Transcript Standing Waves

Hearing Biomechanics
STANDING WAVES
Principle of Superposition

If two or more waves combine at a given point, the resulting disturbance
is the sum of the disturbances of the individual waves.

Two traveling waves can pass through each other without being
destroyed or even altered!
ytotal ( x, t )  y1 ( x, t )  y2 ( x, t )  y3 ( x, t )  .......... yn ( x, t )
Some Results of Superposition

Two waves, same wavelength and frequency, opposite direction:
Standing Wave

Two waves, same wavelength and frequency, similar direction, different
phase: Interference

Two waves, same direction, slightly different frequency and
wavelength:Beats
Formulation of Standing Waves

Pink line represents wave travelling

Blue line represents wave travelling

Black line = sum of left and right-travelling waves = STANDING WAVE
right to left along the string
from left to right along the string
Constructive interference of waves at ANTINODE of standing wave
(max displacement)
 Destructive interference of waves at NODE of standing wave (zero
displacement)
 Distance between successive nodes/antinodes = λ/2

Mathematical formulation of
Standing Waves

Wave moving right to left (pink wave)
y1 ( x, t )   A cos(kx  t )

Wave moving left to right (blue wave)
y2 ( x, t )  A cos(kx  t )

Total wave function (black wave):
y ( x, t )  (2 A sin kx) sin( t )
• Amplitude depends on position
• Zero y-displacement (node) when sin(kx) = 0
• Maximum y-displacement (y=2A) when sin(kx)=+/- 1
String Harmonics
L
Frequency
f1 
1 T
2L m
2f1
3f1
4f1
5f1
6f1
L = Length of string
T = Tension
m = mass of string
Nodes and Antinodes

Standing waves have stationary nodes and anti-nodes
Fundamental
Second Harmonic
1 T
f1 
2L m
Third Harmonic
L = Length of string
T = Tension
m = mass of string
Hearing: Mechanics

Closed-end Air column – Ear Canal
Auditory Canal
Ear Drum
Hearing: Mechanics
Red particles:
extremes of motion
Hearing: Mechanics

Auditory Canal ≅ 2.5cm

Canal closed by eardrum membrane

Incoming acoustic waves of certain frequency can resonate
Auditory Canal
Natural frequency of an air-filled tube of length L, closed at one end
Ear Drum
Thus sensitivity of ear is enhanced in higher frequency range : 2000Hz to 8000Hz
Hearing: Sensitivity of ears

Sensitivity changes with frequency

Measured in Loudness

Constant loudness (isophon) varies
with intensity and frequency

Unit of loudness: phon which is
normalized to intensity at frequency 1000Hz
The solid lines indicate, curves of constant loudness as a function of intensity and frequency. All sounds along the isophone appear equally
loud to the listener. The lowest isophone represents the hearing threshold.
The dip in the isophones at frequencies around 3000Hz and 8000Hz signalize that lower intensities correspond to higher loudness, this results
from the increased sensitivity of the ear due to the resonance effect in the outer ear canal.
Are standing waves only perceived

Healthy ear – all frequencies within audible limit – 20Hz to 20kHz

At frequencies of
Standing waves
Resonance

At resonant frequencies, sound gets amplified
Boundary Behavior
Reflection and Transmission
Reflection
Free boundary
From low to high density
Fixed boundary
From high to low density
Reflection and Transmission
at Eardrum

Partial reflection and transmission

Minimize Reflection and Maximize Transmission
= Optimized Hearing Sensitivity
Reflection and Transmission
at Eardrum
Incident Wave
Reflected Wave
Transmitted Wave
Boundary Conditions
1) Continuous
2) Differentiable (no kink)
A+B=C
A-B = (k2/k1)C
Reflection and Transmission
at Eardrum
𝐴𝑡𝑟𝑎𝑛𝑠
𝐴 𝑖𝑛𝑐
=
𝐴𝑟𝑒𝑓𝑙
𝐴 𝑖𝑛𝑐
=
Z = 𝜌. 𝒗
Impedance = Density . Speed Sound
Intensity ∝ (Amplitude)2