Sound Propagation in Ideal Channels and Tubes

  • Eugen Skudrzyk


Let us first consider an infinitely wide channel, so that the solution depends only on x and z 1. The two-dimensional solution of the wave equation that represents standing waves in the x direction and progressive waves in the z direction is
$$ \tilde p = \bar A\cos ({k_x}x + {\varphi _x}){e^{ - j{k_z}z + j\omega t}}, $$
$$ {k^2} = \frac{{{\omega ^2}}}{{{c^2}}} = {k_{{x^2}}} + {k_{{z^2}}}. $$


Resonant Frequency Wave Front Group Velocity Radiation Resistance Sound Propagation 
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  1. Beranek, L. L.: Precision measurement of acoustic impedance. J.A.S.A. 12 (1940) 3–13;Google Scholar
  2. Beranek, L. L.: Some notes on the measurement of acoustic impedance. J.A.S.A. 19 (1947) 420–427;Google Scholar
  3. Beranek, L. L.: Acoustic measurements. New York: Wiley, London: Chapman and Hall. 1949.Google Scholar
  4. Bolt, R. H., Petrauskas, A. A.: An acoustic impedance meter for rapid field measurements. J.A.S.A. 15 (1943) 79–79.Google Scholar
  5. Brüel, P. V.: Sound insulation and room acoustics. London: Chapman and Hall. 1951.Google Scholar
  6. Casper, L., Sommer, G.: Über Messungen des Schallreflexionskoeffizienten an Materialien bei definierten Schallfeldverhältnissen. Wiss. Veröff. Siemens-Werken 10 (1931)117–127.Google Scholar
  7. Clapp, C. W., Firestone, F. A.: The acoustic wattmeter, an instrument for measuring sound energy flow. J.A.S.A. 13 (1941) 124–136.Google Scholar
  8. Cramer, W. S.: Pulse tube for the measurement of acoustic impedance. J.A.S.A. 25 (1953) 186.Google Scholar
  9. Cremer, L.: Die wissenschaftl. Grundlagen der Raumakustik, Bd. III, Wellentheoretische Raumakustik. Leipzig: Hirzel. 1950.Google Scholar
  10. Davis, A. H., Evans, E. J.: Measurement of absorbing power of materials by the stationary wave method. Proc. Roy. Soc. London (A) 127 (1930) 89–110.ADSMATHCrossRefGoogle Scholar
  11. Dubotjt, P., Davern, W.: Calculation of the statistical absorption coefficient from acoustic impedance tube measurements. Acustica 9 (1959) 15.Google Scholar
  12. Fedorovich, V. N.: Method for measuring acoustical impedance on the basis of measuring the geometric difference between sound pressures. Sov. Phys. Acoust. 1 (1955) 374.Google Scholar
  13. Ferrero, M. A., Sacerdote, G. G.: Measurement of acoustic impedance in a resonant spherical enclosure. Acustica 8 (1958) 325.Google Scholar
  14. Guittard, J.: Impedances terminales de tuyaux sonores sylindriques. Acustica 12 (1962) 313.Google Scholar
  15. Hall, W. M.: An acoustic transmission line for impedance measurement. J.A.S.A. 11 (1939, 1940 ) 140–146.Google Scholar
  16. Hund, M., Kuttruff, H.: Druckkammerverfahren zur Messung von akustischen Impedanzen kleiner Körper (Mikrophone) in Flüssigkeiten. Acustica 12 (1962) 404.Google Scholar
  17. Ingard, U., Bolt, R. H.: Free field method of measuring the absorption coefficient of acoustic materials. J.A.S.A. 23 (1951) 509–516.Google Scholar
  18. Kosten, C. W.: A new method for measuring sound absorption. Appl. scient. Res. B 1 (1950) 35–49;Google Scholar
  19. Kosten, C. W.: A method for measuring sound absorption. Acustica 4 (1954) H. 1.Google Scholar
  20. Kosten, C. W., Zwikker, C.: Die Messung von akustischen Scheinwiderständen und Schluckzahlen durch Rückwirkung auf ein Telefon. Akust. Z. 6 (1941) 124–131.Google Scholar
