The Interaction of Sound and Objects: Scattering and Diffraction

Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J Lecture 4:The Interaction of Sound and Objects: Scattering and Diffraction A. Definitions: P(r, θ) 1. Scattering Scattering refers to the alteration in sound path produced by the interaction of a sound ‘ray’ and an object. Scattering implies that the incident sound wave front is broken into multiple waves by the scatterer. The smearing of propagation directions introduced by interaction with a rough surface is also an example of scattering (AD Pierce “Acoustics: An introduction to its physical principles and applications” Acoustical Society of America,1981). a r Incident Scattering of a plane wave by a rigid immovable object. (From Pierce 1981) Courtesy of Acoustical Society of America. Used with permission. 2. Diffraction Diffraction is another result of interaction of sound waves and objects. Practically, diffraction is the name used to explain the results of such interactions that can not be explained by ray theory. The basic tool for analyzing diffraction is Hyugen’s principle: any wave front can be approximated by a series of in-phase and equal strength simple sources placed along the front. B. Examples of Scattering and Diffraction Image removed due to copyright considerations. Source: Olsen, H. F. Music Physics and Engineering. Dover Press, 1967. Figure 4.1: Scattering and Diffraction by a large slit and a small slit (from Olsen, Music Physics and Engineering). Which of these examples is better fit by ray theory and which is better explained by diffraction? page 1 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J Image removed due to copyright considerations. Source: Olsen, H. F. Music Physics and Engineering. Dover Press, 1967. Figure 4.2 Scattering and Diffraction by a large and small object (From Olsen). Which of these situations is better explained by ray theory and which by wave theory and diffraction? Image removed due to copyright considerations. Source: Olsen, H. F. Music Physics and Engineering. Dover Press, 1967 . Figure 4.3 (a) Scattering from an object positioned at an angle relative to the direction of propagation of the incident wave. The angle of reflection is equal to the angle of incidence. (b) Scattering and transmission by an imperfectly rigid barrier. The energy transmitted equals the incident energy minus the reflected energy. (From HF Olsen “Music, Physics and Engineering” Dover Press 1967) page 2 Lecture 4 Acoustics of Speech and Hearing C. Model’s of Diffraction. 6.551J/HST714J Hyugen’s principle (from Halliday & Resnick 1970) “All points on a wavefront can be considered as point sources for the production of spherical secondary wavelets. After a time, the new position of the wavefront will be the surface of tangency of these secondary wavelets.” This principle allows us to model diffraction by a combination of simple sources. The tools used to analyze diffracted light at a distance far from a slit (called Fruanhofer diffraction) are essentially the tools we used to determine the sound pressure far from an array of sources. To distant screen θ From distant source B A Model for Fraunhofer Diffraction by a slit Figure 4.4 Adapted from Halliday & Resnick. Fundamentals of Physics. Wiley, 1970. D. Diffraction and Scattering of plane waves by a rigid Sphere. While broad-band descriptions of diffraction and scattering get complicated, as you add more and more simple sources, one “simple” system for which there is an analytic solution is the diffraction and scattering of plane waves by a rigid sphere. This solution based on a simple geometric view of spherical symmetry ( only a and θ are significant) is mathematically complicated (Rayleigh, 1986; Morse & Ingard 1968). Figure 4.5A A geometry to describe the sound pressure at a point located on the surface of a spherical artificial head where the angle between the source and the midline of the sphere perpendicular to the measurement point is controlled. page 3 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J Figure 4.5B An alternative geometry to describe the sound pressure at a point located on the surface of a sphere where the angle between the source and the midline of the sphere that contains the measurement point is controlled. page 4 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J Figure 4.6 Mathematical predictions of sound diffraction around a sphere (from Shaw 1974) Image removed due to copyright considerations. Source: Shaw, E. A . G . " The E xternal E ar." In Handbook of Sensory Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff. New York: Springer-Verlag, 1974, pp. 455-490. Image removed due to copyright considerations. Source: Shaw, E. A . G . " The E xternal E ar. " In Handbook of Sensory Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff. New York: Springer-Verlag, 1974, pp. 455-490. A) The sound pressure measured at the surface of a sphere, relative to the free-field pressure in the stimulating plane wave, as a function of ka (the x-scale at the top) or frequency assuming a sphere of the size of the human head (the x-scale at the bottom). The parameter is θ, the angle between the sound source and the mid-line of the sphere. The sound measurement point is located on the surface of the head at θ=90°. The angular position of the