Properties of ultrasonic acoustic resonances for exploitation in comb construction by social hornets and honeybees Jonathan Kadmon * School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, IL-69978 Tel Aviv, Israel Jacob S. Ishay † Department of Physiology and Pharmacology, Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv 69978, Israel David J. Bergman School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, IL-69978 Tel Aviv, Israel and Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA Received 9 December 2008; published 9 June 2009 Physical and mathematical considerations are presented in support of the suggestion that social hornets and bees, which construct brood combs with large arrays of cells in a honeycomb structure, exploit ultrasonic acoustic resonances in those cells in order to achieve the great accuracy of the hexagonal symmetry exhibited by these honeycomb-structured arrays. We present a numerical calculation of those resonances for the case of a perfect-hexagon duct utilizing a Bloch-Floquet-type theorem. We calculate the rate of energy dissipation in those resonances and use that, along with other considerations, to identify the resonance that is best suited for the suggested use by bees and hornets. Previously recorded ultrasonic data on social hornets and honeybees are cited which agree with some of our predictions and thus provide support for the above-mentioned suggestion. DOI: 10.1103/PhysRevE.79.061909 PACS numbers: 87.10.-e, 51.40.+p, 43.80.+p, 43.20.+g I. INTRODUCTION In a previous paper, it was suggested that social hornets and bees might exploit a lateral ultrasonic acoustic resonance in the construction of their brood combs 1. By adjusting such a resonance in one comb cell to be twofold degenerate, they could ensure that the cell has an accurate perfect- hexagon cross section. Also, by adjusting two adjacent cells to have the same resonance frequency, they could ensure that those cells have cross sections of the same size. Quantitative estimates of the possible resonance frequencies in the case of an oriental hornet comb were obtained by taking the cell cross section to be a circle, instead of the actual perfect- hexagon shape. The expectation was that the resonance used by those hornets has a frequency of about 20 kHz. In this paper we present some results which provide more evidence in support of the above-mentioned suggestion: we describe results of computations on cells with the correct perfect-hexagon shape and calculate the energy dissipation from those results. From purely mathematical and physical considerations, we identify one lateral resonance which is best suited for the uses that were suggested in Ref. 1. We present some evidence that the same resonance mode is also used by honeybees in the construction of their brood combs. In that case, the actual resonance frequency will be higher, i.e., in the range of 35–40 kHz. That is because the honeybee comb cells are somewhat smaller than those of the oriental hornet. In Sec. II we briefly describe the main elements of our numerical computation and present its most important results where the air is modeled as an ideal fluid, the ducts are infinitely long, and the duct walls are perfectly rigid and smooth. In Sec. III we discuss a more realistic model where the air is a real fluid, with viscosity and heat conductivity, and the duct walls are rough, and present some calculations of the energy dissipation. In Sec. IV we discuss the conse- quences which follow from those results vis-à-vis the exploi- tation by hornets and bees of ultrasonic resonances in indi- vidual comb cells. In Sec. V we summarize our main results and discuss what needs to be done in order to test our pre- dictions. II. LATERAL ACOUSTIC RESONANCE IN A PERFECT- HEXAGON DUCT FILLED WITH IDEAL FLUID The lateral acoustic resonances in a cell with a constant perfect-hexagon cross section can be described by writing the following form for the space- and time-dependent pres- sure oscillations in the cell, the axis of which is taken to be the z-coordinate axis c is the velocity of sound in air: p mn , , t = e im-ic mn t  mn ,  , 2.1 where m = 0,  1,  2,3, n = 1,2, ... . Here we are using cylindrical coordinates  ,  , z around the cell axis. The appearance of the complex exponential func- tion e im is a consequence of the hexagonal symmetry in the xy plane: the invariance of the perfect hexagon under rota- tion by  / 3 rad around the z axis leads to a Bloch-Floquet- type theorem which forces the wave function  mn  ,  to be periodic in the azimuthal angle  with a period of  / 3 rad. We assume that the air in the cell behaves as an ideal fluid * jona7@post.tau.ac.il † Deceased. PHYSICAL REVIEW E 79, 061909 2009 1539-3755/2009/796/0619097 ©2009 The American Physical Society 061909-1