Sound Waves

An experimental setup demonstrating sound cannot be polarized would include:

• A sound source producing consistent sound waves (e.g., a drum or speaker)
• A barrier with a narrow slit that can be rotated to different orientations
• A sound detector or listener positioned on the opposite side of the barrier
• Measurements of sound intensity at various slit orientations

Results would show:

• Consistent sound intensity regardless of slit orientation
• No preferential filtering based on the angular position of the slit
• Equal transmission whether the slit is vertical, horizontal, or at any angle

This confirms the longitudinal nature of sound waves, where particle displacement occurs only along the propagation direction.

Mathematical Relationships:

For open organ pipes (open at both ends):

• Frequencies: ν = nv/2L where n = 1, 2, 3...
• All harmonics present: ν₀, 2ν₀, 3ν₀, 4ν₀...
• Fundamental: ν₀ = v/2L (when n=1)

For closed organ pipes (closed at one end):

• Frequencies: ν = (2n-1)v/4L where n = 1, 2, 3...
• Only odd harmonics present: ν₀, 3ν₀, 5ν₀...
• Fundamental: ν₀ = v/4L (when n=1)

Physical Principle Explaining Difference: This difference arises from boundary conditions:

• Open ends must be pressure nodes (displacement antinodes)
• Closed ends must be pressure antinodes (displacement nodes)

These boundary constraints force:

1. Open pipes to contain an integer number of half-wavelengths
2. Closed pipes to contain odd multiples of quarter-wavelengths

The richer harmonic content of open pipes (containing all harmonics) explains why they produce a fuller, more complex tone than closed pipes (containing only odd harmonics).

The resonance column apparatus determines sound speed through these steps:

1. A tuning fork vibrates above a water-filled glass tube
2. The water level is adjusted until maximum resonance (loudness) occurs
3. For first resonance, the air column length (l₁) equals λ/4 (plus end correction d)
4. For second resonance, the length (l₂) equals 3λ/4 (plus end correction d)
5. The wavelength is calculated: λ = 2(l₂ - l₁)
6. The speed of sound is determined using: v = λν

where ν is the tuning fork frequency. This method effectively eliminates the end correction factor from calculations.

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

Kundt's tube apparatus works through these key principles:

• A cylindrical tube is closed at both ends, one by a disc and the other by a movable piston
• The tube contains a gas and fine powder sprinkled along its length
• A metal rod attached to the disc is clamped at its middle and set into longitudinal vibration
• When resonance occurs, standing waves form in the gas
• The powder collects at displacement nodes, spaced at regular intervals (Δl)
• The wavelength in the gas equals 2Δl
• The speed of sound is calculated using v = 2Δl × f, where f is the vibration frequency

Example: By measuring the distance between powder heaps and knowing the vibration frequency, one can determine sound velocity in any gas enclosed in the tube.

The amplitude oscillation occurs because:

1. The term 2p₀cos(Δω/2)(t-x/v) represents the varying amplitude of the resultant wave

   • When cos(Δω/2)(t-x/v) = 1, amplitude reaches maximum of 2p₀
   • When cos(Δω/2)(t-x/v) = 0, amplitude becomes zero
   • When cos(Δω/2)(t-x/v) = -1, amplitude reaches -2p₀ (effectively 2p₀ with inverted phase)

2. The actual amplitude follows |A| = |2p₀cos(Δω/2)(t-x/v)|, oscillating between 0 and 2p₀

3. The beat frequency calculation:

   • The cosine completes one full cycle when its argument changes by 2π
   • Time for one beat cycle: 2π/(Δω/2) = 4π/Δω
   • Beat frequency = 1/(4π/Δω) = Δω/4π = |f₁-f₂|

This means the beat frequency equals the absolute difference between the frequencies of the interfering waves.

Step 1: Convert train speed to m/s $u_s = 36 \text{ km/h} = \frac{36 \times 10^3 \text{ m}}{3600 \text{ s}} = 10 \text{ m/s}$

Step 2: Use the Doppler effect formula and solve for actual frequency $\nu_0$: $\nu_0 = \frac{v - u_s}{v} \times \nu'$

Step 3: Substitute values: $\nu_0 = \left(1 - \frac{10}{340}\right) \times 12.0 \text{ kHz}$ $\nu_0 = 0.9706 \times 12.0 \text{ kHz}$ $\nu_0 = 11.6 \text{ kHz}$

Note: The actual frequency is lower than the detected frequency because the train approaches the observer (decreasing separation causes frequency increase).

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.

Key acoustic design principles for auditoriums:

1. Sound reflection optimization:

   • Use curved ceilings to distribute sound uniformly to all seating areas
   • Implement parabolic walls to maintain consistent sound across the width
   • Balance reflections to maintain good loudness without creating echo
   • Design reflection patterns to support, not interfere with, direct sound

2. Reverberation management:

   • Musical venues: Moderate reverberation (0.8-2.0 seconds) enhances quality
   • Speech venues: Shorter reverberation (0.5-1.0 seconds) improves clarity
   • Control reverberation time through strategic material placement
   • Avoid "acoustically dead" spaces (too little reverberation)

3. Echo prevention:

   • Eliminate long-delayed reflections that arrive as distinct echoes
   • Design reflection paths to minimize interference with subsequent sounds
   • Use diffusive surfaces to break up sound waves

4. Material selection:

   • Strategic placement of sound-absorbing materials on walls, ceiling, floor
   • Balance reflective and absorptive surfaces based on venue purpose
   • Consider audience absorption in calculations (filled vs. empty venue)

5. Uniform sound distribution:

   • Design for consistent sound intensity throughout the space
   • Eliminate dead spots and areas of excessive sound intensity

Strategic Sound Reflection vs. Sound Absorption:

• Sound reflection directs sound waves toward audience areas, maintaining loudness and ensuring sound reaches distant seats. Properly designed reflective surfaces (curved ceilings/walls) distribute sound evenly throughout the space.

• Sound absorption reduces reverberation time and prevents echo by dampening reflected sound using specialized materials on surfaces.

Balance is necessary because:

• Too much reflection creates excessive reverberation and echo, causing sound clarity issues
• Too much absorption creates an "acoustically dead" space with insufficient sound intensity
• Optimal design requires controlled reflection for sound distribution while maintaining appropriate reverberation time (typically 1.5-2.5 seconds for music, 0.7-1.2 seconds for speech)
• Different performance types require different acoustic properties (music benefits from some reverberation, speech requires more clarity)

Successful auditorium design strategically positions both reflective and absorptive elements to achieve the desired acoustic environment.