Wave Reflection and Transmission on a String with a Mass in the Middle

A string of uniform density is stretched along the axis and undergoes small transverse oscillations in the plane. We suppose that a wave is incident from and that a mass is fixed to the string at . The incident wave is shown dashed and black, the reflected wave is dashed and blue, the transmitted wave is dashed and red, and the resultant wave is solid and black.

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We suppose that the string is stretched along the axis under tension and that a mass is fixed to the string at . Using force balance, one can show that the equation of motion can be written . We then suppose that we have a sinusoidal wave coming from . This is based on a question from the University of Cambridge "Methods Example Sheet 2" (2009) by Dr. C. Caulfield.
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