This invention relates to an acoustic resonator in which near-Linear
macrosonic waves are generated in a resonant acoustic chamber, having specific
applications to resonant acoustic compressors.
My earlier U.S. Patent 5,020,977 is directed to a compressor for a
compression-evaporation cooling system which employs acoustics for compression.
The compressor is formed by a standing wave compressor including a chamber for
holding a fluid refrigerant. A travelling wave is established in the fluid refrigerant
in the chamber. This travelling wave is converted into a standing wave in the fluid
refrigerant in the chamber so that the fluid refrigerant is compressed.
Heretofore, the field of Linear acoustics was Limited primarily to
the domain of small acoustic pressure amplitudes. When acoustic pressure amplitudes
become large, compared to the average fluid pressure, nonlinearities result. Under
these conditions a pure sine wave will normally evolve into a shock wave.
Shock evolution is attributed to a spacial change in sound speed caused
by the large variations in pressure, referred to as pressure steepening. During
propagation the thermodynamic state of the pressure peak of a finite wave is quite
different than its pressure minimum, resulting in different sound speeds along
the extent of the wave. Consequently, the pressure peaks of the wave can overtake
the pressure minimums and a shock wave evolves.
Shock formation can occur for waves propagating in free space, in
wave guides, and in acoustic resonators. The following publications focus on shock
formation within various types of acoustic resonators.
Temkin developed a method for calculating the pressure amplitude Limit
in piston-driven cylindrical resonators, due to shock formation (Samuel Temkin,
"Propagating and standing sawtooth waves", J. Acoust. Soc. Am. 45, 224 (1969)).
First he assumes the presence of Left and right traveling shock waves in a resonator,
and then finds the increase in entropy caused by the two shock waves. This entropy
loss is substituted into an energy balance equation which is solved for limiting
pressure amplitude as a function of driver displacement. Temkin's theory provided
close agreement with experimentation for both traveling and standing waves of finite
Cruikshank provided a comparison of theory and experiment for finite
amplitude acoustic oscillations in piston-driven cylindrical resonators (D.B. Cruikshank,
"Experimental investigation of finite-amplitude acoustic oscillations in a closed
tube ",J. Acoust. Soc. Am. 52, 1024 (1972)). Cruikshank demonstrated close
agreement between experimental and theoretically generated shock waveforms.
Like much of the literature, the work of Temkin and Cruikshank both
assume piston-driven cylindrical resonators of constant cross-sectional (CCS) area,
with the termination of the tube being parallel to the piston face. CCS resonators
will have harmonic modes which are coincident in frequency with the wave's harmonics,
thus shock evolution is unrestricted. Although not stated in their papers, Temkin
and Cruikshank's implicit assumption of a saw-tooth shock wave in their solutions
is justified only for CCS resonators.
For resonators with non-harmonic modes, the simple assumption of a
sawtooth shock wave will no longer apply. This was shown by Weiner who also developed
a method for approximating the limiting pressure amplitude in resonators, due to
shock formation (Stephen Weiner, "Standing sound waves of finite amplitude", J.
Acoust. Soc. Am. 40, 240 (1966)). Weiner begins by assuming the presence
of a shock wave and then calculates the work done on the fundamental by the harmonics.
This work is substituted into an energy balance equation which is solved for Limiting
pressure amplitude as a function of driver displacement.
Weiner then goes on to show that attenuation of the even harmonics
will result in a higher pressure amplitude limit for the fundamental. As an example
of a resonator that causes even harmonic attenuation, he refers to a T shaped chamber
called a "T burner" used for solid-propellent combustion research. The T burner
acts as a thermally driven 1/2 wave length resonator with a vent at its center.
Each even mode will have a pressure antinode at the vent, and thus experiences
attenuation in the form of radiated energy through the vent. Weiner offers no suggestions,
other than attenuation, for eliminating harmonics. Attenuation is the dissipation
of energy, and thus is undesirable for energy efficiency.
Further examples of harmonic attenuation schemes can be found in the
Literature of gas-combustion heating. (see for example, Abbott A. Putnam,
Combustion-Driven Oscillations in Industry (American Elsevier Publishing
Co., 1971)). Other examples can be found in the general field of noise control
where attenuation-type schemes are also employed, since energy losses are of no
importance. One notably different approach is the work of Oberst, who sought to
generate intense sound for calibrating microphones (Hermann Oberst, "A method for
the production of extremely powerful standing sound waves in air", Akust. Z.
