96:
rarefaction, the vibration is encouraged". This shows that he related thermoacoustics to the interplay of density variations and heat injection. The formal theoretical study of thermoacoustics started by
Kramers in 1949 when he generalized the Kirchhoff theory of the attenuation of sound waves at constant temperature to the case of attenuation in the presence of a temperature gradient. Rott made a breakthrough in the study and modeling of thermodynamic phenomena by developing a successful linear theory. After that, the acoustical part of thermoacoustics was linked in a broad thermodynamic framework by Swift.
1807:
1684:
91:. Feldman mentioned in his related review that a convective air current through the pipe is the main inducer of this phenomenon. The oscillations are strongest when the screen is at one fourth of the tube length. Research performed by Sondhauss in 1850 is known to be the first to approximate the modern concept of thermoacoustic oscillation. Sondhauss experimentally investigated the oscillations related to glass blowers. Sondhauss observed that sound frequency and intensity depends on the length and volume of the bulb.
1616:≈0.66. For typical sound frequencies the thermal penetration depth is ca. 0.1 mm. That means that the thermal interaction between the gas and a solid surface is limited to a very thin layer near the surface. The effect of thermoacoustic devices is increased by putting a large number of plates (with a plate distance of a few times the thermal penetration depth) in the sound field forming a stack. Stacks play a central role in so-called standing-wave thermoacoustic devices.
910:
1625:
to play the intended role in the thermoacoustic effect. The interplay of heat and sound is applicable in both conversion ways. The effect can be used to produce acoustic oscillations by supplying heat to the hot side of a stack, and sound oscillations can be used to induce a refrigeration effect by supplying a pressure wave inside a
1706:. In other words: we have produced a cooler. This is the basis of thermoacoustic cooling as shown in Fig. 2b which represents a thermoacoustic refrigerator. It has a loudspeaker at the left. The system corresponds with the left half of Fig. 1b with the stack in the position of the blue line. Cooling is produced at temperature
1731:
plates but the device works also very well with loosely packed stainless steel wool or screens. It is heated at the left, e.g., by a propane flame and heat is released to ambient temperature by a heat exchanger. If the temperature at the left side is high enough, the system starts to produces a loud
49:
Thermoacoustic effects can be observed when partly molten glass tubes are connected to glass vessels. Sometimes spontaneously a loud and monotone sound is produced. A similar effect is observed if one side of a stainless steel tube is at room temperature (293 K) and the other side is in contact
1666:
in a gas. A standing-wave thermoacoustic engine typically has a thermoacoustic element called the "stack". A stack is a solid component with pores that allow the operating gas fluid to oscillate while in contact with the solid walls. The oscillation of the gas is accompanied with the change of its
1624:
Acoustic oscillations in a medium are a set of time depending properties, which may transfer energy along its path. Along the path of an acoustic wave, pressure and density are not the only time dependent property, but also entropy and temperature. Temperature changes along the wave can be invested
127:
The sound intensity of ordinary speech is 65 dB. The pressure variations are about 0.05 Pa, the displacements 0.2 μm, and the temperature variations about 40 μK. So, the thermal effects of sound cannot be observed in daily life. However, at sound levels of 180 dB, which are
71:
produced heat-generated sound when blowing a hot bulb at the end of a cold narrow tube. This phenomenon also has been observed in cryogenic storage vessels, where oscillations are induced by the insertion of a hollow tube open at the bottom end in liquid helium, called
Taconis oscillations, but the
1754:
The heat exchanging process in TAE is critical to maintain the power conversion process. The hot heat exchanger has to transfer heat to the stack and the cold heat exchanger has to sustain the temperature gradient across the stack. Yet, the available space for it is constrained with the small size
1830:
between gas and material is very good. Ideally, the energy flow in the regenerator is zero, so the main energy flow in the loop is from the hot heat exchanger via the pulse tube and the bypass loop to the heat exchanger at the other side of the regenerator (main heat exchanger). The energy in the
1691:
If we put a thin horizontal plate in the sound field, the thermal interaction between the oscillating gas and the plate leads to thermoacoustic effects. If the thermal conductivity of the plate material would be zero, the temperature in the plate would exactly match the temperature profiles as in
1629:
