Sound Waves

• When a tuning fork prong moves forward (outward):

  • It compresses the adjacent air layer
  • Density and pressure increase above normal
  • A compression pulse forms and travels forward

• When the prong moves backward (inward):

  • It creates a rarefaction in the adjacent layer
  • Density and pressure decrease below normal
  • A rarefaction pulse travels forward

• Key relationships:

  • Maximum pressure occurs at zero displacement (compression)
  • Minimum pressure occurs at zero displacement (rarefaction)
  • Normal pressure occurs at maximum displacement
  • The pressure wave and displacement wave are 90° out of phase

• Each cycle produces alternating compression-rarefaction pulses that propagate at the speed of sound

| Property | Closed Organ Pipe | Open Organ Pipe | |---------|-------------------|-----------------| | Boundary conditions | Closed end: pressure antinode (displacement node)<br>Open end: pressure node (displacement antinode) | Both ends: pressure nodes (displacement antinodes) | | Fundamental mode | $l = \lambda/4$ | $l = \lambda/2$ | | Fundamental frequency | $\nu_0 = v/4l$ | $\nu_0 = v/2l$ | | Harmonic frequencies | $\nu = (2n+1)\nu_0$ where $n=0,1,2...$<br>(Only odd harmonics) | $\nu = n\nu_0$ where $n=1,2,3...$<br>(All harmonics) | | Node/antinode pattern | $n$ internal pressure nodes<br>$(n+1)$ internal pressure antinodes | $(n-1)$ internal pressure nodes<br>$(n-1)$ internal pressure antinodes | | Sound quality | Less rich (missing even harmonics) | Richer (all harmonics present) |

Sources of error in resonance column experiments include:

1. End correction effects:

   • Sound waves extend slightly beyond the open end
   • Minimize by using the two-resonance method (λ = 2(l₂-l₁))

2. Temperature variations:

   • Sound speed changes ~0.6 m/s per °C
   • Control by recording temperature and applying correction formula

3. Tuning fork issues:

   • Frequency uncertainty or incorrect striking technique
   • Use calibrated forks and proper striking on rubber pad

4. Water level determination:

   • Parallax errors in reading position
   • Use precise measurement tools and eye-level readings

5. Air humidity:

   • Water vapor affects sound propagation
   • Record humidity and apply appropriate corrections

To measure speed of sound in a solid rod with Kundt's tube:

1. The rod is clamped at its middle point and attached to a disc at one end

2. When vibrated longitudinally, the rod behaves like an open organ pipe

3. In fundamental mode, the rod's length (l) equals half its wavelength: λrod = 2l

4. The rod vibration creates standing waves in the gas (usually air) inside the tube

5. By measuring distance between powder heaps (Δl), the wavelength in air is found: λair = 2Δl

6. The ratio of speeds is: vrod/vair = λrod/λair = 2l/2Δl = l/Δl

7. Therefore: vrod = (l/Δl) × vair

This technique works because the vibration frequency must be identical in both media, allowing for direct comparison of wavelengths to determine velocity ratios.

Beats are periodic variations in sound intensity that occur when two waves of slightly different frequencies interfere.

Key characteristics:

• Created when two sound waves with frequencies f₁ and f₂ superimpose
• The beat frequency equals |f₁ - f₂|, the absolute difference between the component frequencies
• At certain moments, the waves are in phase (constructive interference) creating maximum amplitude
• At other moments, the waves are out of phase (destructive interference) creating minimum amplitude
• The amplitude varies with the beat frequency while the wave oscillates at the average frequency (f₁+f₂)/2

Example: When a 440 Hz and 444 Hz tuning fork sound together, 4 beats per second are heard as the sound intensity rhythmically varies.

Beats are periodic variations in sound intensity that occur when two sound waves of slightly different frequencies interfere.