5, 27 (1940)). Oberst found that the harmonic content of a finite amplitude
wave was reduced by a resonator which had non-harmonic resonant modes. His resonator
was formed by connecting two tubes of different diameter, with the smaller tube
being terminated and the larger tube remaining open. The open end of the resonator
was driven by an air jet which was modulated by a rotating aperture disk.
With this arrangement, Oberst was able to produce resonant pressure
amplitudes up to 0.10 bar for a driving pressure amplitude of 0.02 bar, giving
a gain of 5 to the fundamental. The driving waveform, which had a 30% error (i.e.
deviation from a sinusoid), was transformed to a waveform of only 5% error by the
resonator. However, he predicted that if more acoustic power were applied, then
nonlinear distortions would become clearly evident. In fact, harmonic content is
visually noticeable in Oberst's waveforms corresponding to resonant pressure amplitudes
of only 0.005 bar.
Oberst attributed the behavior of these finite amplitude waves, to
the non-coincidence of the resonator modes and the wave harmonics. Yet, no explanations
were offered as to the exact interaction between the resonator and the wave harmonics.
Oberst's position seems to be that the reduced spectral density of the resonant
wave is simply the result of comparatively little Q-amplification being imparted
to the driving waveform harmonics. This explanation is only believable for the
modest pressure amplitudes obtained by Oberst. Oberst provided no teachings or
suggestions that his methods could produce linear pressure amplitudes above those
which he achieved, and he offered no hope for further optimization. To the contrary,
Oberst stated that nonlinearities would dominate at higher pressure amplitudes.
A further source of nonlinearity in acoustic resonators is the boundary
layer turbulence which can occur at high acoustic velocities. Merkli and Thomann
showed experimentally that at finite pressure amplitudes, there is a critical point
at which the oscillating laminar flow will become turbulent (P. Merkli, H. Thomann,
"Transition to turbulence in oscillating pipe flow", J. Fluid Mech.,
68, 567 (1975)). Their studies were also carried out in CCS resonators.
Taken as a whole, the literature of finite resonant acoustics seems
to predict that the inherent nonlinearites of fluids will ultimately dominate any
resonant system, independent of the boundary conditions imposed by a resonator.
The literature's prediction of these limits is far below the actual performance
of the present invention.
Therefore, there is a need in the art to efficiently generate very
large shock-free acoustic pressure amplitudes as a means of gas compression for
vapor-compression heat transfer systems of the type disclosed in U.S. Patent 5,020,977.
Further, many other applications within the field of acoustics, such as thermoacoustic
heat engines, can also benefit from the generation of high amplitude sinusoidal
In accordance with the present invention, an acoustic resonator comprises
a chamber containing a fluid, said chamber having a geometry which produces destructive
self-interference of at least one harmonic in said fluid to avoid shock wave formation.
The present invention provides acoustic resonators which eliminate
shock formation by promoting the destructive self-interference of the harmonics
of a wave, whereby near-linear acoustic pressures of extremely high amplitude
can be achieved. The acoustic resonators can also minimise the nonlinear energy
dissipation caused by the boundary layer turbulence of finite acoustic waves.
Other advantages include the ability to minimize boundary viscous
energy dissipation and boundary thermal energy dissipation, to provide an acoustic
driving arrangement for achieving high acoustic pressure amplitudes, and to provide
an acoustic resonator which can maintain near-sinusoidal pressure oscillations
while being driven by harmonic-rich waveforms.
The acoustic resonator of the present invention includes a chamber
containing a fluid. The chamber has a geometry which produces destructive self
interference of at least one harmonic in the fluid to avoid shock wave formation
at finite acoustic pressure amplitudes. Typically, the chamber has a cross-sectional
area which changes along the chamber, and the changing cross-sectional area is
positioned along the chamber to reduce an acoustic velocity of the fluid and/or
to reduce boundary viscous energy dissipation. The chamber may comprise a resonant
chamber for a standing wave compressor used for fluid compression for heat transfer
An acoustic resonator driving system may include a chamber containing
a fluid, wherein the chamber has acoustically reflective terminations at each end.
