A simple mathematical model is constructed and explored. A description of the physics behind the model is included. The mathematical model predicts anomalous conditions under which energy is not conserved.
Let us begin our discussion with some information about waves. A physical wave in air can be considered a moving record of the forces exerted on the air. There are a number of ways that these forces can be created. The simplest and most common cause of pressure waves is a moving object, although nature incorporates other methods of generation. For example, thunder is created when an electric discharge heats the air causing it to expand and deliver the force of that expansion to the surrounding air.
Other methods of generating sound waves include resonant columns (an organ pipe for example) or heating of moist air by means of focused micro waves, or the alternating separation of Von Karman vortex sheets (a whistle for example works on this principle). Typically the sound produced on a 'human scale' represents pressure changes in the microbar region. That is, the pressure variations in the air represent changes is pressure of about one millionth of normal sea level atmospheric pressure. At these small pressure changes sound waves may be treated as linear disturbances in the air. Working at the microbar level, the differences in the property of the air under compression and rarefaction is small enough that we may safely ignore them in the first approximation.
A typical very loud stereo system might produce a radiant field which has a power flow of about one watt. The speaker system then radiates away about one joule per second. The common practice is to consider that the air represents a radiation resistance to the speaker - thus absorbing the power the speaker radiates. In some ways the air may be considered a three dimensional transmission line. Just as a transmission line has a characteristic impedance which is determined by its electrical properties, so does a gas have a mechanical impedance generated by its distributed mass and effective spring constant.
Let us investigate the generation of sound waves by a typical vibrating object. When the object moves first in one direction less space is available for the atoms of air on the side toward which the object moves. Conversely on the side away from the direction of movement of the moving object more space is available to the air in the immediate vicinity of the object. Since air pressure is determined by the number of atoms in a region of space and their temperature, the side toward which the object is moving will have an increased pressure, since the temperature is held constant, but the volume is reduced.
Statistically on the side of increased pressure atoms directly outside of the region of direct movement of the object will feel more impacts in a given period of time on the side toward the object than on their side away from the object. Thus we see a pressure build up propagate through the air, as each succeeding region feels more impacts from one direction than another.
In a similar fashion air on the side away from movement feels more impacts in a given period of time from the side away from the object than on the side toward the object. Thus we can see a rarefaction wave propagated away from the object as each succeeding region moves in response to the rarefaction. Sound waves may be seen then as a statistical phenomenon.
When the object reverses its direction of vibration the side which was experiencing a rise in pressure now experiences a rarefaction wave and vice versa. Once a wave is initiated into free air it continues away from the radiating object. The frequency of the wave in the air is the same as the frequency of vibration of the object.
Experiments have shown that the wave motion progresses through a gas at a speed which depends upon temperature, the molecular weight of the gas, and the pressure. As a first approximation the speed of sound in air at STP may be given as about 1100 ft/sec. Dividing the speed of sound in distance/sec by the frequency in cycles/sec yields a distance/cycle. This is the unit of measurement of the wavelength. Were you to freeze the wave motion in time the wavelength would be the physical distance between repetitions of the wave pattern.
Thus the object in the proceeding description behaves as a dipole radiator. If the wave pattern surrounding a dipole radiator radiating at low frequencies is drawn it may be seen that it consists of two semi circular wave fronts 180 degrees out of phase. The rarefactions from one radiator line up with the increased pressure areas from the other. The molecules from the high pressure side tend to flow into rarefaction area from the other wavefront - weakening the pressure differences on both sides.
This process is repeated for each succeeding wave front - meaning that the pressure of each wave is diminished much more rapidly than would occur did the other wavefront not exist. This circulation of air 'unloads' the dipole at low frequencies decreasing its ability to radiate power away into the air. If the frequency of vibration is low enough very little energy is radiated away from a dipole source since the air from one side of the radiator simply flows to the other side, preventing waves from ever being formed.
If one creates a barrier between the two wave fronts created by a dipole radiator the frequency at which the dipole loses its ability to radiate is lowered - the larger the barrier, the lower the frequency the dipole may effectively radiate. By closing the barrier in on itself, so that the radiation from one side of the dipole is completely enclosed by the barrier, we form what is called in the sound reproduction field an 'infinite baffle'.
An infinite baffle enclosed radiator does not have the low frequency limitations of a dipole. As the frequency being produced by such an infinite baffle enclosed speaker is lowered the volume producing ability of the device begins to decrease at some point. This is because in order to produce waves at a lower frequency it is necessary for the wave producing device to sustain the pressure on the air for a longer period of time.