where a stack is located. In a thermoacoustic prime mover, a high temperature gradient along a tube where a gas media is contained induces density variations. Such variations in a constant volume of matter force changes in pressure. The cycle of thermoacoustic oscillation is a combination of heat
1825:
tube and a loop which contains a regenerator, three heat exchangers, and a bypass loop. A regenerator is a porous medium with a high heat capacity. As the gas flows back and forth through the regenerator, it periodically stores and takes up heat from the regenerator material. In contrast to the
1675:
of the wave. Thermal penetration depth is defined as the distance that heat can diffuse though the gas during a time 1/ω. In air oscillating at 1000 Hz, the thermal penetration depth is about 0.1 mm. Standing-wave TAE must be supplied with the necessary heat to maintain the temperature
95:
gave a qualitative explanation of the
Sondhauss thermoacoustic oscillations phenomena, where he stated that producing any type of thermoacoustic oscillations needs to meet a criterion: "If heat be given to the air at the moment of greatest condensation or taken from it at the moment of greatest
1692:
Fig. 1b. Consider the blue line in Fig. 1b as the temperature profile of a plate at that position. The temperature gradient in the plate would be equal to the so-called critical temperature gradient. If we would fix the temperature at the left side of the plate at ambient temperature
1667:
temperature. Due to the introduction of solid walls into the oscillating gas, the plate modifies the original, unperturbed temperature oscillations in both magnitude and phase for the gas about a thermal penetration depth δ=√(2k/ω) away from the plate, where k is the
729:
1265:
is the thickness of the layer of the gas where heat can diffuse through during half a cycle of oscillations. Viscous penetration depth δv is the thickness of the layer where viscosity effect is effective near the boundaries. In case of sound, the
249:
131:
A full theory of thermoacoustics should account for the propagation of heat in the fluid as it makes compression cycles during the propagation of the sound wave. Good insights can however be gained by making the usual assumption of
1352:
42:. They can use heat available at low temperatures which makes it ideal for heat recovery and low power applications. The components included in thermoacoustic engines are usually very simple compared to conventional
1094:
451:
1449:
1253:
oscillations in temperature that result in no heat transfer to or from the walls, which is undesirable. Therefore, an important characteristic for any thermoacoustic element is the value of the thermal and
1596:
1240:
diagram as shown in Fig. 1c, which applies to a pure traveling wave to the right. The gas moves to the right with a high temperature and back with a low temperature, so there is a net transport of energy.
58:
and Swift and his co-workers. Technologically thermoacoustic devices have the advantage that they have no moving parts, which makes them attractive for applications where reliability is of key importance.
1780:
The frequency of the resultant pressure wave, since this frequency should match the resonance frequency required by the load device, either a thermoacoustic refrigerator/heat pump or a linear alternator.
76:
to diminish and acoustic wave to weaken and then to stop completely. Byron
Higgins made the first scientific observation of heat energy conversion into acoustical oscillations. He investigated the "
1527:
2120:
886:
1249:
The thermoacoustic effect inside the stack takes place mainly in the region that is close to the solid walls of the stack. The layers of gas too far away from the stack walls experience
2154:
913:
Fig. 1. a: Plot of the amplitudes of the velocity and displacements, and the pressure and temperature variations in a half-wavelength tube of a pure standing wave. b: corresponding
611:
588:
150:
1155:
969:
1904:
K. W. Taconis, J. J. M. Beenakker, A. O. C. Nier, and L. T. Aldrich (1949) "Measurements concerning the vapour-liquid equilibrium of solutions of He in He below 2.19 K,"
1845:, so the regenerator acts as a volume-flow amplifier. Just like in the case of the standing-wave system, the machine "spontaneously" produces sound if the temperature
1230:
1194:
54:
are observed which are named "Taconis oscillations". The mathematical foundation of thermoacoustics is by
Nikolaus Rott. Later, the field was inspired by the work of
1745:
The commercially available linear alternators used to convert acoustic energy into electricity currently have low efficiencies compared to rotary electric generators
1831:
loop is transported via a travelling wave as in Fig. 1c, hence the name travelling-wave systems. The ratio of the volume flows at the ends of the regenerator is
991:
plot is a vertical line here. In the middle of the tube the pressure and temperature variations are zero, so we have a horizontal line. It can be shown that the
1751:
TAE uses gases at high pressures to provide reasonable power densities which imposes sealing challenges particularly if the mixture has light gases like helium.
