Abstract
The frequency domain now becomes the lens for our study of plane waves. We will not rely on prior developments in the time domain, such as the d’Alembert solution and the ensuing wave image construction technique. The primary reason for pursuing the analysis in a self-contained manner is that it will prepare us for the study of multidimensional phenomena, for which a time-domain solution might not be available as a guide.
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Notes
- 1.
Hermann Ludwig Ferdinand von Helmholtz (1821–1894), who was German, made many fundamental contributions to the science of acoustics.
- 2.
The modes of a rigid/rigid waveguide occur alternately as even and odd functions relative to the midpoint. The former, which correspond to \(n=0,\) 2, 4, ..., are said to be symmetric modes. They have a maximum value at \(x=L/2.\) The modes for \(n=1,\) 3, 5, ... are antisymmetric modes. A continuous function that is odd with respect to \(x=L/2\) must be zero at that location. Thus, if \(R_{L}\) for this system was exactly one, the midpoint pressure would not have shown any peak at 212.5 and 637.5 Hz.
- 3.
H. Bodén and M. Åbom, “Influence of errors on the two-microphone method for measuring acoustic properties in ducts,” J. Acoust. Soc. Am. 79, pp. 541–549 (1986).
- 4.
A.D. Pierce, Acoustics, McGraw-Hill, Chap. 10 (1981). Reprinted by the Acoustical Society of America (1989).
- 5.
D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley $ Sons, Chap. 2 and Appendix C (2000).
- 6.
D.T. Blackstock, Physical Acoustics, Wiley (2000) p. 321.
- 7.
ANSI S1.26-1995, “American National Standard method for calculation of the absorption of sound by the atmosphere”.
- 8.
D.T. Blackstock, Physical Acoustics, pp. 513–516.
- 9.
D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley & Sons, pp. 322–325 (2000), provides a thorough discussion of boundary layer effects in an otherwise planar oscillating flow. Analysis of this problem may be traced back to G. Kirchhoff’s efforts in 1868.
- 10.
The junction conditions developed here are approximations based on assumptions of planar wave behavior and shortness of the transition distance s relative to the wavelengths \(2\pi c^{\left( n\right) }/\omega .\) An improved approximation for a frequency-domain analysis consistent with the planar approximation replaces pressure continuity at a junction with an impedance relation that asserts that the pressure difference is proportional to the average velocity, \(P_{1}\left( L_{1}\right) -P_{2}\left( 0,t\right) \) \(=Z_{\text {junction} }\left[ V_{1}\left( L_{1}\right) +V_{2}\left( 0\right) \right] /2.\) A good starting point for exploring this refinement is the reference by Karal (1953).
- 11.
Most references multiply the numerator and denominator of the fraction in Eq. (3.5.8) by two, so that only 2ka appears in the equation. Correspondingly, \(\chi \) is considered to be a function of 2ka. Apparently, this representation is done for historical reasons.
- 12.
M.I. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, (1965).
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Ginsberg, J.H. (2018). Plane Waves: Frequency-Domain Solutions. In: Acoustics-A Textbook for Engineers and Physicists. Springer, Cham. https://doi.org/10.1007/978-3-319-56844-7_3
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