At a given volume of sound - at some low frequency - the movement of the membrane becomes large enough to reach the physical limits of the range of motion of the membrane. As the frequency is lowered below this point the volume ( amplitude of the wave) has to be lowered to prevent the membrane from attempting to exceed its physical range of movement limitation.
For the purposes of this paper we may imagine a pulsating sphere - which may be pictured as a dipole which is closed in on itself, forming its own infinite baffle. Such a radiator has no low frequency dipole limitations.
In this paper the mathematical descriptions will be first order algebraic treatments. This is a judgment call on the part of the author; understanding new concepts is difficult enough without having to wade through a paper where someone is attempting to impress everyone with how much math he knows. Carried to its extremes this behavior can make the simplest concepts unintelligible.
The fundamental premise of Fourier analysis is that any repetitive waveform may be described as a series of sine waves. This premise means that instead of having to handle mathematically difficult wave shapes we may safely confine our mathematical analysis of sound to only sine waves, since they may be used to create any other repetitive wave form.
The usual mathematical treatment of sound waves at a given point in space is to model them by where 't' is the time and '' is an angular velocity - usually given in radians per second. Multiplying an angular velocity by a time gives an (angular) distance. It is sometimes useful to describe a sound wave as where '' is a physical distance in space. This representation describes a wave frozen at a moment in time instead of one frozen in a reference position as the representation does.
Measurements of the amplitude of the waves which would be produced by an ideal spherical speaker would show that the amplitude of the wave falls off as 1/r where r is the distance from the center of the speaker. Of course this 1/r relationship fails to obtain inside the speaker itself - much in the same way that Newton's gravitational equation fails to hold its relationship inside of a mass.
The energy transferred by a sound wave obeying the amplitude relationship over a unit of area falls off as . This may be understood by noting the following:
From 1,2, and 3 above we can see that doubling the amplitude of a wave results in the wave containing four times as much energy as before, since the mass of the air reaches twice the velocity it did with the undoubled wave acting on it.
Heat is the disordered random motion of molecules in a gas. Wave behavior is the ordered behavior of groups of molecules. To a first order approximation, if a wave is emitted from a spherical radiator, there is no mechanism for converting the wave - traveling in an obstruction free gas - into heat. If there were such a mechanism the rate at which the amplitude of the wave decreases would necessarily be greater than .
To a first order approximation wave propagation maintains the relative amplitudes and phase relationships of the original emitted wave.
Figure 1 shows the basics of the terminology used in the rest of this paper. The dot labeled (0,0) is the origin (center) of a circle of radius r. The point labeled (X,Y) is used to calculate the radius of the circle from the Pythagorean theorem.
A point source radiator of sound emitting a single frequency may be modeled mathematically by the equation:
Where tda is the Time Dependant Amplitude of the wave at a given position.
Figure 2 shows the representation of a sound wave being emitted from a point source. The x and y axis of the plot represents the two dimensional spatial coordinates, while the z axis of the plot represents the amplitude of the wave at a given (x,y) position. The plot was constructed with Dartmouth University's gnuplot program running under Linux 1.2.1 on a generic clone 386-40 PC with a math co-processor. The equation plotted was
8.
which is the Cartesian form of the polar:
9.
where r is determined by means of the Pythagorean theorem:
10.
Equation 9 is space dependent, time frozen version of equation 7.
The wavelength of the plot in figure 2 is .
When a second radiator is introduced as in figure 3, the situation becomes more complex.
In figure 3 we see two radiators, each displaced an equal distance +/-(b) from the origin on the X axis. In a fashion similar to figure 1 R is the radius of a circle centered on (0,0). The lines A and B represent the distance from the left and right radiators respectively to the point (X,Y) on the circle described by (0,0) and R. By constructing an imaginary right triangle the length of line A is given by:
11.
and the length of B is given by:
12.
The sound produced by the left radiator may be modeled by:
13.
While the sound produced by the right radiator is given by:
14.
The composite sound field from the two radiators may be given by:
15.
The two wave fronts are added if the speakers are in phase, and subtracted if the speakers are out of phase.
Expanding equation 15 by substituting for 'A' and 'B' with equations 11 and 12 yields:
16.
Because the distances 'A' and 'B' may differ, the phase and amplitude of the two sine waves arriving at a point given by (X,Y) may differ. Figure 4 shows the result of adding two sine waves of the same amplitude and frequency, but with a phase shift of 90 degrees. The result is a sine wave with the original frequency, a phase shift of 45 degrees, and an amplitude of times the amplitude of either of the original waves.