1723:
and heat up the left side so that the temperature gradient in the plate would be larger than the critical temperature gradient. In that case, we have made an
128:
normal in thermoacoustic systems, the pressure variations are 30 kPa, the displacements more than 10 cm, and the temperature variations 24 K.
1777:
The onset temperature difference, defined as the minimum temperature difference across the sides of the stack at which the dynamic pressure is generated.
1762:
which dissipates energy due to viscous effects, harmonic generation of different frequencies that carries acoustic power in frequencies other than the
2117:
2090:
K.W.Taconis and J.J.M. Beenakker, Measurements concerning the vapor-liquid equilibrium of solutions of 3He in 4He below 2.19 K, Physica 15:733 (1949).
1926:
N. Rott, Damped and thermally driven acoustic oscillations in wide and narrow tubes, Zeitschrift für
Angewandte Mathematik und Physik. 20:230 (1969).
1635:
1282:
92:
1001:
347:
1399:
1852:
is high enough. The resulting pressure oscillations can be used in a variety of ways, such as in producing electricity, cooling, and
1953:
1538:
1755:
and the blockage it adds to the path of the wave. The heat exchange process in oscillating media is still under extensive research.
1758:
The acoustic waves inside thermoacoustic engines operated at large pressure ratios suffer many kinds of non-linearities, such as
1727:(prime mover) which can e.g. produce sound as in Fig. 2a. This is a so-called thermoacoustic prime mover. Stacks can be made of
2155:
M. Emam, Experimental
Investigations on a Standing-Wave Thermoacoustic Engine, M.Sc. Thesis, Cairo University, Egypt (2013)
2118:
M. Emam, Experimental
Investigations on a Standing-Wave Thermoacoustic Engine, M.Sc. Thesis, Cairo University, Egypt (2013)
1472:
124:). Thermoacoustic machines rely more on the temperature-position variations than the usual pressure-velocity variations.
2160:
M.E.H. Tijani, Loudspeaker-driven thermo-acoustic refrigeration, Ph.D. Thesis, Technische
Universiteit Eindhoven, (2001)
2136:
G.W. Swift, A unifying perspective for some engines and refrigerators, Acoustical
Society of America, Melville, (2002).
1806:
975:. Figure 1a gives the dependence of the velocity and position amplitudes (red curve) and the pressure and temperature
1687:
Fig. 2. a: schematic diagram of a thermoacoustic prime mover; b: schematic diagram of a thermoacoustic refrigerator.
2189:
1818:
1647:
751:
87:
introduced this phenomenon into a greater scale by using a heated wire screen to induce strong oscillations in his
27:
55:
140:
change and will tell the correct direction of heat flow. Under the adiabatic approximation, the one-dimensional
1683:
1612:
is of order 1, so the two penetration depths are about equal. For helium at normal temperature and pressure, P
724:{\displaystyle \delta x={\frac {v_{Ar}}{\omega }}\sin(\omega t-kx)+{\frac {v_{Al}}{\omega }}\sin(\omega t+kx)}
2184:
1875:
1770:
The performance of thermoacoustic engines usually is characterized through several indicators as follows:
462:
1795:
1763:
1655:
1267:
244:{\displaystyle c^{2}{\frac {\partial ^{2}v}{\partial x^{2}}}-{\frac {\partial ^{2}v}{\partial t^{2}}}=0}
2063:
2024:
1982:
1971:"Understanding some simple phenomena in thermoacoustics with applications to acoustical heat engines"
1870:
1362:
73:
39:
2099:
K.T. Feldman, Review of the literature on Rijke thermoacousticphenomena, J. Sound Vib. 7:83 (1968).