A driver mechanically oscillates the chamber at a frequency of a selected resonant
mode of the chamber. The acoustic resonator and drive system of the present invention
may be resonator and drive system of the present invention may be connected to
heat exchange apparatus so as to form a heat exchange system such as a vapour-compression
As described above, the acoustic resonator and acoustic driving arrangement
of the present invention provide a number of advantages and achieve non-linear
acoustic pressures of extremely high amplitude. In particular, the actual performance
of the present invention is far beyond the results predicted in the literature
of finite resonant acoustics.
These and other objects and advantages of the invention will become
apparent from the accompanying specifications and drawings, wherein like reference
numerals refer to like parts throughout.
Shock Elimination via Mode-Alignment-Canceled Harmonics
- FIG. 1 is a graphical representation of a resonator having higher modes which
are harmonics (i.e. integer multiples) of the fundamental;
- FIG. 2 is a graphical representation of a resonator having higher modes which
are not harmonics of the fundamental;
- FIG. 3 is a sectional view of an embodiment of a resonator in accordance with
the present invention, which employs an insert as a means of mode tuning;
- FIG. 4 is a table of measured data for the resonator shown in FIG. 3;
- FIG. 5 is a table of theoretical data for the resonator shown in FIG. 3;
- FIG. 6 is a sectional view of an embodiment of a resonator in accordance with
the present invention which employs sections of different diameter as a means of
- FIG. 7 is a table of measured data for the resonator shown in FIG. 6;
- FIG. 8 is a table of theoretical data for the resonator shown in FIG. 6;
- FIG. 9 is a sectional view of an embodiment of a resonator in accordance with
the present invention showing further optimizations in resonator geometry;
- FIG. 10 is a table of theoretical data for the resonator shown in FIG. 9;
- FIG. 11 is a sectional view of an apparatus used in a resonator driving system
in accordance with the present invention, in which the entire resonator is oscillated
along its longitudinal axis;
- FIG. 12 is a sectional view of the resonator shown in FIG. 9 which employs porous
materials for enhanced cancellation of higher harmonics; and
- FIG. 13 is a sectional view of the resonator and driving system of FIG. 11 as
connected to heat exchange apparatus to form a heat exchange system.
It is well known that "pressure steepening" at high acoustic pressure
amplitudes leads to the classic sawtooth waveform of a shock wave. It is also understood
that a sawtooth waveform implies, from Fourier analysis, the presence of harmonics.
If finite amplitude acoustic waves are generated in a constant cross-sectional
(CCS) resonator, a shock wave will appear having the harmonic amplitudes predicted
by the Fourier analysis of a sawtooth waveform. At first this would not seem surprising,
but it must be understood that a CCS resonator has modes which are harmonic (i.e.
integer multiples of the fundamental) and which are coincident in frequency with
the harmonics of the fundamental. CCS resonators can be considered as a special
case of a more general class of resonators whose modes are non-harmonic. Non-harmonic
resonators hold a previously unharnessed potential for providing extremely high
amplitude linear waves. This potential is realized by non-harmonic resonators which
are designed to promote the self-destructive interference of the harmonics of the
The present invention employs this principle and provides a new resonator
design criterion; to optimize the self-cancellation of wave harmonics. This new
design criterion for mode-alignment-canceled harmonics-(MACH) eliminates shock
formation. MACH resonators have achieved pressure amplitudes of 100 psi peak-to-peak,
with mean pressures of 80 psia, without shock formation. This translates into a
peak acoustic pressure amplitude which is 62% of the mean pressure.
Once the MACH design criterion is understood, many different resonator
geometries can be employed for aligning a resonator's higher modes to promote self-cancellation
of harmonics. A straightforward approach for exploiting the MACH principle is
to align resonator modes to fall between their corresponding harmonics.
The bar graph of FIG. 1 illustrates the relationship between the harmonics
of the fundamental and the resonator modes for a CCS 1/2 wave length resonator.
The vertical axis marks the wave harmonics of the wave, and the bar height gives
the resonant frequency of the mode. At a fundamental frequency of 100 Hz the wave
will have harmonics at 200 Hz, 300 Hz, 400 Hz, etc. From FIG. 1 it can be seen
that the harmonics of the wave are coincident in frequency with the modes of the
resonator. Stated differently, the nth harmonic of the wave is coincident with
the nth mode of the resonator. Consequently, Little or no self-destructive interference
of the wave harmonics will occur, and a shock wave can evolve without restriction.
For a well developed shock wave, the pressure amplitude of the 2nd harmonic will
be within 6 dB of the fundamental's amplitude.
