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.
block-beta
columns 5
CC["Criss-Cross"]:5
%% Condensed Notes
CN["Condensed Notes"]:5
RF["Relevant File"]:5
NV["Narrated Video"]:4 VO["Voice-over"]
PA("Plotting & Analysis")AA("Animation & Analysis")KT("Summary & Interpretation") ID("Illustration & Demo") PO("Polyptych")
%% Proof and Derivation
PD["Proof and Derivation"]:5
AF("Derivation Sheet"):5
NV2["Narrated Video"]:4 VO2["Voice-over"]
PA2("Plotting & Analysis")AA2("Animation & Analysis")KT2("Summary & Interpretation") ID2("Illustration & Demo") PO2("Polyptych")
classDef color_1 fill:#8e562f,stroke:#8e562f,color:#fff
class CC color_1
classDef color_2 fill:#14626e,stroke:#14626e,color:#fff
class CN color_2
class RF color_2
classDef color_3 fill:#1e81b0,stroke:#1e81b0,color:#fff
class NV color_3
class PA color_3
class AA color_3
class KT color_3
class ID color_3
classDef color_4 fill:#47a291,stroke:#47a291,color:#fff
class VO color_4
class PO color_4
%% Proof and Derivation
classDef color_5 fill:#307834,stroke:#307834,color:#307834
class PD color_5
class AF color_5
classDef color_6 fill:#38b01e,stroke:#38b01e,color:#38b01e
class NV2 color_6
class PA2 color_6
class AA2 color_6
class KT2 color_6
class ID2 color_6
classDef color_7 fill:#47a291,stroke:#47a291,color:#47a291
class VO2 color_7
class PO2 color_7
‣
<aside> <img src="/icons/report_pink.svg" alt="/icons/report_pink.svg" width="40px" />
Copyright Notice
All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]
© 2025 Sayako Dean
</aside>