source is the parameter. When the source is pointed directly at the measurements point on the sphere’s surface, θ=90° and the sound pressure is a maximum. When the source is directed at a point directly on the opposite side of the sphere, θ=-90° and the sound pressure is either equal to or a little larger than the pressure of the plane-wave stimulus. When the source is on the other side of the head but not at -90°, the measured sound either equals or is less than the pressure in the incident wave. B) The mathematical predictions illustrated in figure Abut now shown on a directional plot with frequency as a parameter. The radial distance between the center of the plot and the plotted functions codes the level of the measured sound at a point on the surface of the sphere at θ=0°, the direction of the source is varied from -180°< θ < 180°. Directionality functions are shown for 1 & 6 kHz. (From Shaw 1974). These theoretical diffraction patterns are compared to sound pressure measurements on the surface of a sphere on the next page. Again, with a human sized head, ka=1 corresponds to a frequency of about 0.6 kHz. page 5 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J E. Measurements of Sound Pressure on the surface of a sphere P(a,θ): Wiener FM (1947). J. Acoust. Soc. Am. 19: 444-451. Courtesy of Acoustical Society of America. Used with permission. page 6 B Helix (pf) Fossa of Helix Antihelix (pf) Cymba (concha) A Cavum (concha) Antitragus Crus helias A' Tragus L Lobule (pf) Mid-concha position Ear canal entrance position Ear canal insert position Eardrum position Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J 1. Contribution of the external-ear components to the sensitivity of the ear The effect of different external-ear structures on the gain between the sound pressure at the eardrum and the free-field sound pressure for a sound source positioned on the horizontal plane at 45° relative to the midline (the azimuth=45°; elevation = 0°). (After Shaw 1974). Image removed due to copyright considerations. Source: Shaw, E. A. G. "The External Ear." In Handbook of Sensory Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff. New York: Springer-Verlag, 1974, pp. 455-490. page 8 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J 2. The External-Ear and the Directionality of the Ear: Head-Related Transfer Functions The magnitude of the eardrum to free-field sound pressure ratio for the human head for sound sources located on the horizontal plane (Shaw 1974). Image removed due to copyright considerations. Source: Shaw, E. A. G. "The External Ear." In Handbook of Sensory Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff. New York: Springer-Verlag, 1974, pp. 455-490. page 9 Lecture 4 Acoustics of Speech and Hearing Normalized HRTF vs Azimuth sound frequency is the parameter 6.551J/HST714J Interaural Level Differences vs Azimuth sound frequency is the parameter Image removed due to copyright considerations. Source: Shaw, E. A . G . " The External Ear. " In Handbook of Sensory Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff. New York: Springer-Verlag, 1974, pp. 455-490. Measurements and theoretical values for azimuthal dependence (HRTF normalized by HRTF(0)left ) and interaural intensity differences (right) of the magnitude of sound pressure at the ear drum normalized by the stimulus pressure in the plane wave. (SHAW 1974). page 10 Lecture 4 Acoustics of Speech and Hearing Normalized HRTF vs Azimuth sound frequency is the parameter 6.551J/HST714J Interaural Level Differences vs Azimuth sound frequency is the parameter Image removed due to copyright considerations. Source: Shaw, E. A. G. "The External Ear." In Handbook of Sensory Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff. New York: Springer-Verlag, 1974, pp. 455-490. Measurements and theoretical values for azimuthal dependence (HRTF normalized by HRTF(0)left ) and interaural intensity differences (right) of the magnitude of sound pressure at the ear drum normalized by the stimulus pressure in the plane wave. (SHAW 1974) page 11 Lecture 4 Acoustics of Speech and Hearing Normalized HRTF vs Azimuth sound frequency is the parameter 6.551J/HST714J Interaural Level Differences vs Azimuth sound frequency is the parameter Image removed due to copyright considerations. Source: Shaw, E. A. G. "The External Ear." In Handbook of Sensory Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff. New York: Springer-Verlag, 1974, pp. 455-490. Measurements and theoretical values for azimuthal dependence (HRTF normalized by HRTF(0)left ) and interaural intensity differences (right) of the magnitude of sound pressure at the ear drum normalized by the stimulus pressure in the plane wave. (SHAW 1974) page 12 Lecture 4 Acoustics of Speech and Hearing G. Interaural time and phase differences 6.551J/HST714J A simple model of the interaural time differences is presented below: Incident plane waves Left 4.13 The Geometric model of Interaural Time Difference (ITD) produced by a source in the horizontal plane for a spherical head of radius a: ITD = θ θ A A' θ a a sinθ Right a (θ + sin θ). c Note that ITD in this model is independent of Frequency.. In the sinusoidal steady state, the interaural phase difference is: aθ IPD = ω ITD = 2π ITD period Adapted from Shaw, E. A. G. "The External Ear." In Handbook of