  21. Lippert, W. K. R.: The practical representation of standing waves in an acoustic impedance tube. Acustica 3 (1953) 153–160.Google Scholar
  22. Loye, D. P., Morgan, R. L.: Acoustic tube for measuring the sound absorption coefficients of small samples. J.A.S.A. 13 (1942) 261–264.Google Scholar
  23. Mawardi, O. K.: On the generalization of the concept of impedance in acoustics. J.A.S.A. 23 (1951) 571–576;MathSciNetGoogle Scholar
  24. Mawardi, O. K.:The measurement of acoustic impedance of small samples. Acustica 4 (1954) 112–114.Google Scholar
  25. Morrical, K. C.: A modified tube method for measurement of sound absorption. J.A.S.A. 8 (1931) 162–171.Google Scholar
  26. Morton, J. Y.: A measuring set for acoustical and mechanical impedances. Acustica 4 (1954) H. 1.Google Scholar
  27. Neubert, H.: Die Messung winkelabhängiger Schallschluckung in einer zweidimensionalen Hallkammer. Akust. Z. 5 (1940) 189–201;Google Scholar
  28. Neubert, H.:Die Messung winkelabhängiger Schallschluckung in einer zweidimensionalen Hallkammer. Dissertation, Berlin 1940.Google Scholar
  29. Nielsen, A. K.: The construction of acoustic impedance meters. Acustica 4 (1954) H. 1.Google Scholar
  30. Paris, E. T. H.: On the stationary-wave method of measuring sound absorption at normal incidence. Proc. Physic. Soc. 39 (1927) 269–295.ADSGoogle Scholar
  31. Raes, A. C.: Pulse measurements of reflection coefficients in amplitude and phase. Acustica 4 (1954) H. 1.Google Scholar
  32. Richardson, E. G.: Amplitude of stationary waves in tubes. Proc. Roy. Soc. A 112 (1926) 522–541.ADSCrossRefGoogle Scholar
  33. Robinson, N. W.: An acoustic impedance bridge. Philos. Mag. 23 (1937) 665–680.Google Scholar
  34. Sabine, H. J.: Notes on acoustic impedance measurements. J.A.S.A. 13 (1942) 143–150.Google Scholar
  35. Schuster, K.: Eine Methode zum Vergleich akustischer Impedanzen. Physik. Z. 35 (1934) 408 409;Google Scholar
  36. Schuster, K.: Messung von akustischen Impedanzen durch Vergleich. E.N.T. 13 (1936) 164–176.Google Scholar
  37. Scott, R. A.: An apparatus or accurate measurements of sound absorbing materials. Proc. Physic. Soc. London 60 (1948) 253–264.ADSGoogle Scholar
  38. Smith, P. H.: An improved transmission line calculator. Electronics 17 (1944) January 130–133, 318 325.Google Scholar
  39. Steiner, F.: Die Anwendung der Riemannschen Zahlenkugel und ihrer Projektion in der Wechselstromtechnik. Radiowelt 1 (1946) 23–26.Google Scholar
  40. Stewart, G. W.: Direct absolute measurement of acoustical impedance. Physic. Rev. 28 (1926) 1038 1047.Google Scholar
  41. Tamm, K.: Ein-und zweidimensionale Ausbreitung von Wasserschall im Rohr bzw. im Flachbecken. Akust. Z. 6 (1941) 16–34.Google Scholar
  42. Taylor, H. O.: Tube method of measuring sound absorption. J.A.S.A. 24 (1952) 701–704.Google Scholar
  43. Thurston, G. B.: Apparatus for absolute measurement of analogous impedance of acoustic elements. J.A.S.A. 24 (1952) 649–652.Google Scholar
  44. Wente, E. C., Bedell, E. H.: The measurement of acoustic impedance and the absorption coefficient of porous materials. B.S.T.J. 7 (1928) 1.Google Scholar
  45. Westervelt, P. J.: Acoustical impedance in terms of energy functions. J.A.S.A. 23 (1951) 347–348.MathSciNetGoogle Scholar
