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                optical grating type guide electromagnetic fields of the leaky-wave variety as well. The lateral displacement of a light beam reflected from a leaky-wave structure when a Gaussian light beam is incident upon it was studied by Tamir and Bertoni.25 The Tamir and Bertoni25 theory pre- dicts that at a certain critical angle, a reflected beam shift may occur either in the forward or in the backward direc- tion with respect to the incident beam. The early experi- ments of Schoch26,27 using the acoustic analog of the Goos- Hänchen effect for an ultrasonic beam reflected from a liq- uid-solid interface showed a forward lateral displacement of the reflected ultrasound beam. Later, Breazeale and Torbett,1 using a Schlieren photographic technique, observed a backward beam shift of a 6 MHz ultrasonic beam of 10 mm width, reflected from a superimposed peri- odic grating, confirming the backward beam displacement predicted by the theory of Tamir and Bertoni.25 Although the backward shift was observed in acoustics, it was unclear which acoustic phenomenon caused it and above all there was no simulation method available at that time to model it. Much later Declercq et al. developed the necessary theoret- ical model based on a combination of the inhomogeneous wave theory and the plane wave expansion theory of dif- fraction to study the backward beam displacement.3,28 The most important conclusions from that work were, first that the backward beam displacement is caused by a backward propagating Scholte-Stoneley wave generated by the inci- dent beam. Such waves are known for their non-leaky fea- ture when propagating on smooth surfaces, therefore they are used for long-distance nondestructive testing and for acoustic communication through sound propagating on the seafloor. What had not been known was that these waves become slightly leaky when propagating on corrugated sur- faces. The developed theory described the leaky Scholte- Stoneley waves and enabled simulation of the backward dis- placement. Second, the theory also predicted that a bound- ed beam can still be displaced forward or backward at angles of incidence corresponding to leaky Rayleigh wave generation even though this had not been observed by Breazeale and Torbett.1 The clue was that the beam must be sufficiently narrow in order to obtain the effect. Indeed, later experiments have confirmed the theory.29 Backward beam displacement was considered an exotic phenomenon that only exists under very specific laboratory conditions. To verify these assumptions further research was performed and it turned out that practically every angle of incidence causes a backward displacement as long as the incident wave is an ultrasonic pulse. Herbison et al.30 showed through an angular frequency spectrum analysis that prac- tically every ultrasonic pulse contains frequencies that dis- place backward no matter what angle of incidence is used. It was also shown that the backward displacement does not just occur on the liquid side, as in the original experiments of Breazeale and Torbett,1 but also on the solid side.31 The experiments of Herbison et al.30 were not based on the Schlieren technique enabling visualization of sound, but were based on quantitative measurements using a specially designed ultrasonic polar scan system. The technique
allowed us to give the necessary evidence of the physical cause of the backward displacement, namely leaky Scholte- Stoneley waves. (see Fig. 4)
Influence of diffraction on the acoustic performance of theaters and auditoria
Periodic structures, currently very popular as phononic crystals, functioning as acoustic prisms and frequency selective mirrors hold promise for applications ranging from seismic wave deflection to accurate passive filters used in electronics. The Hellenistic theatre of Epidaurus, on the Peloponnese in Greece, which is well known for its extraordinary acoustic qual- ities, attracted Declercq’s attention in the framework of his investigations in this field. The theater, renown for its extraor- dinary acoustics, is one of the best conserved of its kind in the world. It was used for music and poetry contests and theatrical performances.
Many assumptions existed concerning the reasons why the acoustics of this theatre were so extraordinary, yet not a single assumption was satisfactory. Declercq and Dekeyser14 proposed that the acoustic quality of the theatre is due to the seat rows forming a corrugated structure. This research clearly showed and explained a filtering effect whereby the seat rows enhanced the acoustic quality of the theatre.14 This study demonstrated
  Fig.4. Angular spectrograms30 confirming the results obtained by Breazeale and Torbett.1 For a θ of 22.5o, a spectrogram from the region of the specularly reflected
beam (a) shows backward shifted frequencies in the range 5.98-6.12 MHz. The com- plementary spectrogram from a scan to detect the backward surface wave (b) shows it occurring at 6.05 MHz. Propagating bulk modes also detected in (b) and shown the- oretically by a dotted line. At other angles of incidence similar spectrum analysis shows that there is practically always some portion of sound backward displaced.
10 Acoustics Today, January 2013

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