Toroidal Resonance Explorer

This tool imposes a cosine phase on a torus. It does not solve a plasma resonance or an eigenmode. The amplitude is cosine of m times poloidal angle, plus n times toroidal angle, minus two pi frequency times model time. Hold toroidal angle and model time at zero. The blue curve uses m equal one, and the gold curve m equal three. At poloidal angle zero both marked amplitudes are plus one. The full curves are already drawn on identical scales. Now sweep the marked poloidal angle through one full turn. Watch the markers move along the existing curves. The blue marker completes one cosine cycle while the gold marker completes three. This is an angle sweep paced to the explanation, not a change in the wave amplitude scale. At angle two pi both amplitudes return to plus one. The source integer m counts these poloidal cycles; n similarly counts cycles around the central hole at fixed poloidal angle and time. The default values are m equal three and n equal two. The horizontal range is zero to two pi, and both plots use sixty pixels per unit amplitude. Fix both angular coordinates at zero, and compare the source phase at one hertz and two hertz. Each curve is cosine of two pi frequency times model time. At model time zero both markers show amplitude plus one. The horizontal axis is one model second for both plots, and their amplitude scales are identical. Now move the model time cursor from zero through one second. The blue one hertz marker follows one complete cycle. The gold two hertz marker follows two complete cycles. Model time is shown on the axis; this demonstration runs at the pace of the narration. At model time one second both amplitudes are again plus one. Their periods are one second and half a second. Changing frequency changes the prescribed phase speed. The slider does not derive a physical eigenfrequency from geometry, material stiffness, magnetic fields or boundary conditions. With nonzero m or n the single cosine phase travels around the surface; a standing wave would require a different superposition. The source offsets each original torus vertex along its original unit normal by point one eight times cosine amplitude. Consider the outer point of the default torus, with coordinates three point eight, zero, zero, and normal pointing in the positive x direction. Hold its angular coordinates at zero and frequency at one hertz. At model time half a second its amplitude is minus one, so its signed displacement is minus point one eight model units. At seven hundred pixels per model unit the green point is one hundred twenty-six pixels to the left of its original position. Now advance model time from half a second to one second. The signed cosine amplitude crosses zero and becomes positive. Watch the displacement segment shrink to zero, then extend in the opposite direction along the same original normal. At model time one second the amplitude is plus one. The displacement is plus point one eight model units, one hundred twenty-six pixels to the right of the original point. The original point stays fixed as a reference throughout. These model units are a geometric illustration, not measured plasma, DNA or planetary magnetic motion. The default major radius three and tube radius point eight are chosen geometry. Rotate, zoom, pause and export the original tool to inspect this prescribed pattern.