Instead of smooth, fading circles, you will see a complex interference pattern forming box-like geometric peaks near the edges where outgoing and reflected waves crash into each other.

https://youtu.be/aXMpKfl-lpY

<aside> 👏

1. https://www.instagram.com/reel/DeHU53XhacG/?utm_source=ig_web_copy_link&stkn=MzRlODBiNWFlZA==
2. https://via-dean.gitbook.io/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/homogeneous-and-inhomogeneous-helmholtz-waves
3. https://bsky.app/profile/researcherdean.bsky.social/post/3mx52yd3lis2m
4. https://x.com/d54223/status/2107108016549949909?s=20
5. https://pin.it/4MoWY2gWP
6. https://hackmd.io/@bafMjVSBQ6e7vlO7NdkhXA/B1cT3IfKfl </aside>

<aside> <img src="/icons/download_yellow.svg" alt="/icons/download_yellow.svg" width="40px" />

🍄Deliverables 15

Homogeneous and Inhomogeneous Helmholtz Waves

https://payhip.com/b/BacMg

Documents

• Computational Models
• Snippets
• A guide and breakdown distilling key physical takeaways from numerical simulations of the Helmholtz wave equation.

Exploration Frontiers

• Source-free uniform standing waves and resonance modes $\rightarrow$ Eigenvalue Solvers
• Localized point-source driving and concentric ripples $\rightarrow$ Heterogeneous Media
• Dirichlet/Neumann boundaries using the method of images $\rightarrow$ Open Boundaries (ABCs/PML)
• Spatial free-source plane and spherical wave plots $\rightarrow$ Advanced Numerical Methods (FEM)

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graph TD
    %% Nodes representing Current Exploration
    subgraph Current_Exploration ["1. Current Exploration (Helmholtz Dynamics)"]
        H_Stat["A. Homogeneous (Plane Waves / Resonance)"]:::current
        I_Stat["B. Inhomogeneous (Point Source / Radiation)"]:::current
        BC_Ref["C. Bounded Cavities (Reflections / Method of Images)"]:::current
        Dim_3D["D. 3D Visualization"]:::current
        Time_H["E. Time-Harmonic Animation"]:::current
    end

    %% Transition Node
    Progression_Link{"Potential Progression Path"}:::link

    %% Nodes representing Potential Exploration
    subgraph Potential_Orientation ["2. Potential Orientation (Advanced Topics)"]
        Rad_Cond["F. Radiation & Open Boundaries (ABCs / PML)"]:::potential
        Het_Med["G. Heterogeneous Media (Waveguides / Scatterers)"]:::potential
        Eigen_Solv["H. Eigenvalue Solvers (Resonance Modes / Q-Factor)"]:::potential
        Num_Meth["I. Advanced Numerical Methods (FEM Migration)"]:::potential
    end

    %% Connections from Current to Transition
    H_Stat ==> Progression_Link
    I_Stat ==> Progression_Link
    BC_Ref ==> Progression_Link
    Dim_3D ==> Progression_Link
    Time_H ==> Progression_Link

    %% Connections from Transition to Potential
    Progression_Link ==> Rad_Cond
    Progression_Link ==> Het_Med
    Progression_Link ==> Eigen_Solv
    Progression_Link ==> Num_Meth

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    classDef link fill:#581c87,stroke:#eab308,stroke-width:2.5px,color:#fdf4ff;

<aside> 👏

1. https://hackmd.io/@bafMjVSBQ6e7vlO7NdkhXA/B1cT3IfKfl </aside>

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