Journal of Low Temperature Physics, Vol, 102, Nos. 3/'4. 1996 Third Sound in One and Two Dimensional Modulated Structures T. Komuro, H. Kawashima, K. Shirahama, and K. Kono ISSP, Univ. of Tohyo, Ropponqi 7-22-1, Tol~yo, 106 Japan An experimental technique is developed to study acoustic transmission in one and two-dimensional modulated structures by employing third sound of a supeTfluid helium filrlt. In particular, the Penrose lattice, which is a two- dimensional quasiperiodic structure, is studied. In two dimensions, the scat- tering of third sound is weaker than in one dimensions. Nevertheless, we find that the transmission spectT~tm in the Penrosc lattice, which is a two- dimensional prototype of the quasierystal, is observable if the helium film thickness is chosen aTvnnd 5 atomic layers. The transmission spectra in the Penrose lattice are e:r, plained in terms of dynamical theory of diffraction. PACS numbers: 67. 70. +n, ~3.,95. +d, 61.~+p 1. INTRODUCTION Classical wave propagation Oll one-dimensional non-periodic structures such as random, Fibonacci and Time-Morse lattices have been investigated by many authors experimentally and theoretically. ~-'~ We have studied this problem by means of the transmission of third sound. 3'4 Third somld is a surface wave in a superfluid helium fihn adsorbed on a solid surface by van der Waals force. Little dispersion and dissipatiou of third sound provide us aa~ advantage in the analysis of the spectrum. Furthermore, the scattering of third sound due to inhomogeneity of the surface and its characteristic size of the wave length (,-, 0.1 ram) ensure tractability for fabricating modulated structures. The Penrose lattice, which we discuss in this paper, is one of the two- dimensional quasicrystals. It has no trmlslational symmetry but has two- dimensional self-similarity characterized by golden ratio r = (v/5+ 1)/2, and This paper is taken from the Symposium on Quantum Fluids and Solids, Cornell University, Ithaca, June 1995, Part II. 359 0022-2291/96/0200-0359509.50/0 9 1996 Plenum Publishing Corporation