1668:
1114:
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stack, the pores in the regenerator are much smaller than the thermal penetration depth, so the
1949:
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1199:
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331:
133:
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1990:
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2124:
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992:
136:. Even if no heat is exchanged during adiabatic compression, the temperature of the fluid
2067:
2028:
1986:
1662:. A thermoacoustic engine operates using the effects that arise from the resonance of a
1677:
1463:
271:
1945:
1387:
at constant pressure. Viscous effects are determined by the viscous penetration depth
2173:
2108:
Lord Rayleigh, The theory of sound, 2ndedition, Dover, New York (2), Sec.322, (1945).
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1602:
1384:
1104:
972:
141:
77:
23:
909:
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1742:
Very high densities of operating fluids are required to obtain high power densities
1373:
898:
The last two equations form a parametric representation of a tilted ellipse in the
84:
68:
31:
326:
are the average pressure, temperature, and density respectively. In monochromatic
2159:
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1699:(e.g. using a heat exchanger), then the temperature at the right would be below
327:
109:
51:
2076:
2051:
1865:
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1759:
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is the area of the cross section of the sound duct. Since in a standing wave,
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88:
35:
1716:
It is also possible to fix the temperature of the right side of the plate at
1880:
1853:
1822:
1748:
Only expensive specially-made alternators can give satisfactory performance.
1631:
1626:
1347:{\displaystyle \delta _{\kappa }^{2}={\frac {2\kappa V_{m}}{\omega C_{p}}}.}
1255:
976:
286:
80:" phenomena in a portion of a hydrogen flame in a tube with both ends open.
1735:
Thermoacoustic engines still suffer from some limitations, including that:
22:
is the interaction between temperature, density and pressure variations of
983:
plane is reduced to a straight line as shown in Fig. 1b. At the tube ends
2149:
67:
Thermoacoustic-induced oscillations have been observed for centuries.
2165:
Design Environment for Low-amplitude ThermoAcoustic Energy Conversion
2036:
1724:
117:
43:
1994:
1970:
1810:
Fig. 3. Schematic drawing of a travelling-wave thermoacoustic engine
1638:, by the appropriate phasing of heat transfer and pressure changes.
1089:{\displaystyle P={\frac {\gamma p_{0}}{2c}}A(v_{Ar}^{2}-v_{Al}^{2})}
1659:
1270:
for thermal interaction is given by the thermal penetration depth
446:{\displaystyle v=v_{Ar}\cos(\omega t-kx)+v_{Al}\cos(\omega t+kx).}
121:
105:
1940:. Advances in Applied Mechanics. Vol. 20. pp. 135–175.
1444:{\displaystyle \delta _{\nu }^{2}={\frac {2\eta }{\omega \rho }}}
1815:
1651:
1233:
108:
is understood in terms of pressure variations accompanied by an
1591:{\displaystyle \delta _{\nu }^{2}=P_{r}\delta _{\kappa }^{2}.}
113:
2052:"Basic Operation of Cryocoolers and Related Thermal Machines"
1634:. Self-induced oscillations can be encouraged, according to
50:
with liquid helium at 4.2 K. In this case, spontaneous
1794:
The variation of the resultant wave frequency with the TAE
1107:
at fixed pressure to the specific heat at fixed volume and
2150:
Thermoacoustic research at Los Alamos National Laboratory
1236:. In this case, Eqs.(1) and (2) represent circles in the
46:. The device can easily be controlled and maintained.
1541:
1522:{\displaystyle P_{r}={\frac {\eta C_{p}}{M\kappa }}.}
1475:
1402:
1285:
1202:
1166:
1117:
1004:
931:
754:
614:
465:
350:
153:
1787:, indicating the ratio of higher harmonics to the
1590:
1532:The two penetration depths are related as follows
1521:
1443:
1346:
1258:penetration depths. The thermal penetration depth
1224:
1188:
1149:
1088:
963:
880:
723:
582:
445:
243:
1739:The device usually has low power-to-volume ratio.
