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In this session, we do more standing wave problems. The focus is on the role of boundary conditions at the intersection of two continuous media with different physical characteristics.
In this session, we do more standing wave problems. The focus is on the role of boundary conditions at the intersection of two continuous media with different physical characteristics.
An ideal taut string of length \( l \) and mass \( m \) is attached to a fixed point at one end and to a massive ring of mass \( M_R \) at the other end as shown below. The ring is free to move on a horizontal frictionless rod which is perpendicular to the string in its equiibrium position. The tension in the string, \( T \), can be considered constant at all times and gravity can be ignored.
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\begin{eqnarray} \nonumber \omega &=& \frac{T}{M_R}\sqrt{\frac{m}{Tl}}\cot \left( \omega l \sqrt{\frac{m}{Tl}}\right) \\ \nonumber &=& \frac{1}{M_R}\sqrt{\frac{Tm}{l}}\cot \left( \omega \sqrt{\frac{ml}{T}}\right) \end{eqnarray}
This is a transcendental equation with no simple solution.
A system consists of two materials glued together at \(x = L\) and subject to longitudinal (sound) wave propagation. The end at \(x = 0\) is fixed and the end at \(x = 4L\) is free. A tungsten-lead alloy is used for the segment \(0 \leq x \leq L\), with a Young’s modulus of \(Y = 7 \times 10^{10}\) N/m\(^2\) and density \(\rho = 14.4 \times 10^{3}\) kg/m\(^3\), while the segment \(L \leq x \leq 4L\) is comprised of a carbon fiber alloy that is as strong as the tungsten-lead alloy with \(Y = 7 \times 10^{10}\) N/m\(^2\) but nine times less dense, with \(\rho = 1.6 \times 10^{3}\) kg/m\(^3\).
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The phase velocity \(v\) of longitudinal waves \( = \sqrt{\dfrac{Y}{\rho}}\)
\(\therefore \dfrac{v_2}{v_1} = 3\) and the essence of the problem is the boundary conditions at \(x\) = 0, \(L\) and \(4L\).
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[Note that just about any factor other than 3 for speed and length ratios would give an equation which could not be easily solved.]