come strongly into play, because the wavelengths in that tone range are becoming as large as the cabinet face (and the sides and back). This changes how they are reflected back into the room. However, since we hear those 'reflections' as simply making the direct sound from the woofer louder, they can no longer calculated or considered as simple reflections. For example, the presence of a large cabinet face will boost the loudness of the woofer from middle-bass to low-voice, while also throwing a large acoustic shadow behind the cabinet in that same range. These two properties interacting make the speaker audible as the source of those sounds. They also make positioning the speaker difficult for achieving the best tone balance.

To determine the best size and proportions of a cabinet face, the radiation-resistance equation must be modified, both with and without the immediate reflections off the cabinet face, from the floor, the wall behind, and nearby sidewalls. For this, it helps also to study the principles behind the original Klipschorn speaker and what is called the Allison Effect.

Cabinet problems

If a speaker is to make accurate changes in air pressure, then its enclosure/support structure cannot move. Only the cone, dome, or diaphragm should be moving. Yet any enclosure has its own vibrations, as shown in the diagram below. Each drawing shows where perfectly-square panels naturally 'hinge' at higher and higher tones. From them, one can infer where to put braces. These are called Chladni patterns after their discoverer, and interesting images of them are found at these links: a, b, c.

The usual equations for vibration turn out to address only how the cabinet walls flex under the pressure of the woofer's bass tones. Yet, a stethoscope placed on the sides of a wood cabinet reveals the presence of many middle-voice range resonances, audible on that cabinet's corners, its back, even directly over a large interior brace, no matter how thick the wood. These voice range vibrations get into the wood from the woofer or midrange driver touching the wood and from their mounting screws, not from either the sound pressures inside nor outside the cabinet. Rubber-mounting the woofer or midrange driver prevents these vibrations, but then allows that driver to wiggle on each impulse, which it should not.

These voice-range vibrations of the cabinet walls travel out into the air, out-of-phase with the main sound from the cone. Thus, they reduce the clarity and dynamics in the voice range because they are canceling some of the direct sound from the speaker. These vibrations are not explained by a vibration equation because they originate with the actual particles of wood, metal, or plastic moving microscopically -- motions which are literally beyond the reach of braces or any surface damping. Think of them as chaotic surface motions on the cabinet. If you could somehow see them, it would appear like the surface of boiling water.

These particles are vibrating from the nature of their own nearby chemical bonds to the other particles. To learn how to reduce or avoid their vibrations, one studies materials science, touched upon during the first college course in 'Solid-State Physics.' They are not related to acoustical phonons (which are atomic-size), but that helps to visualize what is happening. One also studies the concepts behind shear waves and surface (Rayleigh) waves in materials.

Any cabinet must also 'get out of the way' of the sound pressures leaving the cone or dome. The correct cabinet-face and cabinet-edge curvatures and the overall shape are different at each frequency, which can determined with the aid of four-dimensional vector calculus (x,y,and z directions plus time) to track the sound-pressure front as it moves away from a cone or dome. Combining those results with the radiation-resistance concept and with the psycho-acoustics of cabinet-face reflections determines the size, shape, and curvature of the cabinet's face so that it is neither too large nor too small, and also releases the pressures around to its sides in the least audible manner.

Each cabinet interior shape creates a unique distribution of sound inside, which should never be heard. However, the lower we go down the scale, the more difficult it is to absorb those internal echoes and resonances. Solutions are found when it is understood why acoustic materials absorb at some frequencies but not at others. This is presented in the advanced acoustics books. Do know that non-parallel cabinet walls make very little difference when proper sound-absorbing materials are used inside the cabinet.

Perfect cones, perfect domes

Any diaphragm should move in and out as one unit, not as a quivering mass of uncorrelated vibrations. It should move as a perfectly-solid piston, in and out only -- no excuses. Except there is no direct way to measure this motion on a dynamic impulse. Modern laser scanning does impressively show how much and in what manner the cone 'breaks up,' yet only shows that happening on steady test tones.

Now, since the cone or dome should not break up, any breakups can be electrically seen as a bump or dip in the driver's electrical impedance curve, accurately measured using only a voltmeter and tone generator. What is important to remember is that there should be no irregularities whatsoever to that impedance curve -- it should always be a smooth curve without jumps, dips, or bumps. Any deviation is an indicator of unwanted energy storage and release.

The challenge to designers of raw woofers and tweeters is in finding and fixing the cause of any irregularity, seen in the impedance curve (and usually also seen in the tone-balance curve). For the speaker designer, it helps to discuss the source of an irregularity with an engineer at the woofer or tweeter factory. Once it is known why that irregularity exists, it is possible to imagine how that will sound and determine if it can be avoided, before even trying a sample of that woofer or tweeter.