2017:The Journal of the Acoustical Society of America
1814:Figure 3 is a schematic drawing of a travelling-
2015:Swift, G. W. (1988). "Thermoacoustic engines".
979:(blue curve) for this case. The ellipse of the
881:{\displaystyle \delta T={\frac {cM}{C_{p}}}.}
8:
597:of a gas-particle with equilibrium position
2010:
2008:
2006:
2004:
1676:gradient on the stack. This is done by two
1936:Rott, Nikolaus (1980). "Thermoacoustics".
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1157:, the average energy transport is zero.
908:
1938:Advances in Applied Mechanics Volume 20
1897:
1791:in the resulting dynamic pressure wave.
72:lack of heat removal system causes the
1774:The first and second law efficiencies.
456:The pressure variations are given by
7:
995:, transported by sound, is given by
583:{\displaystyle \delta p=c\rho _{0}.}
63:Historical review of thermoacoustics
1630:transfer and pressure changes in a
741:and the temperature variations are
2056:Journal of Low Temperature Physics
219:
205:
182:
168:
14:
38:and they can be controlled using
1650:(TAE) is a device that converts
16:Study of heat-sound interactions
921:plots of a pure traveling wave.
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1946:10.1016/S0065-2156(08)70233-3
1150:{\displaystyle v_{Ar}=v_{Al}}
971:, we are dealing with a pure
964:{\displaystyle v_{Ar}=v_{Al}}
917:plots of a standing wave. c:
1680:on both sides of the stack.
1671:of the gas and ω=2πf is the
30:can readily be driven using
2050:Waele, A. T. A. M. (2011).
1975:American Journal of Physics
1232:, we have a pure traveling
28:Thermoacoustic heat engines
2206:
2077:10.1007/s10909-011-0373-x
1466:of the gas is defined as
1225:{\displaystyle v_{Al}=0}
1189:{\displaystyle v_{Ar}=0}
1103:is the ratio of the gas
305:. In these expressions,
1969:Wheatley, John (1985).
1802:Travelling-wave systems
1605:, like air and helium,
1876:Thermoelectric cooling
1811:
1688:
1620:Thermoacoustic systems
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1458:the gas viscosity and
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245:
1819:thermoacoustic engine
1809:
1796:operating temperature
1764:fundamental frequency
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1648:thermoacoustic engine
1642:Standing-wave systems
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134:adiabatic compression
1871:Photoacoustic effect
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112:motion of a medium (
74:temperature gradient
40:proportional control
2068:2011JLTP..164..179D
2029:1988ASAJ...84.1145S
1987:1985AmJPh..53..147W
1821:. It consists of a
1785:harmonic distortion
1669:thermal diffusivity
1584:
1556:
1417:
1385:molar heat capacity
1300:
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2123:2013-09-28 at the
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906:as the parameter.
878:
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580:
443:
341:, the solution is
266:the position, and
262:the gas velocity,
241:
2190:Energy conversion
1912: : 733-739.
1673:angular frequency
1514:
1462:its density. The
1439:
1339:
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894:
893:
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2062:(5–6): 179–236.
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1658:in the form of
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2172:
2171:
2168:
2167:
2162:
2157:
2152:
2145:
2144:External links
2142:
2139:
2138:
2129:
2110:
2101:
2092:
2083:
2042:
2000:
1981:(2): 147–162.