  46. White, J. E.: Method for measuring source impedance and tube attenuation. J.A.S.A. 22 (1950) 565.Google Scholar
  47. Willms, W.: Zum Begriff der Schallimpedanz. Acustica 4 (1954) 427.Google Scholar
  48. Wisotzky, W.: Messung akustischer Widerstände. Hochfrequenztechn. u. Elektroak. 53 (1939) 97–104.Google Scholar
  49. Wüst, H.: Untersuchungen über akustische Vierpole. Hochfrequenztechn. u. Elektroak. 44 (1934) 73–79.Google Scholar
  50. Ballantine, S.: J. Franklin Inst. 203 (1927) 85.CrossRefGoogle Scholar
  51. Carisle, R. W.: Method of improving acoustic transmission in folded horns. J.A.S.A. 31 (1959) 1135.Google Scholar
  52. Eisner, E.: Complete solutions of the “Webster” horn equation. J.A.S.A. 41 (1967) 1126.Google Scholar
  53. Goldsmith, A. N., Minton, J. P.: Proc. Inst. Radio Eng 12 (1924) 423. HALL, W. M.: J.A.S.A. 3 (1932) 552.Google Scholar
  54. Hanna, C. R., Slepian, J.: J. Am. Inst. Elect. Eng 43 (1924) 250.Google Scholar
  55. Hoersch, V. A.: Phys. Rev. 25 (1925) 218; Phys. Rev. 25 (1925) 225.ADSCrossRefGoogle Scholar
  56. Kellogg, E. W.: Gen. Elect. Rev. 27 (1924) 556.Google Scholar
  57. Lambert, R. F.: Acoustical studies of the tractrix horn. I. J.A.S.A. 26 (1954) 1024.Google Scholar
  58. Maxfield, J. P., Harrison, H. C.: J. Am. Inst. Elect. Eng. 45 (1926) 243.Google Scholar
  59. Mckinney, C. M., Anderson, C. D.: Experimental investigation of wedge horns used with line hydrophones. J.A.S.A. 26 (1954) 1040.Google Scholar
  60. Mclachlan, N. W.: Loudspeakers. Oxford: Clarendon Press. 1934.Google Scholar
  61. Mohammed, A.: Equivalent circuits of solid horns undergoing longitudinal vibrations. J.A.S.A. 38 (1965) 862.Google Scholar
  62. Northwood, T. D., Pettigrew, H. C.: Horn as a coupling element for acoustic impedance measurements. J.A.S.A. 26 (1954) 503.Google Scholar
  63. Olson, H. F., Massa, F.: Applied acoustics. Philadelphia: Blackiston’s Son and Co. 1934.Google Scholar
  64. Pyle, R. W., Jr.: Solid torsional horns. J.A.S.A. 41 (1967) 1147.Google Scholar
  65. Scibor-MarchochI, R. I.: Analysis of hypex horns. J.A.S.A. 27 (1955) 939.Google Scholar
  66. Stenzel, H.: A. E. G. Mitteil. H. 5 (1931) 310; Z. techn. Physik 12 (1931) 621.Google Scholar
  67. Stewart, G. W.: Phys. Rev. 16 (1920) 313; Phys. Rev. 25 (1925) 230.ADSCrossRefGoogle Scholar
  68. Webster, A. G.: Proc. Nat. Acad. Sci. Washington, 5 (1919) 275.ADSCrossRefGoogle Scholar
  69. Weibel, E. S.: On Webster’s horn equation. J.A.S.A. 27 (1955) 726.MathSciNetGoogle Scholar
  70. Williams, S.: J Franklin Inst. 202 (1926) 413.CrossRefGoogle Scholar
  71. Aggarwal, R. R.: Axially symmetric vibrations of a finite isotropic disk, Part I. J.A.S.A. 24 (1952) 463;MathSciNetGoogle Scholar
  72. Aggarwal, R. R.: Axially symmetric vibrations of a finite isotropic disk,Part II. J.A.S.A. 24 (1952) 663;MathSciNetGoogle Scholar
  73. Aggarwal, R. R.: Axially symmetric vibrations of a finite isotropic disk,Part III. J.A.S.A. 25 (1953) 533;Google Scholar
  74. Aggarwal, R. R.: Axially symmetric vibrations of a finite isotropic disk,Part IV. J.A.S.A. 26 (1954) 341.Google Scholar
  75. Akhmedov, I. A.: Eigenfrequency spectrum of a plate reverberator. Sov. Phys. Acoust. 15 (1970) 455.Google Scholar
  76. Dyer, I.: Response of plates to a decaying and convecting random pressure field. J.A.S.A. 31 (1959) 922.MathSciNetGoogle Scholar
