Virtual Labs in Acoustics

Supported by funding from the National Science Foundation. 

Division of Undergraduate Education: Course, Curriculum & Laboratory Improvement (CCLI) Award # 0230896

The Links below point  to Mathematica  notebooks which contain tutorials and/or workbooks about topics in basic acoustics.  Mathematica is software, from Wolfram, Inc.,  that is well suited to symbolic mathematics and the creation of graphics and sound.These notebooks were written in support of a course in the physics of music and musical instruments that is offered at the University of Vermont.   These notebooks can be  viewed with the free version, MathReader, which is available as a download.  To take full advantage of the notebooks, one should use the standard full-featured  version of Mathematica.

1. Simple vibrations

Goals:   Learn the meanings of frequency,amplitude,phase and
period;learn how to apply these concepts in the description of
graphs of position,velocity and acceleration as functions of
time.

Methods: Guided experiments with a virtual mass/spring linear
oscillator.Students will vary the mass,stiffness,initial
velocity and initial position to see how these quantities affect
frequency,amplitude,phase and period of the resulting motion.

2. Complex vibrations

Goals: Understand superposition of waves,learn the difference
between periodic and nonperiodic waves and listen comparatively to
the simple sounds of pure sine waves,the complex sounds of
harmonic series of sine waves and the complex sounds of sums of
sine waves that are not related harmonically.

Methods: Use Mathematica functions Plot and Play to
display the graph and play the sound of
+++... for different amplitudes and frequencies.

3. Introduction to Travelling Waves

Goals: Learn the meanings of wave velocity, wavelength, frequency and phase; understand the distinctions between transverse waves and longitudinal waves ;learn to
distinguish the difference between the motion of a particle of the
medium through which the wave passes and the motion of the wave
itself; check the dispersion relation: v=fλ.

Methods: Use the Mathematica function Plot to create
one-dimensional transverse wave plots of the form
y(x,t) = A Sin[2πx/λ ± 2πft] for a various values of t
and use Mathematica's animation feature to visualize the moving wave and explain how its shape is depends on f, λ, and A.

4. Introduction to Standing Waves

Goals: Learn how to apply the superposition principle
to the propagation of travelling waves;understand
the concept of interference,and how standing waves are a special case of interference.

Methods: Add one dimensional travelling waves.Observe the
resulting waveforms for different phase differences and different
wavelengths.Illustrate the result of two waves propagating in
different directions and show that this can result in a standing
wave.

5. Interference and diffraction of sound

Goals: Develop a qualitative understanding of interference
and diffraction.

Methods: Use the DensityPlot function to produce a animations of
circular wave fronts emanating from a point or multiple points.Identify nodal curves and antinodal regions. Verify the requirements for constructive and destructive interference. Construct simulations of wavefronts incident on an aperture and wavefronts
incident on a barrier to explore how the size of the barrier/aperture
relative to the wavelength affects the scattered/transmitted wave.

6. Spectrum and timbre

Goals: Show that complex sounds are made by combining simple
vibrations of different frequencies together.Understand how
timbre and spectrum are related.

Methods: Exploit Mathematical's native ability to import audio files and
the Fourier function to calculate the spectrum of
the imported sounds.

7. Damping and driving

Goals: Help students to strengthen their understanding of the real
world behavior of vibration,by introducing them to the concepts of
resonance and damping. Illustrate the exponential nature of the decay of vibrations and sounds. Demonstrate through example the meaning of the "Q" value in
vibration theory.

Methods: Students will be provided with a simulated simple
oscillator,as seen in the "Simple Vibrations" module.Students
will be allowed to vary the frictional force that impedes that
motion of the oscillator and observe the effect of doing so on the
motion.Students will also be allowed to vary the amplitude and
the frequency of a sinusoidal driving force and observe the
results as they construct resonance curves for cases of small, medium and large damping.

8. Room acoustics and reverberation times

Goals: Teach students about room acoustics, attenuation of sound and reverberation times.

Methods: Write Mathematica functions that allow the students to easily import audio files (which the students have recorded earlier, or are otherwise provided with)
that contain the decay of white noise in a selected classroom or
auditorium.The students can then process the signal to produce a
smooth graph of the decay of the sound's intensity in the room as
time elapses and use this information to determine the
reverberation time (defined as the time for a 60 dB fall in sound intensity) or times in the room.

9. Complex vibrations of extended objects.

Goals: Teach students about the multifrequency vibrations that are
evident in the oscillations of stretched strings,beams,drumheads,air columns,chimes and other objects that make complex sounds. Illustrate the concepts of modes, nodes, nodal curves, antinodes and resonant frequencies. Demonstrate the effects of driving these systems on- vs. off- resonances and the effects of driving at a node vs. driving an antinode. Illustrate the important connection between the active vibrational modes and the spectrum of the sound that is produced by the oscillator.

Methods: This task will be accomplished by more than one module.We intend at the very least to construct a simulation of a vibrating string which will be able to be set in motion in any one of its resonant modes as well into the complex motion resulting from either plucking, striking, or bowing the string at different
locations along the string.

Created by Mathematica  (July 19, 2005)