Physics of Sounds
Overview
Properties of vibrating systems
Free and forced vibrations
Resonance and frequency response
Sound waves in air
Frequency, wavelength, and velocity of a
sound wave
Simple and complex sound waves
Periodic and aperiodic sound waves
Fourier analysis and sound spectra
Sound pressure and intensity
The decibel (dB) scale
The acoustics of speech production
Speech spectrograms
Properties of Vibrating Systems
Some terms
• displacement: momentary
distance from restpoint B
• cycle: one complete
oscillation
• amplitude: maximum
displacement, “average”
displacement
• frequency: number of
cycles per second (hertz or
Hz)
• period: number of seconds
per cycle
• phase: portion of a cycle
through which a waveform
has advanced relative to
some arbitrary reference
point
What is the relation between
frequency (f) and period (T)?
How do these differ?
How do these differ?
How do these differ?
Another case of harmonic motion:
tuning fork
Damping
Free vibration
• As we have so far described them, the
mass-spring system and the tuning
fork represent systems in free
vibration. An initial external force is
applied, and then the system is allowed
to vibrate freely in the absence of any
additional external force. It will vibrate
at its natural or resonance frequency.
Forced vibration
• Now assume that the mass-spring
system is coupled to a continuous
sinusoidal driving force (rather than to
a rigid wall).
How will it respond?
Resonance curve
(aka: frequency response or
transfer function or filter function)
• In free vibration, the response
amplitude depends only on the initial
amplitude of displacement.
• In forced vibration, the response
amplitude depends on both the
amplitude and the frequency of the
driving force.
Resonance
Sound waves
Sound waves (cont.)
Frequency, wavelength, and
velocity of sound waves
• Wavelength: the spatial extent of one
cycle of a simple waveform. (Compare
this to period).
• If we know the frequency (f) and the
wavelength (λ) of a simple waveform,
what is its velocity (c)?
Simple vs. complex waves
• So far we’ve considered only sine
waves (aka: sinusoidal waves,
harmonic waves, simple waves, and, in
the case of sound, pure tones).
• However, most waves are not
sinusoidal. If they are not, they are
referred to as complex waves.
Examples of complex waves:
sawtooth waves
Examples of complex waves:
square waves
Examples of complex waves:
vowel sounds
Periodic vs. aperiodic waves
• So far all the waveforms we’ve
considered (whether simple or
complex) have been periodic—an
interval of the waveform repeats itself
endlessly.
• Many waveforms are nonrepetitive, i.e.,
they are aperiodic.
Some examples of aperiodic
waves:
• A sine wave can be described exactly
by specifying its amplitude, frequency,
and phase.
• How can one describe a complex wave
in a similarly exact way?
Fourier analysis
• Any waveform can be analyzed as the
sum of a set of sine waves, each with a
particular amplitude, frequency, and
phase.
How to approximate a square wave
From time-domain
to frequency-domain
Time
Frequency
Periodic vs. aperiodic waves (cont.)
• Periodic waves consist of a set of
sinusoids (harmonics, partials) spaced
only at integer multiples of some
lowest frequency (called the
fundamental frequency, or f0).
• Aperiodic waves fail to meet this
condition, typically having continuous
spectra.
Sound pressure and intensity
•
•
Sound pressure (p) = force per square centimeter
(dynes/cm2)
Intensity (I) = power per square centimeter
(Watts/cm2)
•
I = kp2
•
•
Smallest audible sound = 2 x 10-4 dynes/cm2
= 10-16 Watts/cm2
A problem: Between a just audible sound and a sound at the pain threshold,
sound pressures vary by a ratio of 1:10,000,000, and intensities vary by a ratio
of 1: 100,000,000,000,000! More convenient to use scales based on logarithms.
•
Decibels (dBSPL,IL)
•
= 20 log (p1/p0)
= 10 log (I1/I0)
where p1 is the sound pressure and I1 is the intensity of the sound of interest,
and p0 and I0 are the sound pressure and intensity of a just audible sound.
Decibel scale
Acoustics of speech production
Spectrogram