Allan D. Pierce

Acoustical Society of America

Vern Oliver Knudsen (1893-1974), a founding member and third president of the Acoustical Society of America, had a long and distinguished career as a teacher, acoustician, physicist, academic administrator, and consultant. His career in acoustics began at the Bell Telephone Laboratories, and continued at UCLA, where he was head of the Physics Department, vice-chancellor, and chancellor. Knudsen was born in Provo, Utah, on 27 December 1893. He entered Brigham Young University in 1911 and came under the influence of Professor Harvey Fletcher, whom he credits with steering him toward physics rather than mathematics or engineering as he originally planned. In his senior year he assisted Fletcher in his research on Brownian motion. Following his graduation in 1911, he served as a Mormon missionary and as acting head of the Northern States Mission in Chicago. During World War I he investigated parasitic earth currents using the western segment of the St. Pierre cable, which had broken in mid-Atlantic. The 1700-mile segment terminated at the Chatham, Massachusetts cable station. The purpose of the measurements was to determine what could be done to speed up cable telegraph transmission of messages across the Atlantic during the war to “carry the wording communications between Woodrow Wilson and Lloyd George.” He found that interfering earth currents with frequencies between about 5 to 15 Hz were largely attributable to fluctuations in the Earth’s magnetic field. Knudsen joined Harvey Fletcher at the Bell Telephone Laboratories (then Western Electric) in 1918, where he worked on the development of amplifiers and oscillators, increasing his knowledge of the new technology of vacuum tubes which Fletcher and his colleagues were using in their studies on hearing. In 1919 he began his graduate studies at the University of Chicago as a student of A. A. Michelson. As a suitable subject for his doctoral dissertation, R. A. Millikan proposed a study of the contribution of electrons to the specific heat of metals, a problem previously investigated by Debye, Nernst, and Einstein, but not solved. Recognizing the probable long duration of such a study, Knudsen sought advice from Dean Henry Gordon Gale during a period when Millikan was in Europe. Dean Gale proposed a study of acoustics, which would be well advanced by the time Millikan returned. Knudsen worked rapidly to measure the sensibility of the ear to small differences of intensity and frequency, using vacuum tube techniques acquired from the Western Electric Research Laboratories. When Millikan returned, he approved the study and introduced Knudsen to George Shambaugh, one of the foremost otologists in the U. S. Subsequent association with Shambaugh resulted in studies on the sensibility of pathological ears to small differences in loudness and pitch and to diplacusis. Knudsen received the Ph.D. in physics magna cum laude in 1922. He confounded both colleagues and teachers by turning down offers from the University of Chicago and the Bell Telephone Laboratories to accept the position of Instructor at the newly-formed University of California Southern Branch, later to become UCLA. The Southern Branch occupied a small campus near what is now central Los Angeles; all the buildings together were smaller than Knudsen Hall on the UCLA campus today. Knudsen overcame the lack of facilities by reaching out to his surroundings: he studied the architectural acoustics of local auditoriums and classrooms, most of which were acoustically bad. Volume 14, Number 1 Winter 2004

Extension of the Method of Normal Modes to Sound Propagation in an Almost‐Stratified Medium

A theory is presented that permits the extension of the method of normal modes to guided‐wave propagation in a medium with properties varying slowly with horizontal coordinates in addition to varying with the vertical coordinate. The principal assumption is the neglect of coupling between normal modes. It is predicted that different frequencies in different modes follow different horizontal paths. A ray‐tracing method is described for computing these paths. The theory is then applied to the study of the dispersion of waves from an explosive source in shallow water with slowly varying depth.

Diffraction of sound around corners and over wide barriers

Formulas and procedures are described for the estimation of sound pressure amplitudes at locations partially shielded from the source by a barrier. The analytical development is based on the idealized models of a wave from a point or extended source incident on a rigid wedge or a three‐sided semi‐infinite barrier. Versions of the uniform asymptotic solution for the wedge problem which are convenient for numerical predictions are derived in terms of auxiliary Fresnel functions by means of complex variable techniques previously employed by Pauli from a generalization of the exact integral solution developed by Sommerfeld, MacDonald, Bromwich, and others and are interpreted within the spirit of Kellers geometrical theory of diffraction. The Kirchhoff approximation in terms of the Fresnel number is obtained in the limit of small angular deflections from shadow zone boundaries. An approximate and relatively simple expression for the double‐edge diffraction by a thick three‐sided barrier is given based on the ...

Wave equation for sound in fluids with unsteady inhomogeneous flow

An approximate wave equation is derived for sound propagation in an inhomogeneous fluid with ambient properties and flow that vary both with position and time. The derivation assumes that the characteristic length scale and characteristic time scale for the ambient medium are larger than the corresponding scales for the acoustic disturbance. For such a circumstance, it is argued that accumulative effects of inhomogeneities and the ambient unsteadiness are satisfactorily taken into account by a wave equation that is correct to first order in the derivatives of the ambient quantities. A derivation that consistently neglects second‐ and higher‐order terms leads to a concise wave equation similar to the familiar ordinary wave equation of acoustics. The wide applicability of this equation is established by showing that it reduces to previously known wave equations for special cases and by showing, with the eikonal approximation, that it yields the geometrical acoustics equations for ray propagation in moving i...

