Dan Russell's Acoustics and Vibration Animations

photograph of Dan Russell

Professor of Acoustics
Director of Distance Education
Graduate Program in Acoustics

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: drussell@engr.psu.edu
Follow me on Twitter: @drussellpsu

Today is . The contents (and links) on this page were last updated on February 18, 2014.

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 Pressure and Particle Velocity for Plane Waves - p and u are in phase for forward going waves, but have opposite phase for backward going waves.
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
Scattering from Spheres and Cylinders - Scattering of a Plane Wave from Spheres (rigid and pressure-release) and Cylinders (rigid). NEW Animation (5/17/18)

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.

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. Updated Animations (6/18/18)
Crankshaft: Non-harmonic Oscillator - periodic oscillatory motion that is not sinusoidal. NEW Animation (4/25/18)

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
Vibrational modes of a multi-dof system - progression of mode shapes from single degree-of-freedom to continuous structure
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
Vibration of a Fixed-Fixed String - mode shapes, and frequency spectra for a plucked string
Rectangular Membrane - and degenerate modes of a square membrane New and Improved (10/18/2018)
Circular Membrane - or how a drum head vibrates.Updated Animations (4/29/2018)
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.
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
Bending and Torsional Vibration in a Hockey Stick - flexural bending and torsional twisting vibration in a hockey stick
Vibration Modes of a Glass Beer Bottle - what vibration shapes are responsible for the "clink" of a glass bear bottle?
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

Miscellaneous Animations

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

Interactive Plots Demonstrating Acoustic Phenomena

The interactive plots in these links contain Mathematica Computable Document Format files and require the use of the free Wolfram CDF Player.

Absorption and Attenuation of Sound in Air - interactive plot to calculate and illustrate the absorption of sound in air due to viscous, thermal, and molecular relaxation processes.
Directivity Patterns for an Acoustic Doublet (bipole) as a function of kd - how does the directivity pattern of the sound field radiated by a pair of simple sources depend on the distance between them (or the frequency of the sound)?
Inharmonicity due to Stiffness for Guitar Strings - how does the stiffness of a steel guitar string affect the integer relationship of the resonance frequencies?

Tools used to create these animations

Mathematica -- Almost all of the animations, and all of the interactive CDF plots, were conceived and created with Mathematica. This ensures that the animations are physically and mathematically correct instead of being cutesy cartoons.
COMSOL Multiphysics -- there are a few animations on some of the newer pages that involve problems without nice analytical solutions, but which instead require computational (finite element or boundary element) solutions. I have started using COMSOL Multiphysics to numerically model these problems and to create a few animations.
GIFfun -- While I design and create the animations with Mathematica, the actual animated GIF images on my webpages have been created with GIFfun from Stone Design.