  77. Feit, D.: Pressure radiated by a point-excited elastic plate. J.A.S.A. 40 (1966) 1489.Google Scholar
  78. Gazis, D. C.: Exact analysis of the plane-strain vibrations of thick-walled hollow cylinders. J.A.S.A. 30 (1958) 786.Google Scholar
  79. Goodman, R. R.: Reflection from a thin infinite plate using the Epstein method. J.A.S.A. 33 (1961) 1096.Google Scholar
  80. Gösele, K.: Schallabstrahlung von Platten, die zu Biegeschwingungen angeregt sind. Acustica 3 (1953) 243.Google Scholar
  81. Greene, D. C.: Vibration and sound radiation of damped and undamped flat plates. J.A.S.A. 33 (1961) 1315.MathSciNetGoogle Scholar
  82. Gutin, L. Y.: Sound radiation from an infinite plate excited by a normal point force. Sov. Phys. Acoust. 10 (1965) 369.Google Scholar
  83. Heckl, M.: Schallabstrahlung von Platten bei punktförmiger Anregung. Acustica 9 (1959) 371;MathSciNetGoogle Scholar
  84. Heckl, M.: Wave propagation on beam-plate systems. J.A.S.A. 33 (1961) 640;MathSciNetGoogle Scholar
  85. Heckl, M.: Abstrahlung von einer punktförmig angeregten unendlich großen Platte unter Wasser. Acustica 13 (1963) 182;Google Scholar
  86. Heckl, M.: Körperschalleistung bei flächenhafter Anregung von Platten. Acustica 15 (1965) 332.Google Scholar
  87. Ingard, U.: On the reflection of a spherical sound wave from an infinite plane. J.A.S.A. 23 (1951) 329.MathSciNetGoogle Scholar
  88. Kaekina, T. M.: Damping of transverse normal modes in plates. Sov. Phys. Acoust. 13 (1968) 380.Google Scholar
  89. Klein, B.: Vibration of simply supported flat plates simultaneously tapered in plan-form and thickness. J.A.S.A. 28 (1956) 1177.Google Scholar
  90. Knyazev, A. S., Tartakovskii, B. D.: Abatement of radiation from flexurally vibrating plates by means of active local vibration dampers. Soy. Phys. Acoust. 13 (1967) 115.Google Scholar
  91. Konovalyuk, I. P.: Diffraction of a plane sound wave by a plate reinforced with stiffness members. Sov. Phys. Acoust. 14 (1969) 465–469.Google Scholar
  92. Konovalyuk, I. P., Krasil’nikov, B. H.: Influence of a stiffness member on the reflection of a plane sound wave from a thin plate. LGI Collective Publications No. 4, 1965.Google Scholar
  93. kouzov, D. P.: Resonance effect in the diffraction of an underwater sound wave by a system of cracks in a plate. Prikl. Matem. Mekhan. 28 (1964) 409–417.MathSciNetGoogle Scholar
  94. Krasil’ntkov, V. N.: Refraction of Flecure Waves. Soy. Phys. Acoust. 8 (1962) 58; Scattering of flexure waves on an inhomogeneous elastic plate. Soy. Phys. Acoust. 8 (1962) 141.Google Scholar
  95. kudryavtseva, T. D.: Transmissivity density distribution function for a two-layer system with a random parameter. Soy. Phys. Acoust. 13 (1968) 389.Google Scholar
  96. kuetze, G., bolt, R. H.: On the interaction between plate bending waves and their radiation load. Acustica 9 (1959) 238.Google Scholar
  97. Kur’yanov, B. F.: Spatial correlation of fields emitted by random sources on a plane. J.A.S.A. 9 (1964) 360.Google Scholar
  98. lamb, G. L., jr.: Input impedance of a beam coupled to a plate. J.A.S.A. 33 (1961) 628.MathSciNetGoogle Scholar
  99. Laura, P. A., Saffell, B. F., jr.: Study of small-amplitude vibrations of clamped rectangular plates using polynomial approximations. J.A.S.A. 41 (1967) 836.Google Scholar
  100. Liamshev, L. M.: Diffraction of sound upon a thin bounded plate in liquid. Soy. Phys. Acoust. 1 (1955) 145.Google Scholar
