Acoustics and Vibration Animations

The links below contain animations illustrating acoustics and vibration, waves and oscillation concepts. I started using Mathematica to create animations to help me understand and visualize certain acoustics and vibration phenomena in 1992 while I was a Ph.D. student in the Graduate Program in Acoustics at Penn State. For the next 16 years (1995-2011) I was a physics professor at Kettering University and continued creating animations and using them as educational tools for the courses I was teaching about waves and acoustics. Sometime around 1998 or so I began writing webpages and adding them to this online collection. Now that I'm back at Penn State, teaching graduate level acoustics, I'm continuing to add to my collection of animations. My intent has always been to create physically and mathematically correct animations, accompanied by explanatory text, that illustrate complicated phenomena involving waves and vibration in a manner that aids student understanding. I hope you find these animations useful.

• Please send comments about this page to: dar119@psu.edu
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Sound Waves and Sources

Basic Wave Phenomena

• What is a Wave? - a disturbance which travels through a medium
• Wave Motion in Space and Time - distinguishing between the time behavior and spatial behavior of waves
• Superposition of Two Waves - Interference, Standing Waves, and Beats
• Fourier Decomposition - building a wave shape from sines and cosines
• Phase changes upon reflection from hard, soft, and mixed boundaries
• Reflections from Impedance and the Standing Wave Ratio. - A sinusoidal wave train reflects from an impedance discontinuity. The resulting wave pattern defines the Standing Wave Ratio used to measure impedance.
• Refraction of Sound Waves - how temperature gradients make sound waves change direction
• The Doppler Effect - moving sound sources and sonic booms

Sound Waves and Radiation from Sources

• Longitudinal and Transverse Waves - Particle Motion for Longitudinal, Transverse, Water and Rayleigh Surface Waves
• Phase Between Displacement, Velocity and Pressure for a Longitudinal Sine Wave - p and u are in phase for forward going waves, but have opposite phase for backward going waves. Brand New Animations (11/28/2020)
• Phase Between Pressure and Particle Velocity for a Gaussian Pulse - p and u are in phase for forward going waves, but have opposite phase for backward going waves.
• Displacement vs Pressure -- Reflection from Boundaries - DRAFT page illustrating difference between the phase changes upon reflection for displacement pressure. (displacement inverts upon reflection from a rigid boundary while pressure reflects upright).
• Standing Longitudinal Sound Waves. - Animating the relationship between particle motion and pressure in a standing longitudinal sound wave in a pipe.
• Sound Fields Radiated by Simple Sources - pressure sound fields produced by monopole, dipole, and quadrupole sources
• Directivity Patterns for an Acoustic Doublet (bipole) as a function of kd
• Sound Radiation from Cylindrical Radiators - particle motion and sound fields produced by vibrational modes of a cylinder
• Sound field radiated by a Tuning Fork - animations to accompany a paper published in the American Journal of Physics
• The Baffled Piston - approximation of the sound field radiated by a loudspeaker
• Partial Plane Wave Expansion -- expressing a plane wave in terms of spherical and cylindrical coordinates (for scattering problems).
• Scattering from Spheres and Cylinders - Scattering of a Plane Wave from Spheres (rigid and pressure-release) and Cylinders (rigid).

More Complicated Wave Phenomena

• Plane Waves and Evanescent Waves (Sound Radiation from Plates) - plane wave radiation of sound from bending waves in plates becomes evanescent when plate wave speed is slower than the fluid sound speed.
• Evanescent Modes in Waveguides - higher order modes in ducts driven above and below their cutoff frequency.
• Waves in Dispersive Media - when wave speed depends on frequency
• Examples of Dispersive Waves - Flexural waves and Quasi-longitudinal waves on a rod
• Interaction Between Two Solitary Waves - what is a soliton? and what happens when two solitons collide?

Room Acoustics

• Driving room modes & source location - How does the (resonance) response of a room depend on source location?
• Reverberation Times in a small room - sound files and plots for measurements of T₆₀.

Miscellaneous Animations

• Motion of the Center-of-Mass - not exactly acoustics and vibration, but the animations are cool

Vibration and Structural Waves

Vibration of 1-DOF Simple Oscillators

• Comparing Circular and Sinusoidal Motion - how is circular motion in the complex plane (magnitude and phase) related to sine and cosine functions?
• Simple Harmonic Oscillator - with and without damping, transfer of energy between kinetic and potential forms
• Damped Harmonic Oscillator - underdamped, overdamped, and critically damped
• Phase-Space Diagrams for an Oscillator - plotting velocity v(t) as a function of position x(t)
• Forced Harmonic Oscillator - transient and steady state response to a force applied to the mass
• Base Motion - transient and steady state response of an oscillator to displacement of its support
• The Simple Pendulum - comparing the linear approximation (small angle) with a real pendulum.
• Crankshaft: Non-harmonic Oscillator - periodic oscillatory motion that is not sinusoidal.
• Regions of Resonance - interactive animation of the damping-controlled, stiffness-controlled, and mass-controlled regions in the displacement response of a forced mass-spring system.

Vibration of Multi-DOF Systems

• Coupled Oscillators - energy transfer between two mass-spring systems coupled together
• Dynamic Absorbers - J.P. Den Hartog's classical undamped tuned dynamic absorber
• Multiple-DOF mass-spring systems. - improved animation of the natural modes of a multi-degree-of-freedom mass-spring system.

Vibrational Modes of Continuous Systems

• Vibrational Modes of a Hanging Chain - mode shapes for a hanging chain
• The Plucked String (guitars) - mode shapes, Fourier Coefficients, and time signals for a plucked guitar string
• The Struck String (pianos) - the motion of a piano string after being struck by a hammer.
• The Bowed String (violins) - the Helmholtz motion of a bowed violin string
• Rectangular Membrane - Mode shapes for a rectangular membrane and degenerate modes of a square membrane
• Circular Membrane - or how a drum head vibrates.
• Membranes - vs - Strings: Initial Conditions Comparing the propagation of 2-D waves on a membrane with 1-D waves on a string for initial displacement and initial velocity conditions.
• Torsional Modes -- Twisting oscillation in a rectangular bar.
• Flexural Vibrations of a Beam - how the mode shapes depend on boundary conditions (no text yet, but animations work)
• Vibrational Modes of a Tuning Fork. - how does a tuning fork vibrate?
• Vibration of Baseball Bats - flexural bending modes and cylinder modes

Animations of Experimental Results

• Vibration Modes of a Glass Beer Bottle - what vibration shapes are responsible for the "clink" of a glass bear bottle?
• Bending and Torsional Vibration in a Hockey Stick - flexural bending and torsional twisting vibration in a hockey stick
• Vibrational Mode Shapes of a Racquetball Racket - Bending, Torsional and Membrane modes in a racquetball racket.
• Vibrational Mode Shapes of a Tennis Racket - Bending, Torsional and Membrane modes in a Tennis Racket.
• Vibrational Modes of an Electric Guitar - actual experimental data
• Vibrational Modes of an Acoustic Guitar - actual experimental data
• Vibrational Modes of a (cutlery) Fork -- Bending and Torsional vibration of a fork (the kind you eat with).