Page 9 - Jan2013

ON THE FASCINATING PHENOMENON OF DIFFRACTION BY PERIODIC STRUCTURES
Nico Felicien Declercq
Georgia Institute of Technology
Unité Mixte Internationale (UMI) Georgia Tech –
French Centre National de la Recherche Scientifique (CNRS) 2958 George W. Woodruff School of Mechanical Engineering, Georgia Tech Lorraine Metz, France 57070
Marcus Vitruvius Pollio (first century BC):
“so by the arrangement of theaters in accordance with the science of harmony, the ancients increased the power of the voice.”
Introduction
Although physically very complex,
under a relatively wide range of
conditions the diffraction of
sound by a corrugated structure can be
described and understood under the
principle of a plane wave expansion. A
plane wave expansion was first consid-
ered by Lord Rayleigh and was later
applied by the team of Oswald Leroy
while incorporating the mandatory
mechanical coupling conditions between
the two media separated by the corruga-
tion. In essence the plane wave expan-
sion (i.e., expansion into diffraction
orders) is based on a Fourier series expansion with incorpo- ration of propagation properties given by the dispersion rela- tion and based on the acoustic wave equation. The funda- mental connection between the periodic structure and the decomposition into different diffraction orders is the grating equation as also used in optics. Hence, it is not surprising that research teams in the 1980’s and 1990’s that worked in this area were also very experienced with Raman-Nath diffraction in acousto-optics. Besides curiosity in the area of physics, most researchers during the past century paid attention to dif- fraction of sound by corrugated structures because of the abil- ity to transform bulk acoustic waves into surface acoustic waves. During that time special attention was paid to diffrac- tion spectra and the appearance of anomalies. It was indeed found that certain anomalies (called Wood anomalies) coin- cided with the generation of Scholte-Stoneley surface acoustic waves. The latter are known to be difficult to generate, hence it was hoped corrugated surfaces would be more efficient. Scholte Stoneley waves propagate over large distances and are therefore of high interest for shallow water acoustic commu- nication and also for high speed nondestructive testing of large surfaces and structures. Encouraged by the physics of sound being akin to that of light, Mack A. Breazeale’s team1 developed experiments to verify the existence of acoustic counterparts of diffraction effects in optics. One such effect was the Goos Hänchen effect2 resulting in a backward dis- placed optical beam. The experiments lead to the observation of a similar effect in acoustics under the occurrence of dif- fraction, but could not be explained by the assumptions made at that time. It turned out later3 that the use of inhomogeneous waves in combination with the plane wave expansion tech- nique would be the key to explaining the effect and that the
effect was due to a new type of surface acoustic wave.
As scientific tools such as modern calculus and numerical methods that are applied today were not developed before the end of the 17th century, one must wonder if certain ancient phenomena could possibly be of even higher interest than modern day technology and expe- riences. It was pointed out by others4–11 that Chichen Itza was known for the existence of a transformed echo at the base of the El Castillo pyramid (Kukulkan pyramid) that sounded pretty much like the chirp of a Quetzal Coatl.
Indications based on prediction of Bragg scattering angles as a function of angle of incidence and frequency gave some insight but lacked actual correspondence with the experi- ments. As will be pointed out further Declercq et al.12 made quantitative simulations taking into account mechanical cou- pling and interference at the origin of the diffracted waves and determined reflected features very much in agreement with experiments. An additional discovery found that not only was the staircase itself responsible for the specific feature of the echo, but also the signature of the incident sound.
Declercq,12 while visiting Chichen Itza with fellow stu- dent Goffaux in 2003, discovered the raindrop acoustic effect in addition to the well-known chirp and this phenomenon was further studied experimentally and theoretically by Cruz and Declercq.13
Marcus Vitruvius Pollio, on the other hand, noted in the first century BC that the Greeks built their theatres following nature’s footsteps: “they traced the voice as it rose, and car- ried out the ascent of the theater seats. By the rules of math- ematics and the method of music, they sought to make the voices from the stage rise more clearly and sweetly to the spectators’ ears. For just as organs which have bronze plates or horn sounding boards are brought to the clear sound of string instruments, so by the arrangement of theaters in accordance with the science of harmony, the ancients increased the power of the voice.” New research revealed some interesting features as is described below.14
It turned out that diffraction of sound with incorpora- tion of mechanical coupling and acoustic interference enables the understanding of both phenomena.
A recent re-emergence of interest in the field of diffrac- tion of sound is connected with the development of phonon-
8 Acoustics Today, January 2013