 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2018/paper-2/section-b/b9/d/solution

At an ordinary point or a massless join, the string is continuous, so

Ψ (x_{0}^{-}, t) = Ψ (x_{0}^{+}, t) .

With no concentrated transverse force or point mass, transverse force balance requires

T_{1} Ψ_{x} (x_{0}^{-}, t) = T_{2} Ψ_{x} (x_{0}^{+}, t) .

For the common tension in this problem, the slope

\partial Ψ/ \partial x

is therefore continuous as well. A jump in linear mass density changes wave speed, but introduces no finite inertial force at a zero-length join. A point mass, external support force, or unequal tensions would require modified boundary conditions.

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