When two waves with frequencies f₁ and f₂ interfere:

• They create a resultant wave with amplitude that varies with time
• The resultant wave has a carrier frequency equal to the average: (f₁+f₂)/2
• The amplitude modulation has frequency equal to |f₁-f₂|

Mathematically, for two waves with equal amplitude p₀:

1. The resultant pressure variation can be expressed as: p = 2p₀cos(Δω/2)(t-x/v)·sin(ω̄)(t-x/v) where Δω = |ω₁-ω₂| and ω̄ = (ω₁+ω₂)/2

The human ear perceives this as a single tone (at the average frequency) that gets louder and softer at the beat frequency |f₁-f₂|. Beats are audible when the frequency difference is less than about 16 Hz.

For a stationary observer detecting sound from a moving source:

$\nu' = \frac{v}{v - u_s}\nu_0$

Where:

• $\nu'$ = apparent frequency detected by observer
• $\nu_0$ = actual frequency emitted by source
• $v$ = speed of sound in the medium
• $u_s$ = speed of the source

Key insight: When the source approaches the observer, the apparent frequency increases (higher pitch); when moving away, the frequency decreases (lower pitch).

Note: This formula assumes the source moves directly toward or away from the observer.

Reverberation:

• The persistence of sound after the original sound has stopped
• Results from multiple reflections of sound waves from various surfaces
• Creates a series of increasingly close sound reflections that blend into a continuous decaying sound

Difference from echo:

• Echo: Distinct, separated repetition of sound heard after the original sound
• Reverberation: Overlapping reflections creating a continuous decaying sound

Reverberation time:

• Definition: Time taken for reverberant sound to decrease in intensity by a factor of 10^6
• Significance in building design:
  • Too long: Disturbs clarity as sounds overlap with previous signals
  • Too short: Creates "acoustically dead" spaces with poor sound quality
  • Optimal balance needed between clarity and richness of sound
  • Musical venues typically benefit from moderate reverberation
  • Speech-focused venues require shorter reverberation times

Controlling reverberation requires careful selection of sound-absorbing and reflecting materials.

The simplest rule for determining frequency shifts in the Doppler effect:

• If the distance between source and observer is decreasing → frequency increases
• If the distance between source and observer is increasing → frequency decreases

This applies regardless of whether it's the source moving, the observer moving, or both.

Examples:

1. Source approaching observer: Distance decreasing → frequency increases
2. Source moving away from observer: Distance increasing → frequency decreases
3. Observer moving toward source: Distance decreasing → frequency increases
4. Observer moving away from source: Distance increasing → frequency decreases

Practical test: If you're standing on a platform and a whistling train passes by, you'll hear a higher pitch as it approaches (decreasing distance), then a sudden drop in pitch as it passes (changing from decreasing to increasing distance).

This rule works for all variations of the Doppler effect and avoids having to remember multiple formulas.

Sound intensity patterns during reverberation:

1. Temporal structure:

   • Direct sound arrives first with highest intensity
   • Early reflections follow with decreasing amplitude at increasing frequency
   • Late reflections merge into continuous decay pattern
   • The overall envelope follows exponential decay (linear on logarithmic scale)

2. Key acoustic parameters:

   • Reverberation time (RT60): Time for sound to decay by 60dB
   • Early Decay Time (EDT): Rate of initial decay, critical for perceived clarity
   • Clarity (C50/C80): Ratio of early to late energy, measured in dB
   • Definition (D50): Percentage of energy arriving within first 50ms
   • Bass Ratio: Low-frequency reverberation relative to mid-frequency

3. Quality determinants:

   • Room geometry affects reflection patterns
   • Surface materials determine absorption coefficients
   • Volume-to-surface ratio influences decay rate
   • Diffusion elements prevent flutter echoes and standing waves

Example: Symphony halls typically have RT60 of 1.8-2.2 seconds with balanced frequency response, while lecture halls need shorter RT60 (0.7-1.0s) to maximize speech intelligibility.