Fundamental Structural-Acoustic Idealizations for Structures with Fuzzy Internals

Fundamental issues relative to structural vibration and to scattering of sound from structures with imprecisely known internals are explored, with the master structure taken as a rectangular plate in a rigid baffle, which faces an unbounded fluid medium on the external side. On the internal side is a fuzzy structure, consisting of a random array of point-attached spring-mass systems. The theory predicts that the fuzzy internal structure can be approximated by a statistical average in which the only relevant property is a function m F (Ω) which gives a smoothed-out total mass, per unit plate area, of all those attached oscillators which have their natural frequencies less than a given value Ω. The theory also predicts that the exact value of the damping in the fuzzy structure is of little importance, because the structure, even in the limit of zero damping, actually absorbs energy with an apparent frequency-dependent damping constant proportional to dm F (ω)/dω incorporated into the dynamical description of the master structure. A small finite value of damping within the internals will cause little appreciable change to this limiting value.

Sound diffraction around screens and wedges for arbitrary point source locations

A development is presented for the Green’s function for a point source in the vicinity of a rigid wedge. The diffraction contributions to the Green’s function for arbitrary source and listener location is expressed in a form which can be readily evaluated using the Laguerre technique for numerical integration. The present approach offers the advantages of efficient numerical evaluation and of relatively straightforward reduction to well‐known analytical approximations in limiting cases. Comparisons with previously obtained experimental and numerical results obtained by Ambaud and Bergassoli [Acustica 27, 291–298 (1972)] are presented. The comparison with the experimental results is excellent; the advantages of the present numerical technique, vis a vis that of Ambaud and Bergassoli, are pointed out.

Propagation of Acoustic‐Gravity Waves from a Small Source above the Ground in an Isothermal Atmosphere

A theoretical discussion is presented of the influence of gravity on sound propagation from a small source in an isothermal atmosphere where ambient pressure and density decrease exponentially with height. A solution for the free‐space case is derived that indicates that waves with angular frequency ω between (γ − 1)12g/c and (γ/2)g/c will not be propagated, while those with ω between 0 and (γ − 1)12(g/c) cosθ will not be propagated in a direction making an angle of θ with the vertical axis. A formal solution incorporating appropriate boundary conditions at the ground is derived and discussed. The field along the vertical line passing through the source is found explicitly. A consideration of the energy intensity shows that no energy is propagated within a cone above and below the source if ω < (γ − 1)12g/c, A calculation of the intensity for the case when (γ − 1)12g/c < ω < (γ/2)g/c indicates that the energy flowing from the source tends to concentrate in the lowest layers of the atmosphere. The field fo...

Effects of atmospheric irregularities on sonic-boom propagation.

A review is given of information obtained in recent years concerning the effects on sonic‐boom signatures of departures of the atmosphere from a perfectly stratified time invariant model. These effects include the observed random variations in boom overpressures from those expected for a stratified atmosphere, the anomalously large and variable rise times, and the occurence of spiked or rounded waveforms rather than the characteristic N waves. The extent of the variability in data recorded during actual flight tests is summarized in the form of histograms, representing experimentally obtained probability density functions. The physical mechanisms believed to be responsible for the variations and the anomalous features in the signatures are described. These include refraction and subsequent wavefront rippling by turbulence, the possible focusing or defocusing of rays, the formation of caustics, and the phenomenon of wavefront folding, diffraction, and scattering. Recent statistical theories of shock propag...

Propagation of Acoustic‐Gravity Waves in a Temperature‐ and Wind‐Stratified Atmosphere

A theory is presented that permits the study of the effects of horizontal winds on the dispersion and amplitudes of acoustic‐gravity waves in the atmosphere. It is shown that the effective horizontal group velocity for a given frequency in a given normal mode depends on direction of propagation as well as on frequency and that it is not necessarily in the same direction as the horizontal‐wavenumber vector. A number of useful integral theorems are derived from a variational principle and one is subsequently applied to the development of a perturbation method for the computation of wind effects on dispersion. Application of the method to a realistic example indicates that winds can appreciably alter the dispersion of the normal modes and that they should be considered in any quantitative interpretation of experimental microbarograms.

Guided mode disappearance during upslope propagation in variable depth shallow water overlying a fluid bottom

The adiabatic mode theory for upslope propagation in shallow water breaks down when the depth H(x) decreases to a critical value Hc at which the mode’s phase velocity equals sound speed c2 in the underlying fluid bottom (density ρ2). Matched asymptotic expansion techniques yield the acoustic pressure in the transition region (of lateral extent c2/ωe2) in terms of Airy functions. The expansion parameter e whose smallness justifies the required approximation is the cube root of ( ρ2/ρ1)[(c2/c1)2−1]‖H′‖. During the early approach to the critical range (taken as x = 0), the amplitude decreases within the water layer as (−x)1/2; beyond x = 0, asymptotically as x−5/2. A cylindrically spreading (as r−1/2) wave of narrow beamwidth 0.94 e originates from the transition region (near x = 0) and propagates at angle e with the interface obliquely downward into the bottom fluid. The derived characteristic critical‐depth transition function is discussed in detail. The results are in accord with computations reported by ...