welcome to this tutorial from the ncvs
I'm Brad story a professor in the
department of speech language and
hearing sciences at the University of
Arizona
this tutorial is a brief explanation of
vocal tract resonances and their
Associated standing waves specifically
as they relate to vowel production
typical production of vowels is based on
vibration of the vocal folds providing a
Sound Source that is filtered by the
acoustic characteristics of the vocal
tract
the sound that radiates at the lip
opening contains information of both the
source and filter
an example spectrum of a voice Source
signal is shown here it contains a
fundamental frequency component and many
harmonics whose amplitudes decrease as a
function of increasing frequency
an example spectrum of the output
pressure at the lip opening is shown
here it contains exactly the same
fundamental frequency component and
harmonics except that their amplitudes
have been modified by the filtering
effect of the vocal tract
a spectral envelope is a curve that
encompasses the amplitudes of the
harmonics in a spectrum
a spectral envelope has been constructed
for both The Source spectrum and output
Spectrum
The Source spectral envelope indicates
the decrease in harmonic amplitudes as
frequency increases whereas the envelope
of the output Spectrum clearly shows
distinct regions or Peaks where the
harmonic energy has been enhanced
relative to other regions
these regions of the spectrum
with enhanced energy are called formants
or formant frequencies and result from
the acoustic effect of the vocal tract
on the source Spectrum
to better understand how the vocal tract
enhances the source Spectrum it is often
easiest to begin by imagining the vocal
tract to be a simple tube of uniform
cross-section extending from the vocal
folds to the lips
the tube is considered to be closed at
the end closest to the larynx and open
at the lips
the vibration of the vocal folds does
open and close the glottis that is the
airspace between the vocal folds at the
closed end to inject sound into the
vocal tract
the size of the glottal opening however
is small compared to the cross section
of the vocal tract and there is
potentially much time during each vocal
fold vibratory cycle that the glottis is
actually closed
both of these conditions motivate the
use of a closed open tube as a model of
the vocal tract
note also that the nearly 90 degree Bend
in an actual vocal tract is not
represented by the uniform tube
although the band is biomechanically
significant for providing an efficient
means of deforming the tube shape during
speech it is a much less significance
from an acoustic perspective
the average vocal tract length for an
adult male is about 17 and a half
centimeters
a closed open uniform tube of this
specific length can be characterized
acoustically by the frequency response
function shown here
the frequency response function or
sometimes called a transfer function is
not a spectrum of a particular sound but
rather indicates the effect that the
uniform tube would have on any frequency
component of a sound propagating through
it
the peaks in the frequency response
function correspond to the resonances or
natural frequencies of the particular
tube configuration and they can be
considered the origin of the formant
frequencies that is the resonances of
the vocal tract enhance the amplitudes
of particular regions of frequencies
when a sinusoidal wave with a frequency
matched to the first Resonance of the
uniform tube is injected through a tiny
aperture at the closed end a standing
wave is produced within the tube that
extends along its length
in the case shown here the frequency of
the input sinusoid is 500 Hertz the same
frequency as the first tube resonance
the acoustic pressure along the tube
length is shown as a variation in color
the deepest red indicates large positive
pressure the deepest blue indicates
large negative pressure
and light green represents zero or
atmospheric pressure
the plot just above the tube also
indicates pressure along the tube length
but is shown as a traditional
two-dimensional plot of pressure versus
distance from the closed end or glottis
although the input frequency is 500
Hertz the animation is shown in extreme
slow motion so that the pressure
variations can be visually trapped
note that the pressure oscillates
between large positive values to large
negative values at the closed end and
tapers to zero at the open end
that is the closed end provides
containment of the error and allows for
a buildup of pressure whereas at the
open end the pressure must become equal
to the atmospheric pressure which is
zero
these are known as boundary conditions
that determine the pattern of pressure
variation that can be reinforced by the
specific configuration of the tube
when the input frequency is aligned with
the first Resonance of a closed open
tube the standing wave produced is one
quarter of the wavelength
for this reason closed open tubes are
sometimes referred to as a quarter wave
resonator or a quarter wave tube
in this case a sinusoidal wave with a
frequency matched to the second
Resonance of the uniform tube is
injected at the closed end again
producing a standing wave
this time however the pattern is
different there are now two regions
along the vocal tract length at which
the pressure oscillates between large
positive and negative pressures and two
regions at which the pressure remains at
zero
the regions with zero pressure are
located one-third of the length of the
tube
and also at the lips