Perfect suspensions

Of great importance also is to find out how well a driver behaves at all loudness extremes from very soft to very loud. This is called dynamic linearity or conversely, power compression (at high power). The louder the test signal becomes, the easier it is to measure what might be going wrong. On the other hand, there is almost no way to measure how accurately the cone or diaphragm moves on very small strokes, something the small sounds of music requires. One must discover what prevents motion on tiny signals by studying the principles behind the concept of flexibility, including 'stiction' and 'hysteresis.' Then, the potential for good low-level behavior can be seen by examining the suspension of a raw woofer and tweeter. This quickly leaves only a few candidates to audition using music.

Lightweight cones
and domes

A tweeter's small dome must be light in weight so it can go very high up the scale, far beyond the conventional limits of hearing. However, there are large/long ribbon tweeters that also weigh virtually nothing, yet do not go nearly as high. This is because the real mass to move always includes the boundary layer of air that remains 'stuck' to any dome or diaphragm (and to the cone of any woofer or midrange driver). That layer varies in thickness, becoming several inches thick at the lowest frequencies. This boundary-layer effect is derived from studying fluid mechanics, in which air is considered as an incompressible fluid. But ultimately, air is not quite incompressible which leads to other discoveries about why the dynamic impact from a small woofer stroking very far is not the same as a large woofer moving only a small distance.

A woofer's cone should be as light as possible, for efficiency. However, the lighter the cone, the more supple its suspension must be, to produce an even tone balance from bass to treble. This is because the lighter cone allows higher tones to come out louder, but does not increase the loudness as much in the low tones. What affects the loudness of the low tones is the flexibility of the suspension around and behind the woofer's cone. When the differential equations for the moving cone are examined, it becomes clear that a lighter cone requires a more flexible suspension. Since a suspension can only be made so flexible before it fails (think of tissue paper), then a woofer cone can only be made only so light.

A tweeter's dome can be made only so large before all the highs come out only straight ahead. Since we would like to hear the high notes throughout the room, a tweeter must be small in diameter. A tweeter's dispersion should rightfully be called 'angle of coverage,' but no one does. The chart at left shows a rather narrow dispersion in the high treble, and broader in the low treble, looking down on the top of the speaker.

Why does a larger tweeter not send out highs to the sides very loudly? Because when you stand to one side, you first hear the sound arrive from the nearest side of the dome, then the sound from the far side of the dome, even though you cannot see that side. The sound from the far side arrives just late enough to be increasing the air pressure next to your air while the first arrival is now going down in pressure. These two cancel and so we hear less highs.

When a driver has mass and a suspension, it then introduces its own mechanical time delays in its low range and in its high range, which means its sonic delivery falls behind when attempting to blend it with another, different size of driver, such as a woofer to a tweeter.

Signals at the speed of light, yet still delayed

A signal from the amplifier does not travel down the speaker wires. It moves at the speed of light as a field, and therefore exists everywhere instantaneously in the speaker, the wires and amplifier. Well, not actually all at once when the crossover circuit between woofer and tweeter gets in the way. Most crossover circuits delay the lows more than the highs, and by a different amount at each frequency. Only one type does not, a 'first-order' crossover circuit, which is what we employ.

These electrical time delays are created by the capacitors and inductors of a crossover circuit. How much they delay a signal is presented in any first-year electrical circuits book. Why they delay the signals is presented in the first-year physics books and that lets us begin to understand what makes a better-sounding capacitor and inductor.

Digital timing-correction for speakers is now possible and can certainly be effective. However, these mini-computers have other limitations and consequences, mostly from how the initial timing measurements are made and their limited resolution.

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We hope this gives a good idea of the range of problems that are part of speaker design. Since there is no directed coursework at universities for speaker design, we recommend a budding speaker designer obtain both a BS and an MS degree in Physics, supplemented by other studies. Recommended courses and supplemental texts are given below.

Courses of study

Physics

In USA universities, those who choose to obtain their BS degree in Physics spend their first year learning basic principles and the mathematics to analyze fundamental situations. Even if you do not choose to not be a Physics major, we recommend the following coursework:

• First-year Physics: Divided into several areas of focus, the first semester covers measurements, vectors, motion in one dimension, then motion in three dimensions. Particle dynamics are explored, leading to the concepts of work and energy, linear and angular momentum, rotation of rigid bodies, static equilibrium, oscillation, and then gravity. This gives one the fundamentals of moving (dynamic) systems. The student is expected to already know advanced algebra, trigonometry, and geometry, and be taking the first Calculus courses at this same time. During this time, a Physics laboratory class shows how to develop a working hypothesis and then design meaningful experiments. Results would be analyzed to see if they fit the hypothesis and error magnitudes are calculated. The second half of the first year begins with electrostatics, followed by the concept of the electric field, then Gauss' law, electric potential, capacitors and dielectrics, current and resistance, the magnetic field, Ampere's law, Faraday's and Lenz's laws, inductance, oscillations, and electromagnetic waves. There would also be a lab class. This covers the foundational mathematics of electrical systems, no longer taught to Electrical Engineering majors.

