Source: https://jmrplens.github.io/phonometry/simulation/ # Wave simulation Most of this library predicts a number; this section computes the **wave field itself**. A finite-difference time-domain (FDTD) solver integrates the linear acoustic equations on a 2D grid, so reflection, diffraction, interference, modal behaviour and refraction through inhomogeneous media all emerge from first principles, and its elastic companion carries the same scheme into solids. Both solvers are deterministic (identical inputs give bit-identical outputs on the same platform), validated against analytic oracles, and double as a cross-check engine for the closed-form models of the other sections. The 2D domain is a **cross-section**, and that has one consequence worth settling before any number leaves the solver: a point in the plane is physically an infinite line source, so amplitudes fall as the inverse square root of distance, about 3 dB per doubling instead of 6. Interference and diffraction patterns, arrival times and modal frequencies are faithful; absolute levels and decay rates belong to that geometry and not to a 3D room. A quantitative 3D claim needs a closed form or a 3D solver behind it. The section splits along the media it simulates. The acoustic page explains the numerical method (the staggered leapfrog scheme and its Courant stability bound), the building blocks (sources, probes, obstacles and boundary conditions, including the locally reacting real-impedance edge), the near-to-far-field transformation, when a wave-based simulation is worth its cost, what a 2D domain can and cannot say about a 3D problem, and how numerical dispersion sets the cells-per-wavelength resolution rule. The elastic page extends the same staggered grid to solids: shear waves, free surfaces with Rayleigh waves, and fluid-solid coupling with mode conversion, Scholte interface waves and immersed-plate transmission. A good way to read it is alongside the closed-form pages it cross-checks: the modal frequencies of [room acoustics](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/) reappear as peaks in a simulated room spectrum, the barrier insertion loss of [ground effect and barriers](https://jmrplens.github.io/phonometry/environment/propagation/ground-barriers/) can be re-derived by placing an obstacle in the domain, and the ray bending of [atmospheric refraction](https://jmrplens.github.io/phonometry/environment/propagation/atmospheric-refraction/) emerges from a height-dependent sound-speed profile. When a geometry is too irregular for those models (odd-shaped rooms, multiple barriers, mixed impedance ground), the simulation is the fallback that still gives a quantitative answer; when a closed form exists, prefer it, and use the solver to verify the assumptions it rests on. Those cross-checks are not only arguments: fifteen of the animations in this documentation are output from these two solvers, and they are filed on the guides whose physics they settle rather than here. Room modes growing on and off resonance appear in [room acoustics](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/) and [reverberation prediction](https://jmrplens.github.io/phonometry/buildings/rooms/reverberation-prediction/), which also carries the hall of columns that turns one wavefront into a mixed field; barrier diffraction at two wavelengths in [outdoor propagation](https://jmrplens.github.io/phonometry/environment/propagation/outdoor-propagation/) and [ground effect and barriers](https://jmrplens.github.io/phonometry/environment/propagation/ground-barriers/); downwind and upwind refraction in [atmospheric refraction](https://jmrplens.github.io/phonometry/environment/propagation/atmospheric-refraction/); the ground-effect lobe pattern in outdoor propagation and in [airport noise](https://jmrplens.github.io/phonometry/aircraft/airport-noise/); the standing-wave and transmission tubes in [the impedance tube](https://jmrplens.github.io/phonometry/materials/absorbers/impedance-tube/); the QRD and metadiffuser panels in [diffusers](https://jmrplens.github.io/phonometry/materials/diffusers/diffusers/) and [metadiffusers](https://jmrplens.github.io/phonometry/materials/diffusers/metadiffusers/); the slit absorber in [metamaterial absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/metamaterial-absorbers/); the expansion chamber in [silencers](https://jmrplens.github.io/phonometry/devices/noise-control/silencers/); the wall aperture in [panel sound insulation](https://jmrplens.github.io/phonometry/buildings/design/panel-sound-insulation/); and the SOFAR duct in [underwater propagation](https://jmrplens.github.io/phonometry/underwater/underwater-propagation/). The elastic solver adds two: the bending packet