Consider a sound wave in air that has displacement amplitude 0.1mm.calculate the pressure amplitude for frequensis of:
1) 500 Hz
2) 20,000 Hz
pmax = A*B*k
where k = 2πf/v
v = 344 m/s speed of sound and B = 1.42x10^5 Bulk modulus of air
change amplitude to meters from mm.
To find the pressure amplitude for the given frequencies, we can use the formula:
Pressure Amplitude = Density × Velocity × Displacement Amplitude × Angular Frequency
where:
Density of air (ρ) = 1.2 kg/m³
Speed of Sound in air (v) = 343 m/s (approximately)
Displacement Amplitude (A) = 0.1 mm = 0.1 × 10⁻³ m
Angular Frequency (ω) = 2πf, where f is the frequency
1) For a frequency of 500 Hz:
Angular Frequency (ω) = 2π × 500 = 1000π rad/s
Pressure Amplitude = 1.2 × 343 × (0.1 × 10⁻³) × 1000π ≈ 128π × 10⁻⁵ Pa
2) For a frequency of 20,000 Hz:
Angular Frequency (ω) = 2π × 20,000 = 40,000π rad/s
Pressure Amplitude = 1.2 × 343 × (0.1 × 10⁻³) × 40,000π ≈ 48,000π × 10⁻⁵ Pa
Therefore, the pressure amplitudes for frequencies of 500 Hz and 20,000 Hz are approximately 128π × 10⁻⁵ Pa and 48,000π × 10⁻⁵ Pa, respectively.
To calculate the pressure amplitude of a sound wave, we can use the equation:
Pressure amplitude = Displacement amplitude × ρ × ω × v
Where:
- Displacement amplitude is the given amplitude of the sound wave.
- ρ is the density of the medium (for air at room temperature, ρ ≈ 1.225 kg/m³).
- ω is the angular frequency (ω = 2πf, where f is the frequency in Hz).
- v is the speed of sound in air (for air at room temperature, v ≈ 343 m/s).
Let's calculate the pressure amplitude for the given frequencies:
1) For a frequency of 500 Hz:
- Displacement amplitude = 0.1 mm = 0.1 × 10⁻³ m
- ω = 2πf = 2π × 500 ≈ 3141.59 rad/s
- v = 343 m/s
Pressure amplitude = (0.1 × 10⁻³) × 1.225 × 3141.59 × 343 ≈ 1.118 Pa
So, for a frequency of 500 Hz, the pressure amplitude is approximately 1.118 Pa.
2) For a frequency of 20,000 Hz:
- Displacement amplitude = 0.1 mm = 0.1 × 10⁻³ m
- ω = 2πf = 2π × 20,000 ≈ 125,663.71 rad/s
- v = 343 m/s
Pressure amplitude = (0.1 × 10⁻³) × 1.225 × 125,663.71 × 343 ≈ 527.389 Pa
So, for a frequency of 20,000 Hz, the pressure amplitude is approximately 527.389 Pa.