Question

You live on a busy street, but as a music lover, you want to reduce the traffic noise. (a) If you install special soundreflecting windows that reduce the sound intensity level (in dB) by 30 dB, by what fraction have you lowered the sound intensity (in W\(/\)m\(^2\))? (b) If, instead, you reduce the intensity by half, what change (in dB) do you make in the sound intensity level?

Short answer

(a) Intensity reduced by 1/1000. (b) Change is -3 dB.

Step-by-step solution

Understanding the dB Scale

The decibel (dB) scale is logarithmic and measures sound intensity levels. A change of 10 dB represents a tenfold change in intensity.

Converting dB Reduction to Intensity Reduction

A reduction of 30 dB means \(10^{rac{-30}{10}} = 10^{-3}\).Thus, the intensity is reduced to \( \frac{1}{1000} \) of its original value.

Calculating Fraction of Intensity Reduction

The fraction by which the sound intensity is reduced is \( \frac{1}{1000} \), meaning the sound intensity is reduced to 0.1% of its original level.

Understanding Half Reduction in Intensity

A reduction by half means the new intensity is \( \frac{1}{2} \) of the original intensity.

Calculating dB Change for Half Intensity Reduction

Use the formula for intensity level difference: \(\Delta L = 10 \log_{10} \left( \frac{I_2}{I_1} \right)\),where \( I_2 = \frac{1}{2} I_1 \).Substitute and calculate:\(\Delta L = 10 \log_{10} \left( \frac{1}{2} \right) = -3 \) dB.

Key concepts

Decibel Scale

The decibel (dB) scale is a unit of measurement used to represent sound intensity levels. It is particularly useful because it translates large variations in sound intensity into a more manageable range of numbers. This scale is logarithmic, not linear, meaning each step on the scale represents a multiplicative change in intensity. For instance, an increase in sound intensity by 10 dB equates to a tenfold increase in intensity, allowing us to easily comprehend very large or very small numbers without losing precision or clarity.
When you hear that the sound level is reduced by 30 dB, you can imagine it as dropping down several levels of intensity, specifically reducing the sound power to one-thousandth (\(\frac{1}{1000}\)) of the original level. This little trick of the dB scale enables better communication of sound reductions or increases, especially in fields like acoustics, where sound intensity can vary drastically from quiet whispers to loud jet engines.

Logarithmic Scale

A logarithmic scale is different from a linear scale. Instead of increasing in equal steps, each step in a logarithmic scale is a constant factor, such as tenfold, which simplifies handling large ranges of values. This is the principle behind the decibel system for sound.

• Logarithmic scales help represent hard-to-grasp phenomena like sound and earthquakes.
• They convert multiplicative relationships into additive ones, providing clarity.

In sound measurements, this makes it much easier to handle the wide spectrum from the soft rustling of leaves to the roaring of a jet. By using a logarithm to express changes in intensity, the decibel scale offers a pleasant way to translate these differences into understandable terms. For example, a change of 10 dB means the intensity changes by a factor of 10. Thus, a 20 dB change represents a 100-fold difference due to the power of the logarithm being raised with each jump. This simplifies large multiplicative changes of sound into more manageable additive steps.

Sound Reduction

Sound reduction is the process of decreasing the intensity and therefore the impact of unwanted noise. Understanding sound reduction often involves grasping the concept of decibels and logarithmic scales, as these help us quantify how effective noise reduction measures are.

• For instance, reducing sound intensity by 30 dB is a significant decrease as it represents a reduction by 99.9% to just 0.1% of the original intensity.
• Alternatively, reducing the intensity by half (\(\frac{1}{2}\)) results in a reduction of about 3 dB, a far smaller scale of change.

Each method of sound reduction, whether through special windows, insulation, or soundproofing materials, has its specific level of efficacy. The decibel and logarithmic scales help us appreciate these differences and understand how effective each method might be. By learning how various sound reduction techniques are measured and what those measurements translate to in terms of real-world effect, one can make informed decisions on the best way to mitigate noise in various environments such as homes, offices, or urban spaces.