entering an L-junction, on [junction transmission](https://jmrplens.github.io/phonometry/vibration/structural/junction-transmission/), and the coincidence plate, on panel sound insulation. Both also appear on the elastic page below, where the solver that produced them is explained. Several of those guides do more than illustrate: they run a **whole standardised measurement inside the domain**. The impedance-tube guide performs the ISO 10534-2 and ASTM E2611 reductions on a simulated tube and recovers the sample's analytic absorption and transmission loss; the diffuser and metadiffuser guides drive meshed panels with a plane wave and transform the near field into a polar response; the panel-insulation and plate-junction guides launch bending waves into a plate and watch coincidence and junction splitting happen. In each case the solver is standing in for the laboratory, which is what makes the closed-form comparison a real test rather than a demonstration. Setting up a run is a chain, and each link fixes the next. The highest frequency you need and the **slowest** sound speed anywhere in the domain fix the cell size, through the cells-per-wavelength rule that numerical dispersion sets. The cell size and the **fastest** speed then fix the time step, through the Courant stability bound. The domain has to hold the geometry plus clearance for the absorbing layers, which are themselves sized by the **lowest** frequency. The run must last long enough for the field to cross the domain, and for a steady-state answer long enough for the transient to leave before the analysis window opens. The cost is cells times steps, so halving the cell size costs eight times more in 2D — a factor of four in cells and a factor of two in steps. The acoustic page gives the numbers for each link, and the elastic page adds the extra sampling that free surfaces and interface waves demand. ## Pages in this section Read the acoustic page first: the elastic page assumes its vocabulary and says so in its own opening. - [2D FDTD wave simulation](https://jmrplens.github.io/phonometry/fdtd-simulation/): the staggered-grid pressure-velocity FDTD method following Attenborough & Van Renterghem (2021) chapter 4, its sources, probes, obstacles and boundary conditions, the near-to-far-field chain, the 2D limits and the numerical dispersion rule. - [Elastic waves and fluid-solid coupling](https://jmrplens.github.io/phonometry/elastic-waves/): the P-SV velocity-stress companion solver (Virieux 1986) on the same grid, with stress-imaging free surfaces, Rayleigh waves, mode conversion, Scholte interface waves and immersed-plate transmission, each validated against its exact closed form. ## What this section does not cover **Two dimensions, and no way around it.** Everything a 2D cross-section cannot say about a 3D room, neither solver says, and the cylindrical spreading above is only the most visible consequence. The open boundary is a quadratic-ramp absorbing layer — the simple precursor of a perfectly matched layer, not a PML — so grazing incidence is absorbed less cleanly than normal incidence, and the elastic solver has no elastic PML at all, which shows most on grazing Rayleigh waves. The medium is non-moving: **wind and flow advection are not modelled**, so a refraction study here comes from a height-dependent sound-speed profile and not from a flow field. The only impedance boundary is a frequency-independent real one, so a porous absorber has to be meshed rather than declared. On the elastic side the solid is isotropic and purely elastic: no anisotropy, and no viscoelastic damping beyond the bulk decay rate, so a material loss factor cannot be entered. And nothing here is a room-acoustics package. There is no geometry importer, no material library, no ray tracer, no auralisation and no 3D solver: obstacles are rasterised onto the grid from shapes you define, and the output is a field you analyse yourself. ## Before and after these pages The fields these solvers produce are read back with the same tools as a measurement: the filtering, weighting and level functions of [Signal analysis](https://jmrplens.github.io/phonometry/signals/), with [Build a sound level meter](https://jmrplens.github.io/phonometry/signals/sound-level-meter/) running that chain end to end on one runnable page. There is no theory-reference page for the solvers; the derivations, the stability conditions and the analytic oracles stay inside the two guides above. If you arrived here from a search and want the shape of the whole library, [What do you need to measure?](https://jmrplens.github.io/phonometry/start/tasks/) indexes it by the job and [All guides](https://jmrplens.github.io/phonometry/) lists every page with a line on each.