  101. Lindh, G.: The transmission and reflection of an exponential shock wave impinging on a homogeneous elastic plate immersed in a liquid. Acustica 5 (1955) 257.Google Scholar
  102. Lyapünov, V. T.: Vibration isolation of articulated joints. Soy. Phys. Acoust. 13 (1967) 201.Google Scholar
  103. Lyon, R. H.: Noise reduction of rectangular enclosures with one flexible wall. J.A.S.A. 35 (1963) 1791.Google Scholar
  104. Lysanov, Y. P.: The edge effect in a large radiator. Sov. Phys. Acoust. 10 (1964) 165.Google Scholar
  105. Magrab, E. B., Reader, W. T.: Farfield radiation from an infinite elastic plate excited by a transient point loading. J.A.S.A 44 (1968) 1623.Google Scholar
  106. Maidanrk, G.: Response of ribbed panels to reverberant acoustic fields. J.A.S.A. 34 (1962) 809.Google Scholar
  107. Maidanik, G., Kerwin, E. M., JR.: Influence of fluid loading on the radiation from infinite plates below the critical frequency. J.A.S.A. 40 (1966) 1034.Google Scholar
  108. Mangiarotty, R. A.: Acoustic radiation damping of vibrating structures. J.A.S.A. 35 (1963) 369.Google Scholar
  109. Maslov, V. P.: Oblique incidence of a flexural wave in a plate on a slender obstacle. Sov. Phys. Acoust. 13 (1968) 344.Google Scholar
  110. Meitzler, A. H.: Mode coupling occurring in the propagation of elastic pulses in wires. J.A.S.A. 33 (1961) 435.Google Scholar
  111. Mowbray, D. F., Anderson, G. L.: Free vibrations of an infinite plate based on the linear coupled theory of thermoelasticity. J.A.S.A. 45 (1969) 646.Google Scholar
  112. Nariboli, G. A., Tsai, Y. M.: Asymptotic nature of extensional waves in an infinite elastic plate. J.A.S.A. 47 (1970) 857.Google Scholar
  113. Nayak, P. R.: Line admittance of infinite isotropic fluid loaded-plates. J.A.S.A. 47 (1970) 191–202.Google Scholar
  114. Nikiforov, A. S.: Radiation from damped plates. Soy. Phys. Acoust. 9 (1963) 197;Google Scholar
  115. Nikiforov, A. S.: Excitation of directional flexure waves in plates. Soy. Phys. Acoust. 9 (1964) 311;Google Scholar
  116. Nikiforov, A. S.: Impedance of an infinite plate with respect to a force acting in its plane. Sov. Phys. Acoust. 14 (1968) 247.Google Scholar
  117. Novlkov, A. K.: Spatial correlation of plane bending waves. Soy. Phys. Acoust. 7 (1962) 374.Google Scholar
  118. Onoe, M.: Contour vibrations of thin rectangular plates. J.A.S.A. 30 (1958) 1159;Google Scholar
  119. Onoe, M.: Gravest contour vibration of thin anisotropic circular plates. J.A.S.A. 30 (1958) 634.Google Scholar
  120. Ostergaard, P. B., Cardinell, R. L., Goodfriend, L. S.: Transmission loss of leaded building materials. J.A.S.A. 35 (1963) 837.Google Scholar
  121. Pachner, J.: Pressure distribution in the acoustical field excited by a vibrating plate. J.A.S.A. 21 (1949) 617.MathSciNetGoogle Scholar
  122. Pal’tov, V. A., Pdpyrev, V. A.: Vibration and sound radiation of a plate under random loading. Sov. Phys. Acoust. 13 (1967) 210.Google Scholar
  123. Petritskaya, I. G., Semyakin, F. V.: Correspondence between the theoretical and experimental values of the impedance of a thin layer of air. Sov. Phys. Acoust. 13 (1968) 396.Google Scholar
  124. Price, A. J., Crocker, M. J.: Sound transmission through double panels using statistical energy analysis. J.A.S.A. 47 (1970) 683.Google Scholar