Even when the two wave fronts are added together it is possible for one wave to largely cancel the other at a given position. If the two speakers are wired in phase but the distance 'A' is wavelength longer than the distance 'B' then at that location the waves will tend to cancel each other. The cancellation would not be complete since the amplitudes would in general not be absolutely identical.
Conversely, where the distances of 'A' and 'B' were equal, or one full wavelength different, the waves would tend to reinforce each other. The area of cancellation is called destructive interference, since the amplitude of the composite wave is less than the sum of the two individual waves. Conversely the area where the waves reinforce each other is called an area of constructive interference, since the composite wave has an amplitude which is more than the greater than a single wave.
In areas of destructive interference energy is missing: as the amplitude of the composite wave front approaches 0 the energy in that region approaches 0. Conversely in areas of constructive interference more energy is available in the composite wave front than from the sum of the two individual waves. (The amplitudes add, but energy goes as the square of the amplitude of a wave.) We can see then that the energy missing in the case of destructive interference shows up in regions of constructive interference.
Figure 5 shows two radiators emitting waves of length . The radiators are separated by a distance of 4 units ( wavelengths). The regions of constructive and destructive interference may be seen. Figure 6 shows the same two emitters with the phase of one of them reversed, note that the regions of constructive and destructive interference are reversed.
Figure 5.
Figure 6.
As the sources are brought closer together fewer regions of constructive and destructive interference exist, and those that do exist are larger. The interference anomaly occurs when the distance separating the two emitters is less than of a wave length. At this distance only one type of interference region completely surrounds the speakers. If the speakers are wired in phase the interference of the two wave fronts is everywhere constructive. Conversely if the speakers are wired out of phase the interference region surrounding the speakers is everywhere destructive.
Thus the algorithm for calculating energy which was earlier pointed out is maintained: energy missing in the case of destructive interference shows up in the case of constructive interference. If each speaker is emitting one watt, the power in the surrounding field in the case of destructive interference is approximately 0 watts, while in the case of constructive interference the surrounding field contains approximately 4 watts.
The problem is that energy is apparently not being conserved in either case: either there is energy missing in the case of total destructive interference, or there is too much energy in the case of total constructive interference.
Figure 7 illustrates the case of total constructive interference. In figures 7 & 8 the emitting sources are separated by a distance of one tenth unit ( wavelength. Note that in figure 7 the amplitude of the emitted wave is twice the amplitude of a wave emitted by a single source. This means the energy in the field is four times that of the field generated by a single source. (See figure 2 for comparison).
Figure 8 illustrates the case of total destructive interference.
(Note the scale on figure 8)
To say that this is an unexpected result is an understatement - yet real speakers tested in the real world behave in just this fashion. From these results it would appear that rather than conserving (mass) energy, what nature conserves are her techniques, her algorithms, her mechanical methods of doing things. This statement is very similar in effect to conservation of energy, since those techniques almost always result in energy being conserved, but there exists at least one set of conditions under which those techniques do not result in energy being conserved.
It has been my experience that exposing people to wrong ways of thinking about something is in general a bad idea. Later as they try to think about that target idea, it is difficult for them to remember which way of thinking about it is the more correct approach. However, in this case we have an exception; EVERYBODY thinks about the problem in the wrong way already, so it is helpful to point out those incorrect thought processes and show where they go wrong in detail. I shall do this by means of pseudo question and answer session.
Q. If the interference anomaly you have described existed wouldn't it also show up if you hooked two transmitting nodes onto an Ethernet cable, since these nodes launch waves into the cable much like speakers launch waves into the air?
A. There is a crucial difference between speakers in free air and transmitters tied to a cable. The difference is that a transmitter tied to a cable 'sees' the entire signal produced produced by another transmitter - this other signal effects the transmitter in the same way that the signal it is producing effects it. The effective reflected impedance of the cable is altered so that the total energy in the cable is the sum of the two transmitters output energy.
However, in the case of two speakers radiating into free air, the couple between the two speakers is not 100% as it is in the case of transmitters on a cable. If the two speakers are physically small compared to the distance between them the energy coupling between them is poor.
For example let us assume that we have two speakers of radius one separated by a distance of 10 units. Because of the relationship of a sound field to distance, the signal strength of a speaker would only be of the strength of signal the it produced by the time it reached the other speaker. Even were the speakers efficient at producing and absorbing sound (i.e. they were perfect microphones as well as perfect speakers) the sound energy falling on the second speaker would only be of the energy the first speaker produced. Thus the two speakers are hardly 'aware' of each others signal.
