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The work is devoted to the analysis of the wave field, which is excited by the reflection of the first normal propagation Rayleigh-Lamb wave from the edge of an elastic semi-infinite strip, part of which is rigidly clamped, and part is free from stresses. The boundary value problem belongs to the class of mixed boundary value problems, the characteristic feature of which is the presence of a local feature of stresses at the point of change of the type of boundary conditions. To solve this boundary value problem, the paper proposes a method of superposition, which allows to take into account the feature of stresses due to the asymptotic properties of the unknown coefficients. Asymptotic dependences for coefficients are determined by the nature of the feature, which is known from the solution of the static problem. The criterion for the correctness of the obtained results was the control of the accuracy of the law of conservation of energy, the error of which did not exceed 2% of the energy of the incident wave for the entire considered frequency range. The paper evaluates the accuracy of the boundary conditions. It is shown that the boundary conditions are fulfilled with graphical accuracy along the entire end of the semi-infinite strip, except around a special point ($\epsilon$). In this case, along the clamped end of the semi-infinite strip in the vicinity of a special point of stress remain limited. The presence of the region $\epsilon$ and the limited stresses are due to the fact that the calculations took into account the $N$ members of the series that describe the wave field, and starting from the $N+1$ member of the series moved to asymptotic values of unknown coefficients, the number of which was also limited to $2N$. As the value $N$ increased, the accuracy of the boundary conditions increased, the region $\epsilon$ decreased, and the magnitude of the stresses near the singular point increased.

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The work is devoted to the development of methods for controlling the efficiency of energy transfer in a composite elastic waveguide. Based on the analysis of the scattered field at the boundary of the stepped waveguide, formed by the rigid contact of two half-layers with the same mechanical characteristics, but with the different thicknesses, the main factors that affect the transparency of the interface were established. Harmonic oscillations generated by the first normal wave propagating from infinity in the narrower half-layer were considered. 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 establishing the conditions for changing the transparency 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 propagating waves of higher orders. The quality of energy resonance in the reflected field depends significantly on the symmetry of the oscillations. The established features of the scattered field make it possible to develop recommendations for controlling the transparency of the boundary in a stepped waveguide.