Question

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.

Short answer

Pressure amplitudes: 7.77 Pa (150 Hz), 77.7 Pa (1500 Hz), 777 Pa (15,000 Hz). The 1500 Hz and 15,000 Hz frequencies exceed the pain threshold of 30 Pa.

Step-by-step solution

Understand the Relationship between Displacement Amplitude and Pressure Amplitude

The pressure amplitude \( P_0 \) in a sound wave is related to the displacement amplitude \( s \), the angular frequency \( \omega \), the medium density \( \rho \), and the speed of sound \( v \). The formula is given by \( P_0 = \rho v \omega s \).

Identify Constants and Quantities

We need the density of air \( \rho = 1.21\, \text{kg/m}^3 \) and the speed of sound in air \( v = 343\, \text{m/s} \). The displacement amplitude is given as \( s = 0.0200\, \text{mm} = 0.0200 \times 10^{-3}\, \text{m} \).

Calculate Angular Frequency for Each Frequency

Angular frequency is calculated using \( \omega = 2\pi f \). Compute \( \omega \) for each frequency:(a) For \( f = 150\, \text{Hz} \): \( \omega = 2\pi \times 150 \approx 942 \text{ rad/s} \).(b) For \( f = 1500 \text{ Hz} \): \( \omega = 2\pi \times 1500 \approx 9420 \text{ rad/s} \).(c) For \( f = 15000 \text{ Hz} \): \( \omega = 2\pi \times 15000 \approx 94200 \text{ rad/s} \).

Calculate Pressure Amplitude for Each Frequency

Use the formula \( P_0 = \rho v \omega s \) to compute pressure amplitude:(a) For \( 150 \text{ Hz} \): \[ P_0 = 1.21 \times 343 \times 942 \times 0.0200 \times 10^{-3} \approx 7.77 \text{ Pa} \].(b) For \( 1500 \text{ Hz} \): \[ P_0 = 1.21 \times 343 \times 9420 \times 0.0200 \times 10^{-3} \approx 77.7 \text{ Pa} \].(c) For \( 15000 \text{ Hz} \): \[ P_0 = 1.21 \times 343 \times 94200 \times 0.0200 \times 10^{-3} \approx 777 \text{ Pa} \].

Compare with Pain Threshold

The pain threshold is 30 Pa. Compare calculated pressure amplitudes:(a) For \( 150 \text{ Hz} \), \( 7.77 \text{ Pa} \) is below the pain threshold.(b) For \( 1500 \text{ Hz} \), \( 77.7 \text{ Pa} \) is above the pain threshold.(c) For \( 15000 \text{ Hz} \), \( 777 \text{ Pa} \) is well above the pain threshold.

Key concepts

Displacement Amplitude

When discussing sound waves, displacement amplitude refers to the maximum amount by which particles in the medium move from their rest position due to the wave. Imagine a tiny object floating on a wave. The distance it moves up and down measures the displacement amplitude.
This is important because it helps quantify how "strong" or "intense" the sound wave is in terms of physical movement. In the context of the original exercise, the displacement amplitude was given as 0.0200 mm or 0.0200 x 10^{-3} m.

• Displacement amplitude is directly related to how much energy is delivered by the wave.
• A larger displacement amplitude means the particles are further displaced, often resulting in a louder sound.

Understanding displacement amplitude is crucial for calculating the pressure amplitude, which involves additional characteristics of the medium and the wave.

Frequency and Angular Frequency

Frequency, represented by the letter \( f \), is the number of vibrations or cycles per second of a wave, measured in hertz (Hz). For example, a frequency of 150 Hz means that the wave cycles 150 times per second. Frequency is a fundamental property of waves affecting both the pitch of sound and the wave's energy.
Angular frequency \( \omega \) offers another way to describe how fast the wave oscillates, expressed in radians per second. The angular frequency is calculated using the formula \( \omega = 2\pi f \).

• Angular frequency is a convenient format in physics, especially when working with trigonometric functions.
• Both frequency and angular frequency help characterize the timing and periodicity of wave phenomena.

In the exercise, calculating \( \omega \) forms part of determining the wave's pressure amplitude, a direct measure of the force each cycle of the wave imposes.

Density of Air and Speed of Sound

The density of air \( \rho \) and speed of sound \( v \) are essential parameters affecting how sound waves travel. For most calculations involving air, the density is typically around 1.21 kg/m³, while the speed of sound is about 343 m/s.
These values are crucial: the combination of air density and speed influences the wave's pressure amplitude and energy transfer.

• Density reflects how much matter is present in a given volume of air.
• The speed of sound depends on various factors, such as air temperature, and represents how fast sound travels through the medium.

In the sound wave calculations, these constants determine how efficiently the wave energy is propagated through the air, impacting both the calculated displacement and pressure amplitudes.

Pain Threshold in Acoustics

In acoustics, the pain threshold is the level at which sound becomes physically uncomfortable to hear, typically around 30 Pa (pascals) for humans. This concept helps gauge the intensity of sound pressure that can be tolerated by the ear without causing damage or considerable discomfort.
Sound pressure levels above the pain threshold can cause harm and are considered dangerous. Comparisons against this value are critical, especially in environments where noise exposure risks are evaluated.

• Understanding this threshold helps establish safe listening environments and set regulations for exposure to loud sounds.
• It acts as a benchmark to assess the relative safety of sound pressure levels in surrounding environments.

In the exercise, calculated pressure amplitudes are compared against this standard to elucidate whether the sound waves at various frequencies pose a risk to hearing.