The bar graph of FIG. 2 illustrates one of many possible arrangements
for promoting the destructive self-interference of harmonics. In FIG. 2, the resonator
modes are aligned to fall between the wave harmonics. For this example, the resonator
modes have been shifted down in frequency so that the nth mode lies between harmonics
n and n-1. With this arrangement a large degree of destructive self-interference
of the wave harmonics can occur.
FIG. 3 is a sectional view of a resonator which was constructed and
tested, and whose modes are shifted down in frequency. The resonator in FIG. 3
is formed by a hollow cylindrical chamber 2, an end flange 4, an end flange 6,
and tapered rod insert 8, with all parts being aluminum. Tapered rod insert 8 was
welded to end flange 4 with end flange 4 being welded to chamber 2. End flange
6 was welded to chamber 2, and was drilled to accommodate a process tube and a
pressure transducer. Chamber 2 has an inside diameter of 5.71 cm, and an inside
length of 27 cm. Tapered rod insert 8 has a half-angle end taper of 34.98°, and
a length of 10 cm, measured from end flange 4. Sharp edges on tapered rod insert
8 were rounded off to an arbitrary curvature to reduce turbulence.
Tapered rod insert 8 serves to create a smaller cross-sectional area
along its length inside of chamber 2. In this way, the resonator of FIG. 3 is divided
into two sections of different cross-sectional area, each section having its own
acoustical impedance. This impedance change results in a shifting of the resonator
modes to non-harmonic frequencies. The degree to which the modes are shifted can
be controlled by varying the diameter and length of tapered rod insert 8. The manner
in which the resonator is driven is described below.
FIG. 4 is a table of measured data obtained for the resonator of FIG.
3. The last column provides a relative measure of the degree of mode shift, by
calculating the difference between the frequency "fn" of the nth mode
and n times the fundamental frequency "nf&sub1;." The ideal mode shift, for placing
the resonator modes at the midpoints between neighboring harmonics, is equal to
1/2 the fundamental frequency. For the FIG. 3 resonator, the ideal shift is f&sub1;/2
= 166.97 Hz. For CCS resonators, the mode shift fn-nf&sub1; = 0 for
each mode by definition.
The resonator design of FIG. 3 does not provide ideal mode shifts,
but comes close enough to provide significant results. This is due to the fact
that the Fourier sum of the first few harmonics contributes heavily to shock formation.
Thus, significant cancellation of the 2nd, 3rd, and 4th harmonics will reduce shock
formation greatly. When the resonator of FIG. 3 was pressurized to 80 psia with
gaseous refrigerant HFC-134a, 11.8 Watts of acoustic input power was required to
achieve a 42 psia peak-to-peak pressure amplitude (measured at end flange 4). This
is within 30% of the required driving power predicted by a strictly linear theory
which accounts for only thermal and viscous boundary layer losses. At these operating
conditions the amplitude of the 2nd harmonic was 20 dB down from the fundamental,
with higher harmonics being down 30 dB or more.
FIG. 5 is a table of theoretical data which was generated for the
FIG. 3 resonator. Ideally, fn-nf&sub1; should be approximately equal
to the ideal shift for each of the resonator modes. However, it can be seen in
FIG. 5 that the degree of mode shifting increases with mode number. At the 6th
mode, shifting has increased so much that the mode frequency is now nearly coincident
with the 5th harmonic of the wave. With more advanced resonator designs, many modes
can be simultaneously tuned to lie between the wave harmonics. As the number of
properly tuned modes increases, the resonator's linearity increases.
FIG. 6 is a sectional view of another resonator which was constructed
and tested. The resonator in FIG. 6 has a chamber which is formed by a small diameter
section 10, a conical section 12. a large diameter section 14. a conical taper
16, and an end flange 18. The chamber comprising the small diameter section 10,
the conical section 12, the large diameter section 14, and the conical taper 16
were all machined from a single piece of aluminum. Aluminum end flange 18 was welded
to conical end taper 16. Small diameter section 10 has a length of 7.28 cm and
a diameter of 3.81 cm. Conical section 12 has a half-angle of 25.63° and an inside
length of 3.72 cm. Large diameter section 14 has an inside length of 13.16 cm and
an inside diameter of 7.38 cm. Conical taper 16 has a half-angle of 26.08° and
an inside length of 2.84 cm. Section 10 and section 14 divide the resonator into
two sections of different cross-sectional area, each section having its own acoustical
impedance. This design results in a downward shifting of the resonator modes to
The FIG. 6 resonator eliminates the tapered rod insert of FIG. 3,
thereby reducing the internal surface area of the resonator, which in turn reduces
the thermal and viscous boundary layer losses. The degree to which the modes are
shifted can be controlled by varying the dimensions of section 10, section 14,
conical section 12, and taper 16. Taper 16 compensates for excessive downward shifting
of the higher modes, by shifting primarily the higher modes up in frequency. The
manner in which the resonator is driven is described below.
