Sound Waves

Resonance Column Method:

Procedure:

1. A tuning fork of known frequency is sounded above a water-filled resonance tube
2. Water level is adjusted to find first resonance (maximum loudness)
3. Length $l_1$ of air column is measured (first resonance)
4. Water level is lowered to find second resonance
5. Length $l_2$ of air column is measured (second resonance)

Calculations:

• For first resonance: $l_1 + d = \lambda/4$ (where $d$ is end correction)
• For second resonance: $l_2 + d = 3\lambda/4$
• Subtracting: $(l_2 - l_1) = \lambda/2$
• Therefore: $\lambda = 2(l_2 - l_1)$
• Speed of sound: $v = \lambda\nu = 2(l_2 - l_1)\nu$

End correction ($d$):

• Accounts for the fact that pressure node forms slightly above the open end
• Canceled out in the formula by taking the difference between resonance positions
• Can be calculated if needed: $d = l_1 - \lambda/4$

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

The mathematical relationships in beats:

1. For original waves with frequencies ν₁ and ν₂:

   • Beat frequency = |ν₁ - ν₂|
   • Average wave frequency = (ν₁ + ν₂)/2

2. For amplitude variation:

   • The resultant wave equation: p = 2p₀cos(Δω/2·t)sin(ωt)
   • Where Δω = 2π|ν₁ - ν₂| and ω = 2π(ν₁ + ν₂)/2

3. Key relationships:

   • Frequency of absolute amplitude |A| variation = |ν₁ - ν₂|
   • Frequency of actual amplitude A variation = |ν₁ - ν₂|/2
   • The detected sound has carrier frequency (ν₁ + ν₂)/2 with amplitude modulated at |ν₁ - ν₂|

Example: When tuning a guitar, if a string produces 438 Hz while the reference tone is 440 Hz, you'll hear 2 beats per second. The closer the frequencies get, the slower the beat frequency becomes, helping achieve precise tuning.

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.

When an observer moves toward a stationary sound source:

• The apparent frequency increases: $ν' = \frac{v+u_o}{v}ν_0$ where:

• $ν'$ = apparent frequency
• $ν_0$ = actual source frequency
• $v$ = speed of sound
• $u_o$ = observer speed

• The time between receiving compression pulses decreases to $T' = \frac{vT}{v+u}$

• This occurs because the observer encounters wave crests at a faster rate than they're produced, as the observer moves into upcoming waves.

• Example: A stationary ambulance siren sounds higher in pitch when you move toward it, even though the siren itself maintains constant frequency.

The formulas differ because the physical mechanisms creating the frequency shift are fundamentally different:

• Moving observer, stationary source: $ν' = \frac{v±u_o}{v}ν_0$

• The wave pattern in the medium remains unchanged
• The observer intercepts waves at a different rate due to their motion
• The wavelength in the medium stays constant at $λ = vT$
• Only the encounter rate changes

• Moving source, stationary observer: $ν' = \frac{v}{v∓u_s}ν_0$

• The source motion physically compresses/expands wavelengths in the medium
• The actual wavelength in the medium becomes $λ' = \frac{v-u_s}{ν_0}$ or $λ' = \frac{v+u_s}{ν_0}$
• The compression/rarefaction pattern itself is altered in the medium
• This creates a fundamentally different equation form

The mathematical difference reflects that observer motion changes perception without altering the wave pattern, while source motion physically changes the wave pattern in the medium.

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.

When a source moves at supersonic speed (faster than the wave speed in the medium):

1. Wavefronts pile up and form a cone-shaped shock wave
2. The source remains at the apex of this cone as it moves
3. Spherical wavefronts intersect on the surface of this cone
4. Constructive interference creates pressure waves of very large amplitude
5. When an observer is intercepted by this cone, they hear a sonic boom

The Mach number is defined as: $\text{Mach Number} = \frac{u_s}{v}$

Where:

• $u_s$ = speed of the source
• $v$ = speed of waves in the medium

The semi-vertical angle $\theta$ of the shock wave cone is related to the Mach number by: $\sin\theta = \frac{1}{\text{Mach Number}}$

Key insight: The sonic boom continues as long as supersonic speed is maintained, not just when breaking the sound barrier.

The physical mechanism of constructive interference in shock waves:

• When a source moves supersonically, it outpaces its own sound waves
• Multiple spherical wavefronts from different emission points exist simultaneously
• These wavefronts intersect precisely along a conical surface
• At these intersection points, pressure waves from many different moments arrive simultaneously
• This creates constructive interference where wave amplitudes add together

Why this creates intense pressure changes:

• Normal sound wave: single compression/rarefaction cycle
• Shock wave: multiple compression cycles stack at exactly the same point in space and time
• The pressure gradient becomes extremely steep (nearly discontinuous)
• Air molecules have insufficient time to gradually respond to pressure changes
• This causes the characteristic "N-wave" pressure profile with sudden compression followed by rarefaction
• Energy that would normally disperse spherically becomes concentrated on the cone surface
• The pressure change can exceed 100 times that of normal conversation sound

Example: A supersonic bullet creates a shock wave strong enough to be heard as a distinctive "crack" even when the bullet itself makes minimal noise during flight.

Geometric Principles of Curved Reflective Surfaces:

• Law of Reflection - Sound waves reflect at the same angle to the normal as the incident angle (angle of incidence = angle of reflection)
• Focal properties - Curved surfaces concentrate or disperse sound based on their geometry:
  • Concave surfaces focus sound toward specific points
  • Convex surfaces scatter sound over wider areas
  • Parabolic reflectors direct parallel sound waves to a focal point (or from a source at the focus to parallel waves)

Influence on Listening Experience:

• Sound intensity distribution - Properly curved surfaces ensure uniform sound levels throughout seating areas
• Time delay management - Strategic reflection creates beneficial early reflections (arriving within 50ms of direct sound) that enhance perceived loudness
• Frequency response - Different curvatures affect different wavelengths, influencing tonal balance
• Spatial perception - Well-designed reflections create a sense of envelopment and source localization

Improper curvature can cause acoustic defects like sound focusing (hot spots), echoes, or flutter echoes that degrade the listening experience.