Sensory Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff. New York: Springer-Verlag, 1974, pp. 455-490. ITD (ms) Estimates of ITD based on scattering and diffraction theory match the simple spherical model predictions at ka > 2 but underestimates ITD (and IPD) at frequencies where k< 0.3 . Angle of Incidence Courtesy of Acoustical Society of America. Used with permission. Figure 4.14 The symbols are measurements made of the interaural time difference at different angles of incidence at high (o,x) and low frequencies (the square, dot and triangle). The line labeled ‘for ka>>1’ is the prediction of the simple spherical model for all frequencies, and scattering and diffraction theory for ka >> 1. The line labeled for ka <<1 is based on scattering and diffraction around a rigid sphere for that condition. (From Kuhn GF (1977). Model for the interaural time differences in the azimuthal plane. J. Acoust. Soc. Am. 62: 157167). page 13 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J The root of the simple model’s underestimation of interaural time is that, ITD varies with frequency, being longer at low frequencies than at higher. This frequency-dependent interaural time is consistent with models of diffraction about a rigid sphere. From Kuhn 1977. Courtesy of Acoustical Society of America. Used with permission. While the measured and calculated ITDs at frequencies above 1000 Hz in man ka > 2 are consistent with the spherical model of Figure 4.13: the Normalized ITD,: Π= ITD , (a /c)sin θ approximate is 2 in the spherical model and in the measurements in that high-frequency range. However. the measurements show it equals about 3 for ka < 1 (f < 600 Hz in man). This 50% increase in interaural time at low frequency is consistent with an effective increase in head dimensions by 50%. From Kuhn 1977. Courtesy of Acoustical Society of America. Used with permission. page 14 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J H. The ‘Duplex Theory’ for the localization of tones in the horizontal plane. The differences in interaural intensity produced by low frequency sounds ka <0.3 are too small to be useful in judging the location of sound in the horizontal plane. Instead, it is clear that some animals, including man, utilize interaural time (or phase) differences to locate sources of lowfrequency tones in the horizontal plane. (Mills 1972). 1. Measurements of Minimum Audible Angle (Mills 1958) Minimum Audible Angle is the smallest angular deviation from a base azimuthal position θ that is detectable by human listeners. The stimuli used in its quantification are long duration tone bursts, and you can consider the system in the sinusoidal steady state. Localization performance is best (MAA is smallest) for sources on the midline. (From Mills AW (1958). On the minimum audible angle. J. Acoust. Soc. Am. 30: 237246.) Courtesy of Acoustical Society of America. Used with permission. page 15 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J Measurements of MAA From Mills AW (1958). On the minimum audible angle. J. Acoust. Soc. Am. 30: 237-246.) Courtesy of Acoustical Society of America. Used with permission. Low frequency (< 1000 Hz) MAA are consistent with the ability of the auditory system to detect interaural phase, while above 1000 Hz, the MAA are consistent with the ability of the listener to detect interaural intensity differences. (A jnd is a just notable difference.) page 16 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J From Mills AW (1958). On the minimum audible angle. J. Acoust. Soc. Am. 30: 237-246.) Courtesy of Acoustical Society of America. Used with permission. EXTERNAL-EAR PRESSUE GAIN (dB) I. The Effect of Sound Elevation on the Sound Pressure Input to the Auditory System. 20 10 0 -10 -20 0° Azimuth Varied Elevation CAT 2 4 18° -18° 90° 0° 36° 6 8 10 20 FREQUENCY (kHz) Courtesy of Acoustical Society of America. Used with permission. While the difference between elevation and azimuth is irrelevant when discussing the symmetric sphere, asymmetries in the placement of the external ear and in the structures of the external ear, generate differences in the frequency dependence of the ear with sounds of varied elevation. These differences take the form of “interferencenotches” produced by interactions of small wave-length sounds with the pinna flange and the different structures within the concha. On the left are measurements of external-ear gain made at varied elevations in the cat from Musicant et al. 1990. page 17 Lecture 4 Acoustics of Speech and Hearing 6.551J/HST714J References Kuhn, GF, Model for the interaural time differences in the azimuthal plane. J. Acoust. Soc. Am. 62 (1977) pp. 157-167. Mills, AW, On the minimum audible angle. J. Acoust. Soc. Am. 30 (1958) pp. 237-246. Mills, AW, Auditory localization. In J.V. TOBIAS (ed.) Foundations of Auditory Theory (Academic Press, New York, 1972) pp. 301-348. Musicant AD, Chan JCK and Hind JE (1990). Direction-dependent spectral properties of cat external ear: New Data and cross-species comparisons. J. Acoust. Soc. Am. 87: 757-781. Shaw, EAG, Transformation of sound pressure level from the free field to the eardrum in the horizontal plane. J. Acoust. Soc. Am. 56 (1974) pp. 1848-1860. Wiener, FM, Sound diffraction by rigid spheres and circular cylinders. J. Acoust. Soc. Am. 19 (1947) pp. 444-451. page 18