1961:
1954:
1928:
1914:
1896:
1895:
1893:
1890:
1889:
1888:
1883:
1878:
1873:
1868:
1861:
1858:
1849:
1842:
1835:
1803:
1800:
1799:
1798:
1792:
1783:The degree of
1781:
1778:
1775:
1768:
1767:
1756:
1752:
1749:
1746:
1743:
1740:
1720:
1710:
1703:
1696:
1643:
1640:
1621:
1618:
1613:
1609:
1603:working fluids
1599:
1598:
1587:
1582:
1577:
1573:
1567:
1563:
1559:
1554:
1549:
1545:
1530:
1529:
1518:
1512:
1509:
1502:
1498:
1494:
1488:
1483:
1479:
1464:Prandtl number
1452:
1451:
1437:
1434:
1429:
1426:
1420:
1415:
1410:
1406:
1391:
1380:
1369:
1355:
1354:
1343:
1335:
1331:
1327:
1320:
1316:
1312:
1309:
1303:
1298:
1293:
1289:
1274:
1262:
1246:
1243:
1221:
1218:
1213:
1210:
1206:
1185:
1182:
1177:
1174:
1170:
1144:
1141:
1137:
1133:
1128:
1125:
1121:
1097:
1096:
1085:
1080:
1075:
1072:
1068:
1064:
1059:
1054:
1051:
1047:
1043:
1040:
1034:
1031:
1024:
1020:
1016:
1010:
1007:
958:
955:
951:
947:
942:
939:
935:
896:
895:
892:
891:
888:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
842:
839:
835:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
799:
796:
792:
788:
781:
777:
772:
769:
763:
760:
757:
739:
738:
735:
734:
731:
720:
717:
714:
711:
708:
705:
702:
699:
696:
691:
686:
683:
679:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
641:
636:
633:
629:
623:
620:
617:
593:The deviation
591:
590:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
549:
544:
541:
537:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
501:
498:
494:
490:
485:
481:
477:
474:
471:
468:
454:
453:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
410:
407:
403:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
367:
364:
360:
356:
353:
322:
315:
308:
292:
281:
277:
272:sound velocity
252:
251:
240:
237:
229:
225:
221:
216:
211:
207:
200:
192:
188:
184:
179:
174:
170:
161:
157:
101:
98:
64:
61:
24:acoustic waves
15:
13:
10:
9:
6:
4:
3:
2:
2202:
2191:
2188:
2186:
2185:Heat transfer
2183:
2181:
2178:
2177:
2175:
2166:
2163:
2161:
2158:
2156:
2153:
2151:
2148:
2147:
2143:
2133:
2130:
2126:
2122:
2119:
2114:
2111:
2105:
2102:
2096:
2093:
2087:
2084:
2078:
2073:
2069:
2065:
2061:
2057:
2053:
2046:
2043:
2038:
2034:
2030:
2026:
2022:
2018:
2011:
2009:
2007:
2005:
2001:
1996:
1992:
1988:
1984:
1980:
1976:
1972:
1965:
1962:
1957:
1955:9780120020201
1951:
1947:
1943:
1939:
1932:
1929:
1923:
1921:
1919:
1915:
1911:
1907:
1901:
1898:
1891:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1863:
1859:
1857:
1855:
1848:
1841:
1834:
1829:
1824:
1820:
1817:
1808:
1801:
1797:
1793:
1790:
1786:
1782:
1779:
1776:
1773:
1772:
1771:
1765:
1761:
1757:
1753:
1750:
1747:
1744:
1741:
1738:
1737:
1736:
1733:
1730:
1726:
1719:
1714:
1709:
1702:
1695:
1685:
1681:
1679:
1674:
1670:
1665:
1664:standing-wave
1661:
1657:
1653:
1649:
1641:
1639:
1637:
1636:Lord Rayleigh
1633:
1628:
1619:
1617:
1608:
1604:
1585:
1580:
1575:
1571:
1565:
1561:
1557:
1552:
1547:
1543:
1535:
1534:
1533:
1516:
1510:
1507:
1500:
1496:
1492:
1486:
1481:
1477:
1469:
1468:
1467:
1465:
1461:
1457:
1435:
1432:
1427:
1424:
1418:
1413:
1408:
1404:
1396:
1395:
1394:
1390:
1386:
1379:
1375:
1368:
1364:
1360:
1341:
1333:
1329:
1325:
1318:
1314:
1310:
1307:
1301:
1296:
1291:
1287:
1279:
1278:
1277:
1273:
1269:
1261:
1257:
1252:
1244:
1242:
1239:
1235:
1219:
1216:
1211:
1208:
1204:
1183:
1180:
1175:
1172:
1168:
1158:
1142:
1139:
1135:
1131:
1126:
1123:
1119:
1110:
1106:
1105:specific heat
1102:
1078:
1073:
1070:
1066:
1062:
1057:
1052:
1049:
1045:
1038:
1032:
1029:
1022:
1018:
1014:
1008:
1005:
998:
997:
996:
994:
990:
986:
982:
978:
974:
973:standing wave
956:
953:
949:
945:
940:
937:
933:
920:
916:
911:
907:
905:
901:
889:
875:
866:
863:
860:
857:
854:
848:
845:
840:
837:
833:
829:
823:
820:
817:
814:
811:
805:
802:
797:
794:
790:
779:
775:
770:
767:
761:
758:
755:
748:
747:
744:
743:
742:
732:
715:
712:
709:
706:
703:
697:
694:
689:
684:
681:
677:
671:
665:
662:
659:
656:
653:
647:
644:
639:
634:
631:
627:
621:
618:
615:
608:
607:
604:
603:
602:
600:
596:
577:
568:
565:
562:
559:
556:
550:
547:
542:
539:
535:
531:
525:
522:
519:
516:
513:
507:
504:
499:
496:
492:
483:
479:
475:
472:
469:
466:
459:
458:
457:
440:
434:
431:
428:
425:
422:
416:
413:
408:
405:
401:
397:
391:
388:
385:
382:
379:
373:
370:
365:
362:
358:
354:
351:
344:
343:
342:
340:
336:
333:
329:
325:
318:
311:
304:
300:
296:
288:
284:
273:
269:
265:
261:
257:
238:
235:
227:
223:
214:
209:
198:
190:
186:
177:
172:
159:
155:
147:
146:
145:
143:
142:wave equation
139:
135:
129:
125:
123:
119:
115:
111:
107:
99:
97:
94:
93:Lord Rayleigh
90:
86:
81:
79:
78:singing flame
75:
70:
69:Glass blowers
62:
60:
57:
56:John Wheatley
53:
47:
45:
41:
37:
33:
29:
25:
21:
2132:
2113:
2104:
2095:
2086:
2059:
2055:
2045:
2020:
2016:
1978:
1974:
1964:
1937:
1931:
1909:
1905:
1900:
1854:heat pumping
1846:
1839:
1832:
1813:
1769:
1734:
1717:
1715:
1707:
1700:
1693:
1690:
1645:
1623:
1606:
1600:
1531:
1459:
1455:
1453:
1388:
1377:
1374:molar volume
1366:
1358:
1356:
1271:
1259:
1248:
1237:
1159:
1108:
1100:
1098:
988:
984:
980:
924:
918:
914:
903:
899:
897:
740:
601:is given by
598:
594:
592:
455:
338:
334:
320:
313:
306:
298:
290:
275:
267:
263:
259:
255:
253:
137:
130:
126:
103:
85:Pieter Rijke
82:
66:
52:oscillations
48:
32:solar energy
19:
18:
1886:Thermophone
1652:heat energy
987:=0, so the
902:plane with
328:plane waves
110:oscillating
2174:Categories
1892:References
1866:Cryocooler
1760:turbulence
977:amplitudes
303:molar mass
89:Rijke tube
83:Physicist
36:waste heat
2180:Acoustics
1881:Pyrophone
1823:resonator
1627:resonator
1601:For many
1576:κ
1572:δ
1548:ν
1544:δ
1511:κ
1493:η
1436:ρ
1433:ω
1428:η
1409:ν
1405:δ
1326:ω
1311:κ
1292:κ
1288:δ
1251:adiabatic
1063:−
1015:γ
855:ω
849:
830:−
818:−
812:ω
806:
756:δ
704:ω
698:
690:ω
660:−
654:ω
648:
640:ω
616:δ
557:ω
551:
532:−
520:−
514:ω
508:
480:ρ
467:δ
423:ω
417:
386:−
380:ω
374:
337:and with
287:ideal gas
285:. For an
274:given by
220:∂
206:∂
199:−
183:∂
169:∂
2121:Archived
1860:See also
104:Usually
2064:Bibcode
2025:Bibcode
1983:Bibcode
1906:Physica
1732:sound.