FIG. 7 and FIG. 8 are tables of the measured data and theoretical
data, respectively, for the resonator of FIG. 6. In comparison with the FIG. 3
resonator, the FIG. 6 resonator has improved the tuning of the 2nd, 3rd, and 4th
modes, as well as reduced the excessive shifting of higher modes. The FIG. 6 resonator
brings the 2nd, 3rd, and 4th modes much closer to the ideal shift, and results
in improved performance.
When the resonator of FIG. 6 was pressurized to 80 psia, with gaseous
refrigerant HFC-134a, pressure amplitudes of up to 100 psi peak-to-peak (measured
at an end 10a of small diameter section 10) were achieved without shock formation.
However, turbulence was evident, indicating that the acoustic velocity was high
enough to cause non-Laminar flow. As shown below, resonator geometry can be altered
to greatly reduce acoustic velocity. At 60 psi peak-to-peak (measured at the end
10a of small diameter section 10) all harmonics were more than 25 dB down from
the amplitude of the fundamental, for the FIG. 6 resonator.
In general, the modes of a given resonator geometry can be calculated
from the general solution of the wave equation written for both pressure and velocity:
P(x) = Acos(kx) + Bsin(kx)
V(x) = i/(ρc)(Acos(kx) + Bsin(kx))
where i = (-1)1/2, ρ = average fluid density, c = speed
of sound. The arbitrary complex constants A and B are found by applying the boundary
conditions of the resonator to the above equations for P(x) and V(x). Resonators
embodying the present invention were designed by iterating P(x) and V(x) in the
frequency domain across finite elements of the resonator, until zero velocity is
reached at the resonator's end. As demonstrated above, the mid-harmonic placement
of resonator modes provides one of many ways to exploit the MACH principle. For
more exact predictions of harmonic cancellation, the harmonics can be treated as
waves traveling within the boundaries of the resonator, while accounting for their
self-interference. The goal of which is to show harmonic self-cancellation as a
function of changes in the resonator geometry.
Importance of the MACH Principle
It is revealing to compare the performance of MACH resonators with
that of CCS resonators which do not restrict shock formation. As a comparison,
consider the normal evolution to shock formation which occurs as a finite amplitude
wave propagates. Using the method of Pierce, it is possible to calculate the distance
a 60 psi peak-to-peak pressure wave must travel for a fully developed shock wave
to evolve (Allan D. Pierce, Acoustics, p.571 (Acoustical Society of America
1989)). For a mean pressure of 80 psia (in gaseous HFC-134a), the waveform will
evolve from a sinusoid to a shock after traveling only 22 cm, which is less than
one traverse of the 27 cm length of the FIG. 6 resonator! From this it is easy
to appreciate the longstanding assumption that at extremely high amplitudes, intrinsic
nonlinearities of a gas will dominate any resonator design considerations.
Other Resonator Design Parameters
To efficiently create high amplitude resonant acoustic waves, it is
important to keep the resonator boundary layer viscous and thermal losses as low
as possible. Also, the acoustic velocity associated with a desired pressure amplitude
should be minimized to avoid excessive turbulence.
For a pure sinusoidal standing wave in a resonator of constant cross-sectional
area, the peak acoustic velocity is equal to P/(ρc), where P ≡ peak acoustic
pressure amplitude, ρ ≡ average fluid density, and c ≡ speed of
sound at the average pressure. In practice, the peak acoustic velocity can be decreased
by the proper resonator geometry. For example, the resonator of FIG. 6 has a peak
acoustic velocity equal to 0.82(P/(ρc)) (P being measured at the end 10a of
small diameter section 10), due to the expansion at the center of the chamber provided
by conical section 12. This increase in cross-sectional area occurs just before
the velocity maxima at the center of the chamber, thereby lowering the acoustic
Expansions, like those of the FIG. 6 resonator, have other advantages
as well. When the acoustic velocity is reduced, boundary layer viscous losses are
reduced. Also, the expansion reduces the peak acoustic pressure amplitude at end
flange 18, thereby reducing boundary layer thermal losses at this end of the resonator.
