Two sections of a long horizontal rope are joined together. Section 1 has linear mass density μ1=0.020kgm−1; section 2 has μ2=0.080kgm−1. The entire rope is under a uniform tension of 50N. A continuous sinusoidal transverse wave of amplitude 4.0cm, frequency 5.0Hz, and vertical polarization travels from section 1 toward section 2. Several meters beyond the junction a ring admits only displacements that lie along a straight line making 60° with the vertical.
Neglect energy losses except as specified.
(a) Derive expressions for the wave speeds v1 and v2 in the two rope sections in terms of the given quantities, then calculate their numerical values.
(b) Applying appropriate boundary conditions at the junction, derive symbolic expressions for the amplitudes of the reflected and transmitted waves. Evaluate the numerical amplitude of the transmitted wave.
(c) Indicate whether the reflected wave in section 1 is inverted or non-inverted, and justify your answer.
(d) After the transmitted wave passes through the ring, determine its amplitude and its intensity as a fraction of the original incident intensity on section 1.