Q. Doesn't the air simply churn back and forth between the two radiators like a dipole if the speakers are wired out of phase?
A. To a certain extent the back and forth movement of the air does occur. If you look at the movement of the air under these circumstances you realize that it does not look like the normal loading which occurs under different conditions. Indeed if the speakers are closer than wave apart instead of looking like a resistive load, the air between the speakers begins to resemble a simple mass. That is; there is movement, but very little compression which takes place.
Conversely if the speakers are wired in phase under these circumstances, the impedance of the air between the speakers resembles a spring, There is compression, but little movement. However, this is once again a second order effect, as the rest of the surrounding air continues to have primarily a resistive transmission line component to its impedance.
Neither a spring nor a mass (capacitive and inductive impedances in electrical terms) can serve as an energy sink - they do not absorb energy - it takes a combination of mass and spring to act as a transmission line, and thus as an infinite sink for energy.
Q. Exactly, doesn't this mass like quality of the air under out of phase conditions account for the missing energy - that is, doesn't the inductive property of the air prevent energy from being pumped into it?
A. No, please note that the capacitive spring like property of air during in phase conditions also won't act like an energy sink. Electrically the air surrounding a speaker could be modeled as a low impedance resistor in parallel with a high impedance, variable, inductor and capacitor network. Under out of phase conditions, the capacitor is adjusted to its highest impedance setting, and the inductor to its lowest.
Conversely during in phase operation the inductor is set to its highest impedance setting, while the capacitor is set to its lowest impedance setting. These adjustments have little effect on the over all behavior of the circuit because of the relative differences in the impedance of the reactive section to the resistive component of the circuit. The circuit remains primarily resistive in nature.
The real first order issues to be considered are:
At the current time I am only entertaining questions or confirmations of the above first order issues. Don't write to me about second order effects like the near field antenna effect, or questions of speaker efficiency. First things first. It is only after everyone is satisfied on the first order issues that we may move on to second order cases.
Postscript:
One of the only things remaining from the certainty of nineteenth century physics is conservation of energy. Yet even that is not unchanged; first Einstein's discovery that mass and energy are two sides of the same coin changed conservation of energy into conservation of mass-energy. Secondly the way we look at conservation of energy has changed. What started as a conjecture, then became an important principle, then an inviolate law, has now become an item of religious dogma. Anyone who dares to question that dogma becomes the scientific equivalent of a heretic - a pariah to be cast from the house of physics.
Yet nothing in physics is properly dogma: all is hypothesis - to be discarded as soon as evidence to the contrary is presented.
There was once a movement in computer programming to construct programs only of algorithms which were provably correct; the idea being that a combination of provably correct routines would itself be free of error. That movement collapsed when it was realized that it was still possible to write programs full of errors even when using only provably correct routines. After all - each of the instructions for a typical microprocessor is free of error and provably correct - but it is still possible to write programs with those error free instructions which are themselves full of mistakes.
I believe that the case in physics is very similar: each of the laws of physics can be demonstrated to conserve energy - but that does not necessarily mean that combinations of those laws will be bug free.
In the years that this web page has been up only one professional physicist has bothered to investigate the claims made here. What he found was deeply disturbing to him: he found no error in my math or physics; the plots of interference patterns did indeed show a violation of conservation of energy - his independent reworking of the equations gave the same results. I furnished him with data on a simple experiment conducted with two real speakers, an amplifier and a sine wave generator. After inspecting my crude data gathered with a commercial sound pressure meter he agreed that the data tended to support the predictions of the equations. Even in a crude experiment such as I was able to run he was forced to admit that the effect did seem to exist.
He finally decided that the 'error' I was making had to lie in the fact that I was using simple superposition of waves instead of a rigorous 'exact' solution for the production of sound - since it was 'impossible' that a real violation of conservation of energy could occur. And here he found 'the problem'; the 'exact' solution for the emission of sound from a point 'guaranteed' that energy would be conserved mathematically; and that was enough for him to decide that he now had proven that all was right with the world. I did not think that it was my place to point out to him that the simple method I had used also conserved energy for a single speaker - that the whole point of the discussion was that it was only when more than one speaker interacted that the anomaly appeared. We parted ways cordially and no one since has done anything concerning the paper.
If there is anyone else who would care to look at this paper I would be more than happy to provide him with my admitidly crude experimental results to inspect.