1. Hearing Biomechanics
Standing waves

2. 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!

3. 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

4. Formulation of Standing Waves
Pink line represents wave travelling right to left along the string
Blue line represents wave travelling from left to right along the string
Black line = sum of left and right-travelling waves = STANDING WAVE
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

5. Mathematical formulation of Standing Waves
Wave moving right to left (pink wave)
Wave moving left to right (blue wave)
Total wave function (black wave):
Amplitude depends on position
Zero y-displacement (node) when sin(kx) = 0
Maximum y-displacement (y=2A) when sin(kx)=+/- 1

6. String Harmonics L Frequency 2f1 3f1 4f1 5f1 6f1 L = Length of string
T = Tension
m = mass of string
5f1
6f1

7. Nodes and Antinodes L = Length of string T = Tension
Standing waves have stationary nodes and anti-nodes
Fundamental
Second Harmonic
Third Harmonic
L = Length of string
T = Tension
m = mass of string

8. Hearing: Mechanics Closed-end Air column – Ear Canal Ear Drum
Auditory Canal
Ear Drum

9. Hearing: Mechanics
Red particles: extremes of motion

10. 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

11. 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

12. 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.

13. 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

14. Boundary Behavior Reflection Reflection and Transmission Free boundary
From low to high density
Fixed boundary
From high to low density

15. Reflection and Transmission at Eardrum
Partial reflection and transmission
Minimize Reflection and Maximize Transmission = Optimized Hearing Sensitivity

16. Reflection and Transmission at Eardrum
Incident Wave
Reflected Wave
Transmitted Wave
Boundary Conditions
Continuous
Differentiable (no kink)
A+B=C
A-B = (k2/k1)C

17. Reflection and Transmission at Eardrum
𝐴𝑡𝑟𝑎𝑛𝑠 𝐴 𝑖𝑛𝑐 =
𝐴𝑟𝑒𝑓𝑙 𝐴 𝑖𝑛𝑐 =
Z = 𝜌. 𝒗
Impedance = Density . Speed Sound
Intensity ∝ (Amplitude)2