Here are small animations, illustrations, or comments on some examples and wave phenomena in general. They are offered in various forms.
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Resonance phenomenon. Elastic and absorption amplitude.
Potential function for transverse oscillations of one mass on two springs. Small oscillation approximation.
Longitudinal oscillations of two masses with adjustable masses, spring stiffnesses, and mode amplitudes. Configuration space.
Longitudinal oscillations of an adjustable number of masses, fixed ends. All masses have the same mass m and all springs have the same stiffness k.
Calculation of the potential energy matrix using small oscillation approximation -- using second-order Taylor expansion. Download Mathematica notebook.
This example is fully contained (except for mode superposition) in the interactive spring system with N masses above.
Longitudinal oscillations of three masses with four springs, fixed ends. Superposition is created from all modes with the same (half) amplitude. Exactly the same equations of motion result for currents in example 1.14; the angular frequencies and amplitude ratios will therefore be completely identical.
How does a truss bridge oscillate? Up, down, left, or right? Lots of modes, lots of oscillations.
Animation of travelling standing (undamped and damped) waves on a chain of atoms.
Animation of d'Alembert's solution (superposition of counter-propagating waves). Representation of energy quantities on a string.
Composition of travelling waves radiated from two sources at distance 2d. In interactive files, it is possible to change distance d and observe the change in amplitude of the resulting travelling wave.
Standing waves on a string – fixed and free ends. Possibility to set individual amplitudes. As can be seen, superposition of standing waves is no longer a standing wave.
Wave packet – Fourier transform for a given rectangular spectrum.
Rectangular pulse – Fourier transform for a given rectangular signal course.
Damped harmonic wave – Fourier transform for exponentially damped harmonic wave – atomic radiation.
Superposition of frequency spectra of two close tones sounding for a short time. With too short tone duration, their sound merges into an indistinguishable mixture.
Demonstration of phase and group velocity on the composition of two harmonic travelling waves.
Demonstration of wave packet spreading in a dispersive medium.
Demonstration of reflections (and transmissions) of waves at medium termination and medium interface.
In the case of massive coupling of two strings, the transmission and reflection coefficients are complex and frequency-dependent. The interactive graph shows the magnitude of these coefficients as a function of angular frequency with adjustable string and coupling parameters.
Adjustable animation of transmissions and reflections of waves at two interfaces between three media.
Graphs of complex reflection coefficient for two interfaces (between three media) taking into account all reflections and interference.
Animation of radiated shell of acceleration field connecting the field of static and moving charge. Animation of electric field of wave radiated by harmonic oscillation of charge.
Visualization of travelling wave in a waveguide for the first four modes independent of x coordinate.
Animation of planar harmonic travelling EM wave passage through planar interface. Application of Snell's law, visualization of evanescent wave when exceeding critical angle.
Visualization of planar, circular, and elliptical polarization.
Visualization of rotation of the plane of linearly polarized light for superposition of phase-shifted circular polarizations.
Demonstration of diffraction patterns for diffraction on two point apertures, rectangular and circular slit.