Alexander Silbiger

Nonaxisymmetric Modes of Vibration of Thin Spherical Shells

Discussions of modes of vibration of thin spherical shells usually are restricted to axisymmetric vibrations. While nonaxisymmetric modes do exist, they are degenerate, that is, the frequencies are identical to the frequencies of the axisymmetric modes. This is shown to be a consequence of the spherical symmetry of the shell. The modes can be divided into three classes; for large mode numbers these classes correspond to extensional, shear, and flexural vibrations.

Scattering of Sound by an Elastic Prolate Spheroid

The scattering of a plane wave by an elastic prolate spheroid at arbitrary angle of incidence is formulated by means of an expansion in spheroidal wave functions. The structural response of the spheroid is expressed in terms of a surface impedance operator. Even for a homogeneous elastic solid or shell, the impedance operator generally couples the spheroidal modes of the acoustic medium, and hence numerical solutions involve the inversion of an infinite set of linear equations. However, a number of special cases are considered here for which direct solutions can be obtained. These include resonant mode scattering and low‐frequency scattering.When one of the structural modes is excited by the incident wave at its resonant frequency, it may radiate with such intensity as to mask radiation by other modes. The conditions under which this type of scattering can take place are shown to be very restrictive.When the acoustic wavelength is large and the frequency is small compared to the lowest natural frequency o...

Radiation from Circular Pistons of Elliptical Profile

While extensive information is available on flat pistons and on spherical radiators, no data have been given in the literature for the intermediate case of convex circular pistons of elliptical profile. In order to fill this gap, this paper presents information on the radiation loading and on the near and far field of such pistons. For the case of a free piston it is found that at small ka (where 2a is the diameter or major axis) the reactance decreases with increasing thickness (minor axis), while the resistance increases at small ka and decreases at large ka with increasing thickness. In the intermediate ka‐region, that is, when the wavelength in the fluid is of the order of magnitude of the dimensions of the piston, the dependence of the impedance on the thickness is more complicated. For the case of a piston in an infinite baffle the behavior of the reactance is similar to that for the free pistons. The resistance however is independent of the thickness in the limit of small ka and decreases with incr...

Focusing and Refraction of Harmonic Sound and Transient Pulses in Stratified Media

In media for which the speed of sound is position dependent, propagating sound will be refracted and, in some cases, focused. In the focusing regions, usually referred to as caustics or convergence zones, significant amplification of the pressure levels above those predicted by spherical spreading has been observed for continuous waves as well as for explosive pulses. In addition, the waveforms of explosive pulses undergo drastic distortion. In the present paper, an asymptotic theory of the refraction and focusing of sound originating from a point source in a stratified medium is presented. It is applicable to realistic velocity profiles and encompasses both transient pulses and harmonic waves. A comparison with Barashs and Goertners recent experiment involving explosive pulses indicates that the theory gives reliable estimates of the peak pressure levels at caustics, but reproduces only qualitatively the shape of the focused pulse. The discrepancy is attributed mainly to the neglect of finite‐amplitude effects in the theorys formulation. The inaccuracies inherent in the high‐frequency asymptotic methods employed in the theory are discussed in some detail.

Liber secundus diversarum modulationum: Singulis, binis, ternis, quaternisque vocibus (1627) by Girolamo Frescobaldi (review)

of Girolamo Frescobaldi’s Opere complete (complete works) has been eagerly awaited by musicians devoted to this early baroque master. It appears six years after the twelfth volume of the set, which started in 1975, and which now includes all of Frescobaldi’s works originally published during the composer’s lifetime or shortly thereafter (with a few small exceptions––see below). The eleventh volume presents what is probably the least known, and until now least accessible part of his compositional output: the concertato motets (or sacred concertos) for one to four voices and organ continuo, originally published in 1627 as a set of partbooks with the unwieldy title Liber secundus diversarum modulationum singulis, binis, ternis, quaternisque vocibus. What makes the appearance of this volume particularly welcome is that while thirteen of the thirty-one motets have been available in print for some years (Mottetti a 1. 2 e 3 voci con continuo, ed. Christopher Stembridge, Capolavori musicali dei secoli XVIIo e XVIIIo [Padua: G. Zanibon, 1987]), the remaining eighteen, which survive incomplete due to the loss of the cantus secundus partbook (also containing altus parts), appear here for the first time, in a reconstructed version by the editors. To add newly composed parts to these works was a bold undertaking––one that might even seem to contradict the essential nature of a critical scholarly edition, which attempts to present to us, to the extent possible, the original work, with a minimum of editorial intrusion. Of course that is an illusion, because any editorial work, even on small matters like accidentals, barlines, beaming, or continuo figures, involves interpretation. But it takes quite a few steps from there––giant steps––to add new parts to a contrapuntal texture. Before discussing how successful the editors have been as stand-ins for the composer, let’s take a closer look at the Liber secundus and its background, as presented in this edition. Presumably the volume was preceded by a Liber primus, but of such a publication no trace has been found. In fact, if it were not for the unique surviving copy of the Liber secundus in the British Library––even if incomplete––we would not even have known of Frescobaldi’s extensive engagement with the concertato motet. Only four other motets credited to his name have come down to us in anthologies published in Rome between 1616 and 1625. One of these, Iesu Rex admirabilis, included in the collection Sacri affetti (Rome: L. A. Soldi, 1625), does also appear with some variants in the 1627 Liber secundus (no. 24). If we assume that, similarly, the other three pieces, surviving in anthologies from 1616, 1618, and 1622, were subsequently incorporated MUSIC REVIEWS

