# AOpen | AX4L-N | Resonance in acoustic tubes 1. wavelength 2. plain wave

```Resonance in acoustic tubes
1. wavelength
2. plain wave propagation
3. reflection
4. phase matching
Two ways to measure the
period of a sine wave.
Time
Frequency = 1 / Time
Sound: pressure fluctuation
that travels through space.
Speed of sound = 35,000 cm/s
Space
Wavelength = spatial period
Wavelength = speed of sound * period duration
λ=c*T
λ=c/f
because f = 1 / T
Space
Wavelength = spatial period
1. wavelength. Sine wave has a spatial
period, peaks and valleys located in space.
Space
sound propagates from source in a
sphere.
However, sound in a tube propagates in
a plane – effectively, no curvature
However, sound in a tube propagates in
a plane – effectively, no curvature
2. Plane wave propagation
3. Reflection
Sound reflects off of surfaces
- more reflection off of hard surfaces
- less reflection off of soft surfaces
- scattered reflection off of uneven surfaces
hard
soft
uneven
Sound traveling in a closed tube
reflects off the ends of the tube.
Sound traveling in an open tube
also reflects off the ends of the tube.
Reflection off of a soft surface
The vocal tract is a tube that is open
at one end and closed at the other.
has two kinds of reflection:
1. hard surface at closed end
2. soft surface at open end
Sound waves traveling though space
interfere with each other.
A
direction
direction
B
Destructive interference: A + B = 0
A
direction
direction
B
A+B
Constructive interference: A + B = AB
A
direction
direction
B
A+B
Constructive interference: A + B = AB
1
A
direction
1
direction
B
2
A+B
Constructive interference: A + B = AB
A
direction
-1
direction
B
-1
A+B
-2
Reflected waves in a tube interfere with
each other.
constructive interference = resonance
destructive interference = nonresonance
Q: What frequencies will resonate in a
tube?
= Q: What sine waves will show
constructive interference?
two factors wavelength and tube length
key: wave must “fit” in tube
fit = reflect in phase
An example of reflecting “in phase”
- a sine wave that “fits” in a closed tube
wavelength = tube length
An example of reflecting “in phase”
- a sine wave that “fits” in a closed tube
wavelength = tube length
the reflected
wave is in phase
Δ constructive
interference
An example of reflecting “in phase”
- a sine wave that “fits” in a closed tube
wavelength = tube length
the reflected
wave is in phase
Frequency of this
resonance:
Δ constructive
interference
f = c/λ
Another example of reflecting “in phase”
- a sine wave that “fits” in a closed tube
wavelength = ½ * tube length
A general formula for calculating the
resonant frequencies of sine waves that will
resonate in a tube closed at both ends:
Fn = nc/2L
n = resonant frequency number (1,2,3, ...)
c = speed of sound (35,000 cm/s)
L = tube length (in cm)
Now consider a tube that is open at
one end, and closed at the other.
Now consider a tube that is open at one
end, and closed at the other.
Reflection from the open end is different.
Phase shift!
A sine wave that “fits” in a tube that is
open at one end, and closed at the other.
A sine wave that “fits” in a tube that is
open at one end, and closed at the other.
phase shift at
open end
A sine wave that “fits” in a tube that is
open at one end, and closed at the other.
λ
phase shift at
open end
A sine wave that “fits” in a tube that is
open at one end, and closed at the other.
λ
phase shift at
open end
resonant frequency is: f = c/(4/5*L)
Another sine wave that “fits” this tube.
resonant frequency:
f = c/(4/3L)
λ
A general formula for resonant frequencies
of tubes open at one end and closed
at the other:
fn = (2n-1)c/4L
n = resonance number (1,2,3...)
c = speed of sound (35,000 cm/s)
L = tube length (in cm)
the vowel schwa [ә]:
a tube open at one end (lips)
and closed at the other (glottis)
Vocal tract length: ~ 17.5 cm
F1 = c/4L = 35,000/70 = 500 Hz
F2 = 3c/4L = 1500 Hz
F3 = 5c/4L = 2500 Hz
Peter Ladefoged saying [ ә ]:
2500 Hz
1250 Hz
400 Hz
```