--- jupyter: description: 'Physics examples with planex2d in Python: the eigenvalues of drums, plates, bars and resonators, beams, columns, pipes and cracks against their classic formulas, boat stability, conic mirrors, the brachistochrone and transition curves.' --- # Physics examples - [Isospectral drums](isospectral-drums.md) - [The Helmholtz resonator](helmholtz-resonator.md) - [Chladni figures](chladni-figures.md) - [Tuning a marimba bar](marimba-bar-tuning.md) - [The tuning fork](tuning-fork.md) - [Boat stability](boat-stability.md) - [Flow through sand](flow-through-sand.md) - [Solar shading](solar-shading.md) - [The catenary arch](catenary-arch.md) - [Euler buckling](euler-buckling.md) - [Cantilever deflection](cantilever-deflection.md) - [Stress in a thick-walled pipe](thick-walled-pipe.md) - [The stress intensity factor of a crack](crack-stress-intensity.md) - [Mirrors from conic sections](conic-mirrors.md) - [The fastest slide](fastest-slide.md) - [Transition curves](transition-curves.md) These examples are classic problems of physics and engineering mechanics, each with a known answer to check against. Many are eigenvalue, elasticity or flow problems. For these, the geometry is drawn in planex2d, netgen meshes it, NGSolve solves it, and the result is compared with a formula, a table or a published value, from Euler's buckling load to the stress intensity factor of a crack. Others need only the geometry, as the waterline of a heeled boat at constant displacement, rays reflected by conic mirrors or a bead racing down a ramp, an arc and a cycloid.

In this section

Isospectral drums
Isospectral drumsTwo drums with the same eigenvalues, so the shape cannot be heard.
The Helmholtz resonator
The Helmholtz resonatorThe note of a bottle from an acoustic eigenvalue solve, against the formula with its end corrections.
Chladni figures
Chladni figuresThe modes of a free square plate against the tabulated frequencies, and the nodal lines where sand gathers.
Tuning a marimba bar
Tuning a marimba barThe arch under a bar that puts its first overtone two octaves up, from its bending modes.
The tuning fork
The tuning forkTwo tines swinging against each other leave the stem at rest; the clang mode and the length for 440 Hz.
Boat stability
Boat stabilityThe waterline at constant displacement and the righting lever curve.
Flow through sand
Flow through sandStokes flow through packed grains against the Kozeny-Carman law.
Solar shading
Solar shadingThe shaded share of a roof panel over a day, every quarter of an hour, at the solstices and the equinox.
The catenary arch
The catenary archHooke's inverted chain in compression, a semicircle opening at its haunches, and the thrust w a.
Euler buckling
Euler bucklingThe critical load of a column for the four end conditions, against Euler and Engesser's shear correction.
Cantilever deflection
Cantilever deflectionA beam as a plane solid against Euler–Bernoulli and Timoshenko, from 1 to 20 times as long as high.
Stress in a thick-walled pipe
Stress in a thick-walled pipeLamé's hoop stress, which no wall thickness brings below the pressure, and a shrink-fitted ring that lowers it.
The stress intensity factor of a crack
The stress intensity factor of a crackA central crack from the energy it releases, against Griffith and Feddersen.
Mirrors from conic sections
Mirrors from conic sectionsA whispering gallery and a Cassegrain telescope from exact rational B-splines, traced ray by ray.
The fastest slide
The fastest slideA ramp, an arc and the cycloid raced from rest, and the same time from any height.
Transition curves
Transition curvesThe Euler spiral between a straight and an arc: its curvature, the shift of the circle and the ride.
See [Examples](index.md) for the other collections.