Fantasy and craft: the solo instrumentalist

You would hardly believe, sir, the high regard which the Italians have for those who excel on instruments, and how much more importance they attach to instrumental music than to vocal, saying that one man can produce by himself more beautiful inventions than four voices together, and that it has charms and liberties that vocal music does not have. Andre Maugars, Response faite a un curieux sur le sentiment de la musique d’Italie (Rome, 1639) In Lorenzo Bianconi’s Music in the Seventeenth Century (1987), arguably the most original and provocative recent study of the period, solo instrumental music is treated so marginally that it is not even given a chapter of its own: ‘In “practical” and statistical terms, the role of seventeenth-century instrumental music is essentially modest and of minority significance – not at all what its relatively profuse cultivation on the part of modern “baroque” musicians would suggest’. It is true that the percentage of purely instrumental volumes among surviving seventeenth-century musical editions is relatively small. Instrumental performances also left much less of a paper-trail than operas, oratorios or major civic or ecclesiastical celebrations. For a historian eager to embed music into larger political, social and cultural frameworks, they tend to remain below the horizon, in part because their functions and meanings are harder to assess. Nevertheless, the wide popularity of Baroque instrumental music in modern times should be reason enough for giving it attention. In fact, interest in early instrumental repertories, particularly of solo music, is hardly a recent phenomenon. Pianists have been broadening their scope with anthologies of ‘early keyboard music’ since the later nineteenth century, if not before, and most young piano students have been exposed to the easier pieces of Byrd, Purcell and Pachelbel. Similarly, guitarists have been extending their repertory with adaptations of sixteenth- and seventeenth-century lute gems, while chorale settings and preludes from Sweelinck to Buxtehude have long been an important if not indispensable part of the church organist’s working capital.

Music and the Crisis of Seventeenth-Century Europe

It has been just over 200 years since the completion of Charles Burney’s monumental General History of Music, a pioneering work of musical historiography.l The final volumes cover the sixteenth century to “the present period,” that is, to 1789. It is interesting to look at Burney’s response to some of Galileo’s great and innovative musical contemporaries, who were no more remote from him temporally than Beethoven and Schubert are from us.

Phase Shift at Caustics and Turning Points

The phase shift observed in recent underwater pulse propagation tests is shown to originate at the caustic rather than at the turning point.

Focusing of Underwater Step‐Waves near a Caustic

The pressure field near a caustic (convergence zone) is analyzed for transient pulses with vanishing rise time. On the caustic, linear acoustic theory predicts a singularity of the order of (t−ta)−1/6 at the shockfront, ts being the arrival time of the front at the caustic. The corresponding total impulse remains finite. Factors limiting the peak pressure are discussed. Theoretical results are compared with data obtained by the Naval Ordnance Laboratory with explosive sources in a flooded quarry. [This work was supported by the U. S. Office of Naval Research.]

The Scattering of Sound by a Thin Prolate Spheroidal Shell

This paper deals with the scattering of a plane wave incident on a prolate spheroidal shell along its axis of revolution. The procedure can be easily generalized to an arbitrary direction of incidence. Contrary to the case of spherical and cylindrical shells, the spheroidal wave functions in terms of which the incident and scattered waves are expressed, do not coincide with the normal modes of the shell. As a result, when the coefficients of the expression of the shell motion in the spheroidal functions are used as generalized coordinates, all the Lagrangian equations will be coupled. Expressions suitable for computer calculations are given, and an example is worked out in which the frequency of the incident wave is close to one of the natural frequencies of the shell. Under these conditions the scattering action of the shell differs strikingly from that of a rigid spheroid. (This paper is based on work sponsored by the Office of Naval Research.)