Similarly, the expansion provided by end taper 16 of FIG. 6 further reduces the
boundary layer thermal losses. When the position of an expansion, like conical
section 12 of FIG. 6, is varied along the length of the resonator, the boundary
layer thermal losses and the boundary layer viscous losses will vary. It has been
found theoretically that the sum of these losses reaches a minimum when the expansion
is centered at approximately 0.3 of the length of the resonator.
In general, practical energy efficient resonator designs require a
compromise between mode tuning for harmonic cancellation, minimizing acoustic velocity,
and minimizing thermal and viscous losses. FIG. 9 is a sectional view of a resonator
which represents one of a vast number of possible compromises between these design
The FIG. 9 resonator chamber has a conical expansion section 20, a
curved expansion section 22, a curved end taper section 24, and an end flange 28.
Ports 21a, 21b, such as an inlet and outlet or valves, are provided at an end 20a
of the resonator. Although not shown, such ports are also provided in the resonators
of FIGs. 3 and 6. The resonator chamber is preferably formed by a low thermal conductivity
material such as fiberglass, since this will reduce the boundary layer thermal
losses. However, any material, such as aluminum, which can be formed into a desired
configuration can be used. The FIG. 9 resonator is similar in principle to the
FIG. 6 resonator in its method of modal tuning, except for the curved sections
which provide greater mode tuning selectivity. This selectivity is due to the varying
rate of change of cross-sectional area provided by the curved sections, which is
explained as follows. The magnitude of frequency shift of a mode, caused by a given
area change, depends on which part of the standing wave pattern encounters the
area change. Each of the many superimposed standing wave patterns in a resonator
will encounter a fixed area change at a different point along its wave pattern.
Thus, an area change which tunes one mode properly may cause unfavorable tuning
for another mode. Curved sections can provide compensation for this unfavorable
tuning by exposing different modes to different rates of area change. The term
"curved section" is not intended to refer to a specific mathematical surface. Rather,
the term "curved section" is understood to mean in general any section which provides
a rate of change of area, as a function of the longitudinal dimension, whose derivative
is non-zero. Any number of mathematical surfaces can be employed. It is contemplated
that one possible set of equations for the curved expansion section 22 and curved
end tapered section 24 could be as follows.
In FIG. 9 the constant diameter section at end 20a of the resonator
has an inner diameter of 2.54 cm and is 4.86 cm Long. Conical expansion section
20 is 4.1 cm long and has a 5.8° half-angle. Curved expansion section 22 is 3.68
cm long. To the right of curved section 22, the diameter remains constant at 5.77
cm over a distance of 11.34 cm. Curved end taper 24 is 2.16 cm long. To the right
of curved end taper 24, the diameter remains constant at 13 cm over a distance
of 0.86 cm. Curved expansion section 22 was described in a finite element program
by the equation Dn = Dn-1 + 0.00003(7+n), and curved end
taper 24 was described by the equation Dn = Dn-1 + 0.00038(n),
≡ the diameter of the current element, and Dn-1
≡ the diameter of the previous element, and with each element having a length
FIG. 10 is a table of theoretical data for the FIG. 9 resonator, which
shows that the point at which modes and harmonics overlap in frequency has been
significantly extended to higher frequencies.
The FIG. 9 resonator also reduces the acoustic velocity to a value
of 0.58(P/(ρc)) (P being measured at a small diameter end 20a of the resonator),
which represents a significant reduction in acoustic velocity for the desired pressure
amplitude. In addition, the FIG. 9 resonator reduces the total thermal and viscous
energy dissipation of the FIG. 6 resonator by a factor of 1.50. Neglecting turbulent
losses, the total rate of thermal and viscous energy loss, at a given pressure
amplitude, is equal to the acoustic input power required to sustain that pressure
amplitude. Thus, reducing thermal and viscous energy losses will increase energy
Half-Peak Entire-Resonator Driving
The odd modes of a resonator can be effectively driven by mechanically
oscillating the entire resonator along its longitudinal axis. This is the preferred
method used by the resonators of the present invention. Althrough the resonators
of FIG. 3, FIG. 6, and FIG. 9 could be driven by coupling a moving piston to an
open-ended resonator, this approach has certain disadvantages which are avoided
by the entire resonator driving method.
