Exact solutions of equation of transverse vibrations for a bar with a specific cross section variation law

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• 作者：M. A. Mironov
• 关键词：bar ; vibrations ; propagation time ; black hole
• 刊名：Acoustical Physics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：63
• 期：1
• 页码：1-6
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Acoustics;
• 出版者：Pleiades Publishing
• ISSN：1562-6865
• 卷排序：63

文摘

Flexural wave propagation along a bar whose thickness smoothly decreases down to zero within its end piece is considered. The propagation velocity tends to zero as the tapered end of the bar is approached, and the time of wave propagation to the tapered end is infinite. As a consequence, waves propagating along the bar are not reflected from the end. Previous quantitative study of the effect in the WKB approximation shows that, in the case of parabolic tapering, the WKB approximation yields a uniform asymptotics, which is valid (or invalid) for any of the bar’s cross sections. In the case of a bar with parabolic tapering, the equation of flexural vibrations of the bar has exact analytic solutions in the form of power functions. Based on these solutions, a modified WKB approximation is proposed to solve equations for bars with nonparabolic thickness variation laws. The input impedance of a bar with a parabolic tapering is calculated and analyzed.