even though the pattern is different the
boundary conditions at the closed and
open ends of the tube are met just as in
the previous animation
in this case three quarters of the
wavelength comprise the standing wave
in this case a sinusoidal wave with a
frequency matched to the third Resonance
of the uniform tube is injected at the
closed end producing a standing wave
with three zero pressure nodes
the pattern of pressure is different
than the two previous animations but
again the boundary conditions at the
closed and open ends of the tube are met
for comparison to the previous standing
wave animations the sinusoidal wave
injected at the closed end in this case
has a frequency 750 Hertz that is not
aligned with a resonance
a standing wave is not produced at this
frequency rather the amplitude of the
input sound is highly attenuated as it
propagates in the tube
thank you
the boundary condition of zero pressure
at the open end of the tube is an
idealization
in reality the section of air and
vibration at the lips transmits a small
amount of acoustic energy that radiates
outward away from the end of the tube
while typically far less than one
percent of the total energy produced
during speech or singing
this is the sound that is actually heard
by a listener
this correction
also slightly increases the effective
length of the tube and results in a
lowering of each resonance frequency as
can be seen in the frequency response
function
standing waves corresponding to each of
the first three resonances of the
uniform tube will be shown again but
with the correction for sound radiation
the patterns of pressure variation
appear to be nearly identical to those
in the idealized cases
the difference is simply that the
frequencies at which the standing waves
occur are slightly lower than before
note that in each case the upward and
downward movement of the small black dot
at the right side of each pressure plot
indicates the radiated pressure
foreign
although the uniform tube is useful for
understanding the basic Acoustics of the
vocal tract it is not a realistic
configuration
in these two cases the uniform tube will
morph into a tubular version of an e
vowel on the left
and an ah vowel on the right both of
which were measured from a real person
using magnetic resonance imaging
as the vocal track shape changes the
corresponding frequency response
functions will also change
the deformation of the uniform tube into
realistic vocal tract configurations
shifts the resonance frequencies upward
or downward into a pattern that is
representative of each respective vowel
standing waves will now be demonstrated
for the first three resonances of the
vowels e and ah
the first Resonance of this particular e
vowel is 227 Hertz as can be seen
however the standing wave associated
with this frequency is similar to the
standing wave observed for the first
Resonance of the uniform tube
which had a frequency of 500 Hertz
that is the pressure oscillates between
large positive and large negative values
in the back portion of the vocal tract
and tapers to nearly but not quite zero
at the lips the difference is that the
pressure variation along the vocal tract
length is modified by the expansions and
constrictions that form the
e-configuration
the first Resonance of this particular
ah vowel is 811 Hertz but again the
standing wave associated with this
frequency is similar to the standing
wave observed for the first Resonance of
both the uniform tube and the E vowel
the pressure variation along the vocal
tract length however is modified by the
constrictions and expansions that form
the ah configuration
the standing waves for the second and
third resonances of the E and ah vowels
will now be played consecutively
watch carefully for similarities in
locations of zero pressure and
differences of pressure amplitudes in
other regions of the vocal tract
you may have observed that the locations
of zero pressure or nodes is quite
similar for standing waves associated
with each resonance even though the
frequency of those resonances is quite
different due to the specific vocal
tract configuration
as we produce speech or sing the shape
of the vocal tract undergoes almost
continuous change in order to shift the
frequencies at which standing waves will
be supported
this allows for easy passage of those
frequency components aligned with a
resonance
through the vocaltrex system and to the
outside world
the continual change in the frequency
bands that are enhanced or suppressed by
this process
is the means by which talkers encode a
message that is transmitted to a
listener
all of the examples shown thus far have
been based on a vocal tract length of
17.5 centimeters
as mentioned previously this is a
typical length of an adult male vocal
tract
for comparison
a typical length of an adult female
vocal tract is about 15 and a half
centimeters whereas a two-year-old child
has a vocal tract that is about
10.5 centimeters long
this animation shows the frequency
response function of the uniform tube as
the length is shortened from 17 and a
half to ten and a half centimeters
as can be observed the shortening of
vocal tract length systematically
increases the frequency of each
resonance
had shorter vocal tracts been used for
animating the standing waves at each
resonance frequency the patterns of
pressure variation along the vocal tract
would have appeared essentially the same
the difference would simply be that the
standing waves occur at higher
frequencies
this concludes the tutorial on vocal
tract resonances and their Associated
standing waves