Entire resonator driving can be understood as follows. If the entire
resonator is oscillated along its longitudinal axis, then the end caps will act
as pistons. The odd mode pressure oscillations at the two opposite ends of a double-terminated
resonator will be 180° out of phase with each other. Consequently, when the entire
resonator is oscillated, its end caps, or terminations, can be used to drive an
odd mode in the proper phase at each end of the resonator. In this way, the fundamental
mode can be effectively driven.
FIG. 11 is a sectional view of one of many approaches which can be
used to drive an entire resonator. In FIG. 11 an electrodynamic shaker or driver
29 is provided, having a current conducting coil 26 rigidly attached to end flange
28 of resonator 34, and occupying air gap 30 of magnet 32. Magnet 32 is attached
to end flange 28 by a flexible bellows 36. Bellows 36 maintains proper alignment
of coil 26 within air gap 30.
When coil 26 is energized by an oscillating current, the resulting
electromagnetic forces will cause resonator 34 to be mechanically oscillated along
its Longitudinal axis. Magnet 32 can be rigidly restrained so as to have infinite
mass relative to resonator 34. In the preferred embodiment, magnet 32 is left unrestrained
and thus free to move in opposition to resonator 34. In either case, an appropriate
spring constant can be chosen for bellows 36 to produce a mechanical resonance
equal to the acoustic resonance, resulting in higher electro-acoustic efficiency.
Bellows 36 could be replaced by other components such as flexible diaphragms, magnetic
springs, or more conventional springs made of appropriate materials.
Entire resonator driving reduces the mechanical displacement required
to achieve a given pressure amplitude. When driving the entire resonator, both
ends of the resonator act as pistons. In most cases, entire resonator driving requires
roughly half the peak mechanical displacement which would be needed for a single
Half-Peak Entire-Resonator (HPER) driving provides the following advantages.
As discussed above, the proper tuning of modes of a chamber is critical to efficiently
achieving high acoustic pressure amplitudes. It follows that this tuning must remain
constant during operation. Resonators which are terminated on both ends will maintain
precise tuning during operation and throughout the Lifetime of the resonator.
A further advantage relates to the use of HPER driving for acoustic
compressors. Since HPER driven chambers are sealed, there are no oil-dependant
moving parts that come in contact with the fluid being compressed; resulting in
an inherently oil-free compressor. The suction and discharge valves needed for
acoustic compressors would typically be placed at the narrow end of a resonator,
where the pressure amplitudes are the greatest. For example, valve placement for
the resonator of FIG. 9 would be positioned at ports 21a, 21b at end 20a. The ratio
of pressure amplitudes at the two ends of the FIG. 9 resonator is approximately
3:1 (left to right).
As discussed above, a properly designed MACH chamber will cause the
higher harmonics of its fundamental to be self canceling. For the same reason,
a MACH chamber will tend to cancel out harmonics which may be present in the driver's
displacement waveform. Thus, MACH chambers can convert a non-sinusoidal driving
displacement into a sinusoidal pressure oscillation. In addition, any mechanical
resonance present in a driver, like the driver of FIG. 11, would tend to convert
a non-sinusoidal driving current into a sinusoidal displacement waveform.
In some applications, the use of non-sinusoidal driving signals can
result in greater overall efficiency. For example, the power amplifiers needed
for driving linear motors can be designed to operate very efficiently in a pulsed
output mode. Current pulses can be timed to occur once every acoustic cycle or
to skip several acoustic cycles.
Another type of non-sinusoidal driving, which MACH chambers can facilitate,
is a fluid's direct absorption of electromagnetic energy, as disclosed in U.S.
Patent 5,020,977, the entire content of which is hereby incorporated by reference.
Pulsed microwave and infrared energy, when passed through an absorptive fluid,
will create acoustic waves in the fluid. This electromagnetic-to-acoustic conversion
will tend to result in very harmonic-rich acoustic waves. MACH chambers will tend
to cancel the resulting harmonics, thereby promoting a sinusoidal pressure oscillation.
Electromagnetic pulses can be timed to occur once per acoustic cycle, or to skip
several acoustic cycles.
Porous materials, such as sintered metals, ceramics, and wire mesh
screens are commonly used in the field of noise control. Porous materials can provide
acoustic transmission and refection coefficients which vary as a function of frequency
and acoustic velocity. Properly placed within a resonator, these materials can
be used as an aid to mode tuning.
