Dan Russell's Acoustics and Vibration Animations

photograph of Dan Russell

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
Follow me on Twitter: @drussellpsu

Today is . The contents (and links) on this page were last updated on January 17, 2024.

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

Longitudinal Sound Waves

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.
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).
How does sound reflect from the open end of a pipe? -- mass-spring model to illustrate how a longitudinal sound wave reflects from the open end of a pipe. New Animation => 2023 => (supported NYU Teaching Advancement Grant with Susie Levi)
Standing Longitudinal Sound Waves. - Animating the relationship between particle motion and pressure in a standing longitudinal sound wave in a pipe.

Sound Sources, Radiation, and Scattering

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 (This page is undergoing repair/updates)
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 Time in a Small (but very reverberant) Room - audio files and plots for measurements of T₆₀ in a small, empty room. [Updated: 1/17/2024]
Reverb Times for a small Auditorium (page needs to be updated, but has measurements (and sound recordings) of T₆₀ for a small theater at Kettering University)

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 in a Rectangular Beam -- Twisting oscillation in a rectangular bar.
Flexural Vibrations in a Rectangular Beam - flexural bending mode shapes and frequencies for a rectangular bar (New COMSOL animations 1/16/2024)
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).

Miscellaneous Animations

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

A New Animation Creator

My current Ph.D. student, Noah Parker, is beginning to develop his own interactive animations to help undergraduate students learn about acoustics, engineering, and physics.

Noah Parker's "Math of Sound" website with interactive animations

Giving Credit Where Credit is Due

I learned how to use Mathematica to create animations of waves and oscillation phenomena from Dr. Victor Sparrow, my PhD advisor at Penn State. Here is collection of several of his earliest animations from 1992-2000.

Vic Sparrow's Acoustics and Vibration Animations.

Videos of Acoustics Demonstrations on my YouTube Channel

How the Speed of Sound changes with Temperature (comparing resonance frequencies of the same length tube with room temperature, hot air, and cold air.)
An Acoustic Rocket: Reproducing Dvorak's 1878 Acoustic Repulsion Demonstration
The Softball Bat Piano -- playing "Take Me Out To The Ball Game" on a collection of softball bats with different barrel frequencies.
Classroom demonstration of longitudinal waves in a rod.
Vibration of a compound string -- half thick and half thin (a partially unwound guitar string)
Classroom demonstration of evanescent modes in a waveguide
Sound Radiation from Dipoles and Quadrupoles - comparing sound radiation (especially low frequency) from dipoles and quadrupole sources.
Bat Vibration with and without a vibration absorber - proof that a vibration absorber can reduce vibration.
The Hot Chocolate Effect - demonstrating how the speed of sound depends on the elastic and inertia properties of the medium.
Longitudinal Wave Propagation on a Slinky - High Speed Video of two demonstrations: (1) Propagation speed depends on the number of slinky coils and (2) Propagation time is constant because wave speed is linearly proportional the stretch slinky length.
The Coupled Oscillator - Two pendulums connected by a soft spring exhibit a coupled behavior
Multiple-DOF Oscillations - Whiffle Balls and Rubber bands demonstrate mode shapes for 1-, 2-, 3-, and 4-degree-of-freedom mass-spring systems.
The Friction Oscillator - A simple harmonic oscillator for which kinetic friction forces provide the restoring force that actually causes the oscillation to occur.
- "The Friction Oscillator" tied for First Place in the 2012 Gallery of Acoustics at the 164th meeting of the Acoustical Society of America.
Motion of a Plucked String - High Speed Video (1000fps) showing the motion of a plucked string.
Mode Shapes of a Circular Membrane #2 - High Speed Video (1000fps) showing the first several mode shapes of a circular membrane.
Mode Shapes of a Circular Membrane #1 - The first several mode shapes of a circular membrane (captured with a strobe light)
Regions of Forced Response - Comparing the stiffness-controlled, damping-controlled, and mass-controlled regions of the response curve for a forced, damped simple harmonic oscillator.
Acoustic Propulsion - nonlinearities in Helmholtz resonators driven at very high pressure levels (greater than 125dB) create an acoustic thrust, similar to a rocket.

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.

Acoustics and Vibration Animations

by Dr. Daniel A. Russell