1361:is the
1256:viscous
1238:δT – δx
989:δT – δx
981:δT – δx
919:δT – δx
915:δT – δx
900:δT – δx
330:, with
44:engines
1952:
1725:engine
1376:, and
1099:where
319:, and
258:time,
118:liquid
1654:into
1454:with
1357:Here
993:power
297:with
291:c=γRT
254:with
122:solid
106:sound
100:Sound
1950:ISBN
1816:wave
1656:work
1646:The
1383:the
1372:the
1234:wave
890:(2)
733:(1)
339:ω=kc
301:the
276:c=γp
270:the
138:does
2072:doi
2060:164
2033:doi
1991:doi
1942:doi
1196:or
1160:If
925:If
846:cos
803:cos
695:sin
645:sin
548:cos
505:cos
414:cos
371:cos
120:or
114:gas
34:or
2176::
2070:.
2058:.
2054:.
2031:.
2021:84
2019:.
2003:^
1989:.
1979:53
1977:.
1973:.
1948:.
1917:^
1910:15
1908:,
1856:.
1713:.
1365:,
985:δx
595:δx
312:,
295:/M
289:,
280:/ρ
116:,
26:.
2127:.
2080:.
2074::
2066::
2039:.
2035::
2027::
1997:.
1993::
1985::
1958:.
1944::
1850:H
1847:T
1843:a
1840:T
1838:/
1836:H
1833:T
1766:.
1721:a
1718:T
1711:L
1708:T
1704:a
1701:T
1697:a
1694:T
1614:r
1610:r
1607:P
1586:.
1581:2
1566:r
1562:P
1558:=
1553:2
1517:.
1508:M
1501:p
1497:C
1487:=
1482:r
1478:P
1460:ρ
1456:η
1425:2
1419:=
1414:2
1392:ν
1389:δ
1381:p
1378:C
1370:m
1367:V
1359:κ
1342:.
1334:p
1330:C
1319:m
1315:V
1308:2
1302:=
1297:2
1275:κ
1272:δ
1263:κ
1260:δ
1220:0
1217:=
1212:l
1209:A
1205:v
1184:0
1181:=
1176:r
1173:A
1169:v
1143:l
1140:A
1136:v
1132:=
1127:r
1124:A
1120:v
1109:A
1101:γ
1084:)
1079:2
1074:l
1071:A
1067:v
1058:2
1053:r
1050:A
1046:v
1042:(
1039:A
1033:c
1030:2
1023:0
1019:p
1009:=
1006:P
957:l
954:A
950:v
946:=
941:r
938:A
934:v
904:t
876:.
873:]
870:)
867:x
864:k
861:+
858:t
852:(
841:l
838:A
834:v
827:)
824:x
821:k
815:t
809:(
798:r
795:A
791:v
787:[
780:p
776:C
771:M
768:c
762:=
759:T
719:)
716:x
713:k
710:+
707:t
701:(
685:l
682:A
678:v
672:+
669:)
666:x
663:k
657:t
651:(
635:r
632:A
628:v
622:=
619:x
599:x
578:.
575:]
572:)
569:x
566:k
563:+
560:t
554:(
543:l
540:A
536:v
529:)
526:x
523:k
517:t
511:(
500:r
497:A
493:v
489:[
484:0
476:c
473:=
470:p
441:.
438:)
435:x
432:k
429:+
426:t
420:(
409:l
406:A
402:v
398:+
395:)
392:x
389:k
383:t
377:(
366:r
363:A
359:v
355:=
352:v
335:ω
323:0
321:ρ
316:0
314:T
309:0
307:p
299:M
293:0
282:0
278:0
268:c
264:x
260:v
256:t
239:0
236:=
228:2
224:t
215:v
210:2
191:2
187:x
178:v
173:2
160:2
156:c
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