Consider a sound wave in air that has displacement amplitude 0.0200 mm. Calculate the pressure amplitude for frequencies of (a) 150 Hz; (b) 1500 Hz; (c) 15,000 Hz. In each case compare the result to the pain threshold, which is 30 Pa.

Verified step by step guidance

1. Understand that the pressure amplitude of a sound wave is related to its displacement amplitude, frequency, and the properties of the medium (air in this case). The formula to calculate pressure amplitude (ΔP) is ΔP = ρvωs₀, where ρ is the air density, v is the speed of sound in air, ω is the angular frequency, and s₀ is the displacement amplitude.
2. Convert the displacement amplitude from millimeters to meters: s₀ = 0.0200 mm = 0.0200 × 10⁻³ m.
3. Calculate the angular frequency (ω) for each given frequency using the formula ω = 2πf, where f is the frequency. For example, for 150 Hz, ω = 2π × 150.
4. Use the known values for air density (ρ ≈ 1.21 kg/m³) and speed of sound in air (v ≈ 343 m/s) to calculate the pressure amplitude for each frequency using the formula ΔP = ρvωs₀.
5. Compare the calculated pressure amplitude for each frequency to the pain threshold of 30 Pa to determine if the sound wave exceeds this threshold.

Key Concepts

Sound Waves

Sound waves are longitudinal waves that propagate through a medium, such as air, by compressions and rarefactions. The displacement amplitude refers to the maximum change in position of particles in the medium from their equilibrium position, which is crucial for determining the wave's intensity and pressure amplitude.

Pressure Amplitude

Pressure amplitude is the maximum change in pressure caused by a sound wave as it travels through a medium. It is directly related to the displacement amplitude and frequency of the wave, and is a key factor in determining the loudness of the sound. Calculating pressure amplitude involves understanding the relationship between these variables.

Frequency and Pain Threshold

Frequency refers to the number of oscillations or cycles per second of a sound wave, measured in Hertz (Hz). Different frequencies affect the pressure amplitude, which can be compared to the pain threshold of 30 Pa, the level at which sound becomes physically painful to humans. Understanding this comparison helps assess the potential impact of sound waves at various frequencies.