# Ring Mass Controls String Wave Reflection

The mass-loaded boundary condition for an oscillating string attached to a ring of mass ($$m$$) is mathematically derived by equating the ring's inertial force, $$m u\_{t t}\left(x\_0, t\right)$$, to the transversal force exerted by the string, which is approximated as $$-S u\_x\left(x\_0, t\right)$$ for small oscillations. This derivation yields the boundary condition $$m u\_{t t}\left(x\_0, t\right)=-S u\_x\left(x\_0, t\right)$$ at the string endpoint. This dynamic boundary condition is numerically implemented in simulations, where the mass of the ring acts as a tunable parameter that significantly influences wave reflection. Both the derivation and simulation demonstrate that if the mass is negligible, the condition simplifies to the Neumann boundary condition ($$u\_x\left(x\_0, t\right)=0$$), leading to a free end reflection; conversely, increasing the mass causes the wave to reflect as if the end were fixed.

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