• Second-year Physics: Beginning with fluid mechanics, the math for waves is developed, followed by temperature, heat and the first law of thermodynamics, the kinetic theory of gases, entropy and the second law of thermodynamics. In the second semester, the math of geometrical optics is developed, along with the principles behind interference, diffraction, light, and quantum physics. In these is recognized the simple equations often adapted to acoustics and for speakers.

• Classical Electrodynamics: This upper level Physics course concerns the deeper principles and mathematical theories of electricity and magnetism, including electrostatics, magnetostatics, polarized media, direct and alternating current, followed by electromagnetic fields and waves and their interaction with physical objects such as insulators and conductors.

• Modern Physics: Concepts are introduced for Einstein's Special Relativity, wave-particle duality, atomic structure, Schroedinger's wave equation, the hydrogen atom, atomic and molecular spectra, and solid state and band theory. Seemingly unrelated to speaker design, these subjects are necessary for what follows.

• Solid State Physics: This presents the theory of solids, especially of metals, covering crystal structure, x-ray diffraction, phonons, thermal and electrical properties of insulators, band structure, semiconductor impurities, superconductivity, and magnetism. This coursework is necessary to understand conduction in metals, the behaviors of magnetic fields inside materials, the properties of insulators, and the vibrations of materials (along with any damping of those).

• Graduate-level Physics: Classical Mechanics re-examines the validity of Newtonian mechanics and oscillations. Lagrange's and Hamilton's equations are developed for calculating central forces, scattering, and rigid body motion. A student will have already mastered vector analysis, differential equations, and calculus.

• Mathematical Methods of Physics is a survey of the applications for vector calculus, partial differential equations, special functions such as the very useful Green's Functions, Fourier analysis, and generalized functions such as the Dirac delta function. This is where the advanced calculus for wave propagation across space and time are learned, which can be then be manipulated for acoustics.

• Theoretical Mechanics brings new insight into the behaviors of moving systems, including non-linear ones.

Other studies

• A course in Acoustics is always good, but with a Physics education, one can simply study the best books on Acoustics to gain an understanding. We recommend The Master Handbook of Acoustics by F. Alton Everest, all of the books written by Leo L. Berenak, and Elements of Acoustics by Samuel Timkin. The latter takes graduate-level math to comprehend.

• An excellent text is Acoustics and Psychoacoustics by Howard and Angus.

• A marvelous author for mechanical systems and structures is J.E. Gordon. Fun reading for the non-technical and yet highly educational for those with a physics background!

• Do know that when faced with difficulty in understanding what is taught, it may be the fault of the instructor, but far too often it is the textbook being used. Other textbooks may explain the math and physics in more-understandable ways, and we recommend a first-year general Physics book, Fundamentals of Physics by Halliday and Resnick. Solid State Physics by Ashcroft and Mermin is also excellent for understanding electrical and magnetic behaviors.

Other tips

Work in groups for homework. One 'buddy' is not enough. Expect homework for each class to consume 4-5 hours per week.

Consider apprenticing for several months in a professional wood shop to learn what cabinetmakers can and cannot do (or would rather not do).

Upon completion of the undergraduate Physics courses, you will recognize in any speaker-design book the equations that came from approximations and know how to correct many of them. You will be able to examine the specifications of any raw driver and reject 95 percent of them without purchasing samples.

A few final words

Music and sound are all about delivering the changes in pressure as time passes. In the rush to market however, most speakers are designed and analyzed using test tones which do not change in time. Their designers thus ignored the requirements for making music, which of course, must change as time passes.

The coursework above is the only way to learn the math of dynamic systems such as loudspeakers, to gain the knowledge of materials' properties, electricity, and magnetism, and to understand how the air and our ears behave. Unfortunately, most of these subjects are far too complex to learn on one's own.

Remember, you are learning how to calculate what happens when things start and stop moving, whether that 'thing' is wood, Kevlar, the air, an electron, or a magnetic field, and how it is moving. These 'changes in momentum' take both time and energy, so it is important to understand how to incorporate these 'changes as time passes' in all of the math. You can indeed determine what is required to make the air pressure change at any given instant, by any amount, over there, where your ears are. Yes, this may be a lot to study, and fortunately there is both challenge and reward, especially in hearing the music come alive.

• Recorded Sound - The Ideal Soundfield - Time and Phase Coherence - Math and Physics - Drivers - Adjustability - Cabinets - Ports - Circuits - Measurements

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