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

A string of uniform density is stretched along the x axis and undergoes small transverse oscillations in the (x,y) plane. We suppose that a wave is incident from x=-∞ and that a mass is fixed to the string at x=0. 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 x axis under tension T and that a mass M is fixed to the string at x=0. Using force balance, one can show that the equation of motion can be written M(Subscript[((((∂^2)y)/(∂(t^2)))), x=0])= T ((Subscript[(((∂y)/(∂x))), x=(SubPlus[0])])-(Subscript[(((∂y)/(∂x))), x=(SubMinus[0])])). We then suppose that we have a sinusoidal wave coming from x=-∞. This is based on a question from the University of Cambridge "Methods Example Sheet 2" (2009) by Dr. C. Caulfield.

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Contributed by: Will Rayward-Smith

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