  125. Pretlove, A. J.: Note on the virtual mass for a panel in an infinite baffle. J.A.S.A. 38 (1965) 266.Google Scholar
  126. Raske, T. F., Schlack, A. L., JR.: Dynamic response of plates due to moving loads. J.A.S.A. 42 (1967) 625.Google Scholar
  127. Rybak, S. A.: Sound transmission through a periodically inhomogeneous plate in a liquid. Soy. Phys. Acoust. 8 (1962) 83.Google Scholar
  128. Rybak, S. A., Tartakovskii, B. D.: On the vibrations of thin plates. Soy. Phys. Acoust. 9 (1963) 51.MathSciNetGoogle Scholar
  129. Scnocn, A.: Der Schalldurchgang durch Platten. Acustica, Beihefte 2 (1952) 1;Google Scholar
  130. Scnocn, A.: Seitliche Versetzung eines total reflektierten Strahls bei Ultraschallwellen. Acustica, Beihefte 2 (1952) 18.Google Scholar
  131. Shahady, P. A., PASSARELLI, R., LAURA, P. A.: Application of complex-variable theory to the determination of the fundamental frequency of vibrating plates. J.A.S.A. 42 (1967) 806.MATHGoogle Scholar
  132. Sheinman, L. E., Shenderov, E. L.: Transmission of a sound pulse through a thin plate at oblique incidence. Sov. Phys. Acoust. 15 (1970) 368–374.Google Scholar
  133. Shenderov, E. A.: Transmission of sound through a thin plate with interjacent supports. Sov. Phys. Acoust. 9 (1964) 289.The Wave Equation in Spherical Coordinates and Its Solutions 741Google Scholar
  134. Sirotyuk, M. G.: Transformation of longitudinal acoustic oscillations into shear or torsional oscillations. Sov. Phys. Acoust. 5 (1959) 259.Google Scholar
  135. Skudrzyk, E. J.: Sound radiation of a system with a finite or an infinite number of resonances. J.A.S.A. 30 (1958) 1152.Google Scholar
  136. Smith, P. W., Jr.:Minimum axial phase velocity shells. J.A.S.A. 30 (1958) 140.Google Scholar
  137. Thompson, W., JR., Rattayya, J. V.: Acoustic power radiated by an infinite plate excited by a concentrated moment. J.A.S.A. 36 (1964) 1488.Google Scholar
  138. Torvik, P. J.: Reflection of wave trains in semi-infinite plates. J.A.S.A. 41 (1967) 346.Google Scholar
  139. Tsakonas, S., Chen, C. Y., Jacobs, W. R.: Acoustic radiation of an infinite plate excited by the field of a ship propeller. J.A.S.A. 36 (1964) 1708.Google Scholar
  140. Twersky, V.: On the nonspecular reflection of sound from planes with absorbent bosses. J.A.S.A. 23 (1951) 336.MathSciNetGoogle Scholar
  141. Tyutekin, V. V.: Flexural oscillations of a circular elastic plate loaded at the center. Sov. Phys. Acoust. 6 (1961) 389;Google Scholar
  142. Tyutekin, V. V.: Reflection and refraction of flexure at the boundary of separation formed by two plates. J.A.S.A. 8 (1962) 180.Google Scholar
  143. Tyutekin, V. V., Shxvarnikoy, A. P.: Propagation of flexural waves in an inhomo geneous plate with smoothly varying parameters. Soy. Phys. Acoust. 10 (1965) 402.Google Scholar
  144. Usoskin, G. I.: Scattering of flexural waves by a system of point obstacles. Soy.Phys. Acoust. 13 (1967) 83.Google Scholar
  145. Young, J. E.: Transmission of sound through thin elastic plates. J.A.S.A. 26 (1954) 485.Google Scholar

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© Springer-Verlag/Wien 1971

Authors and Affiliations

  • Eugen Skudrzyk
    • 1
  1. 1.Ordnance Research Laboratory and Physics DepartmentThe Pennsylvania State UniversityUniversity ParkUSA

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