FIG. 12 is a sectional view of a resonator 34 illustrating one of
many possible uses of porous materials. In FIG. 12 a porous material 38 is rigidly
mounted near end flange 28 of resonator 34. Porous material 38 will have a minimal
effect on the fundamental of the resonator, whose acoustic velocity becomes small
near the surface of end flange 28. The higher modes of the resonator can have velocity
maxima near the position of porous material 38. Thus, the higher harmonics of the
wave can experience larger reflection coefficients at the porous material and be
reflected so as to promote destructive self-interference. Tuning can be adjusted
by varying the position of porous material 38 along the length of resonator 34.
In this way, a porous material can be used as an aid in optimizing
the destructive self-interference of harmonics. The design flexibility provided
by porous materials allows more aggressive optimization of specific resonator parameters,
such as reducing the fundamental's acoustic velocity, without losing the desired
For microwave driven resonators, porous material 38 could also act
together with end flange 28 to form a microwave cavity for the introduction of
microwave energy into resonator 34. FIG. 12 illustrates an electromagnetic driver
39 coupled to the resonator 34 by a coaxial cable 41 having a loop termination
41a inside the resonator 34 in the area between the porous material 38 and end
flange 28. The microwave energy would be restricted to the area between porous
material 38 and end flange 28.
FIG. 13 is a sectional view of resonator 34 and drive apparatus 29
as used in a heat exchange system. In this case, ports 34a and 34b of resonator
34 are connected to a heat exchange apparatus 45 via conduits 47 and 49. Port 34a
is provided with a discharge valve 52, and port 34b is provided with a suction
valve 54. Discharge valve 52 and suction valve 54 will convert the oscillating
pressure within resonator 34, into a net fluid flow through heat exchange apparatus
45. The heat exchange apparatus may include, for example, a conventional condenser
and evaporator, so that the heat exchange system of FIG. 13 may form a vapor-compression
While the above description contains many specifications, these should
not be construed as limitations on the scope of the invention, but rather as an
exemplification of one preferred embodiment thereof. This preferred embodiment
is based on my recognition that acoustic resonators can provide significant self-cancellation
of harmonics, thereby providing extremely high amplitude acoustic waves without
shock formation. The invention is also based on my recognition that other nonlinearities
associated with finite amplitude waves, such as turbulence and boundary layer losses,
can be reduced by proper resonator design.
Application of the MACH principle can provide nearly complete cancellation
of wave harmonics. However, the present invention is not Limited to resonators
which provide complete cancellation. As shown in the above specifications, cancellation
of a harmonic need not be complete to obtain shock-free high amplitude acoustic
waves. Nor do all harmonics need to be canceled. There is a continuous range of
partial harmonic cancellation which can be practiced. Harmonics can be present
without shock formation, as long as their amplitudes are sufficiently small. Resonators
which cancel one, two, or many harmonics could all be considered satisfactory,
depending on the requirements of a particular application. Thus, the scope of the
invention is not limited to any one specific resonator design.
There are many ways to exploit the basic features of the present invention
which will readily occur to those skilled in the art. For example, shifting resonator
modes to the midpoint between adjacent harmonics is only one of many ways to exploit
the MACH principle. Resonator modes can be shifted to any degree as long as adequate
self-destructive interference is provided for a given application.
In addition, many different resonator geometries can support standing
waves and can be tuned to exploit the MACH principle. For example, a toroidal resonator
can be tuned by using methods similar to the embodiments of the present invention.
Although the present specification describes resonators whose modes are shifted
down in frequency, similar resonator designs can shift modes up in frequency. For
example, if the diameters of section 10 and section 14 in FIG. 6 are exchanged,
then the resonator's modes will be shifted up in frequency rather than down. Furthermore,
resonators can be designed to operate in resonant modes other than the fundamental,
while still exploiting the MACH principle. Still further, the shock suppression
provided by MACH resonators will occur for both liquids and gases.
Also, it is understood that the application of MACH resonators to
acoustic compressors is not limited to vapor-compression heat transfer systems,
but can be applied to any number of general applications where fluids must be compressed.
For example, there are many industrial applications where oil-free compressors
are required in order to prevent contamination of a fluid. Finally, many different
drivers can be used with HPER driven resonators. For example, electromagnetic and
piezoceramic drivers can also provide the forces required for entire resonator
driving. In short, any driver that mechanically oscillates the entire resonator
and provides the required forces can be used.