• Natalia Gorodetska Institute of Hydromechanics of the NAS of Ukraine
  • Inna Starovoit Institute of Hydromechanics of the NAS of Ukraine
  • Tetiana Sobol Institute of Hydromechanics of the NAS of Ukraine
  • Tetiana Shcherbak Institute of Hydromechanics of the NAS of Ukraine



elastic waves, interface transparency, energy analysis


The article is devoted to the analysis of the scattered field at the boundary of a stepped waveguide formed by the rigid contact of two half-layers with the same mechanical characteristics, but with different widths. The wave field is excited by the first normal wave propagating from infinity in the narrower half-layer. Mathematical difficulties of the posed boundary problem are due to the presence of a local singularity in the stresses at the point of change of the boundary conditions at the boundary of the two hemispheres. The solution is built by the superposition method, which allows taking into account the local singularity due to the asymptotic features of the unknowns. The quality criterion of the obtained solution was the control of the accuracy of the fulfillment of the conjugation conditions at the boundary of the two half-layers. The main attention in the work is focused on the established conditions for changing the transpar[1]ency of the boundary depending on the frequencies, the symmetry of the oscillations, and the ratio of the half-layer widths. It was shown in the work that for both symmetric and antisymmetric oscillations of a stepped waveguide, there are frequency ranges in which the transparency of the boundary changes significantly. For both types of symmetry, in the frequency range up to the critical frequency for the third propagating normal wave, there are two frequency ranges in which the transparency of the boundary increases rather sharply. The frequencies at which local energy maxima are observed in the reflected field are different for symmetric and anti-symmetric oscillations. For symmetric oscillations, the first energy maximum in the reflected field is observed at the frequency when only one wave can propagate in both half-layers. This effect is due to the increase in the role of inhomogeneous waves in the transmitted field. The second energy maximum in the reflected field is due to the transformation of the energy of the incident wave into propagating waves of higher orders. In the case of antisymmetric oscillations, both maxima are due to the energy features of propa[1]gating waves of higher orders. The quality of energy resonance in the reflected field was also significantly dependent on the symmetry of the oscillations. The established features of the scattered field make it possible to develop recommenda[1]tions for controlling the transparency of the boundary in a stepped waveguide

Author Biographies

Natalia Gorodetska, Institute of Hydromechanics of the NAS of Ukraine

Doctor of Physical and Mathematical Sciences, Professor

Inna Starovoit, Institute of Hydromechanics of the NAS of Ukraine

PhD of Physical and Mathematical sciences

Tetiana Sobol, Institute of Hydromechanics of the NAS of Ukraine

PhD of Physical and Mathematical Sciences

Tetiana Shcherbak, Institute of Hydromechanics of the NAS of Ukraine

PhD of Physical and Mathematical sciences


Miller G. K. Axisymmetric stress‐wave propaga tion across the common end face between two semi‐infinite Cylinders, solid to fluid, The Journal of the Acoustical Society of America, 1968, Vol. 44, №4. Р. 1040-1051. DOI: 10.1121/1.1911194

Городецкая Н. С., Недилько Е. А. Влияние механических характеристик контактирующих сред на отражающие свойства границы в составном упругом волноводе. Акустичний віс ник, 2012. Том 15, № 4. С. 14–24.

Tamine M. Scattering and transmission of elastic waves from an interface between two planar waveguides. Surface Review and Letters, 2003, Vol. 10, № 5, P. 727-736. DOI: 10.1142/ S0218625X0300558X

Pagneux V., Maurel A. Lamb wave propagation in elastic waveguides with variable thickness. Pub lished By: Royal Society, 2006, Vol. 462, P. 1315- 1339. DOI:

Городецкая Н. С., Недилько Е. А. Распростране ние антисимметричных волн в ступенчатом упру гом волноводе. Акустичний вісник, 2013–2014. Том 16, № 1. С. 16–27.

Benmeddour F., Grondel S., Assaad J., Moulin E. Study of the fundamental Lamb modes interaction with symmetrical notches. NDT&E International, 2008, Vol. 41, P. 1–9. DOI: j.ndteint.2007.07.001

Cegla F.B., Rohde A., Veidt M. Analytical predic tion and experimental measurement for mode con version and scattering of plate waves at non symmetric circular blind holes in isotropic plates. Wave Motion, January 2008, Vol. 45, Iss. 3, P. 162–177. DOI: j.wavemoti. 2007.05.005

Городецька Н. С., Неділько О. О. Трансформація енергії згинної хвилі на сходинці при різних механічних параметрах контактуючих середовищ. Наукоємні технології. Фізика, 2015, Том 25, № 1, С. 52–56. DOI: 5461.25.8229

Koji Hasegawa, Masanori Koshiba, Michio Su zuki. Analysis of finite periodic waveguides for elastic waves using finite-element method. Elec tronics and Communications in Japan, Part 2, 1987, Vol. 70, Iss. 6, P. 27-36. DOI: –

Грінченко В.Т., Городецька Н.С. Метод суперпо зиції стосовно граничних задач для неоднорідних хвилеводів. Математичні методи та фізико механічні поля, 2006, Том 49, № 1, С. 20–30.

Cebrecos A.; Picó R.; Sánchez-Morcillo V. J.; Staliunas K.; Romero-García V.; Garcia-Raffi L. M. Enhancement of sound by soft reflections in exponentially chirped crystals. AIP Advances, December 2014, Vol. 4, Iss. 12. DOI: 10.1063/1.4902508

Hussein M. I., Leamy M. J., and Ruzzene M. Dynamics of phononic materials and structures: historical origins, recent progress, and future out look. Applied Mechanics Reviews, July 2014, 66(4): 040802 (38 pages). DOI:10.1115/ 1.4026911

Khanikaev A. B., Mousavi S. H., Tse W.-K., Kargarian M., MacDonald A. H., Shvets G. Photonic topological insulators. Nature Materials, 2012, Vol. 12, P. 233–239.

Yang Z., Gao F., Shi X., Lin X., Gao Z., Chong Y., Zhang B. Topological acoustics. Physical Re view Letters, 20 March 2015, Vol. 114, Iss. 11. DOI:

De Ponti J. M., Iorio L., Riva E., Ardito R., Braghin F., Corigliano A. Selective mode conver sion and rainbow trapping via graded elastic waveguides. Physical Review Applied, 15 Sep tember 2021, Vol. 16, Iss. 3. DOI: 10.1103/PhysRevApplied.16.034028

De Ponti J. M., Iorio L., Ardito R. Graded elastic meta-waveguides for rainbow reflection, trapping and mode conversion. EPJ Applied Metamaterials, January 2022 9:6. DOI: 10.1051/epjam/2022004





Ecology, chemical technology, biotechnology, bioengineering