What are the sound intensity levels for sound waves of intensity (a) 3.0 x 10⁻⁶ W/m²?

Verified step by step guidance

1. Identify the formula to calculate the sound intensity level (SIL) in decibels (dB). The formula is: \( SIL = 10 \log_{10} \left(\frac{I}{I_0}\right) \), where \( I \) is the intensity of the sound wave and \( I_0 \) is the reference intensity, typically \( 1.0 \times 10^{-12} \, W/m^2 \).
2. Substitute the given intensity \( I = 3.0 \times 10^{-6} \, W/m^2 \) into the formula.
3. Use the reference intensity \( I_0 = 1.0 \times 10^{-12} \, W/m^2 \) in the formula.
4. Calculate the ratio \( \frac{I}{I_0} \) by dividing the given intensity by the reference intensity.
5. Compute \( 10 \log_{10} \left(\frac{I}{I_0}\right) \) to find the sound intensity level in decibels (dB).

Key Concepts

Sound Intensity

Sound intensity is defined as the power per unit area carried by a sound wave. It is measured in watts per square meter (W/m²) and quantifies how much sound energy passes through a given area in a specific time. Higher intensity levels correspond to louder sounds, while lower levels indicate quieter sounds.

Decibel Scale

The decibel (dB) scale is a logarithmic scale used to measure sound intensity levels. It expresses the ratio of a particular sound intensity to a reference intensity, typically the threshold of hearing (1 x 10⁻¹² W/m²). The formula to convert intensity (I) to decibels is L = 10 log10(I/I₀), where I₀ is the reference intensity.

Threshold of Hearing

The threshold of hearing is the minimum sound intensity level that the average human ear can detect, approximately 1 x 10⁻¹² W/m². This reference point is crucial for calculating sound intensity levels in decibels, as it allows for a standardized comparison of different sound intensities relative to what humans can perceive.