Question

Applications. Decibel Level Decrease with Distance: The decibel decrease \(G\) when moving from a sound source at a distance \(d_{n}\) to a distance \(d_{f}\) is given by $$ G=20 \log _{10} \frac{d_{f}}{d_{n}} \mathrm{d} \mathrm{B} $$ Find the decibel drop when moving twice as far from a sound source.

Short answer

The decibel drop when moving twice as far from a sound source is approximately 6.02 dB.

Step-by-step solution

Identify the Initial Distance

Let the initial distance from the sound source be represented by the variable \(d_n\). In this problem, we are not given a specific value for \(d_n\) but we know we'll be calculating for when the distance is doubled.

Find the Final Distance

To find the final distance \(d_f\), we assume we are moving twice as far from the original position. Therefore, \(d_f = 2 \times d_n\).

Substitute Distances into the Decibel Drop Formula

Use the formula for decibel decrease, \(G=20 \log_{10} \frac{d_f}{d_n} dB\), to find \(G\) when the final distance is twice the initial distance: \[ G = 20 \log_{10} \frac{2d_n}{d_n} = 20 \log_{10}(2) \approx 6.02 \quad \text{dB}. \] The value of \(\log_{10}(2)\) is approximately 0.3010.

Calculate Decibel Drop

Multiply 20 by the log value: \[ G = 20 \times 0.3010 = 6.02 \quad \text{dB}. \] This is the decibel drop when moving twice as far from the sound source.

Key concepts

Logarithms in Decibel Calculation

When it comes to understanding sound levels, logarithms play a crucial role in measuring changes in decibel (dB). Essentially, the decibel scale is logarithmic. This means it represents the ratio of a particular sound intensity to a reference level using a logarithmic scale.

In practical terms, a logarithm tells you how many times one number called the 'base' of the logarithm needs to be multiplied by itself to get another number. The equation for calculating a decibel increase or decrease typically involves the base-10 logarithm, denoted as \( \log_{10} \).

To calculate the change in decibel level when the distance from a sound source changes, you use the formula \( G = 20 \log_{10}(\frac{d_{f}}{d_{n}}) \), where \( d_{f} \) is the final distance and \( d_{n} \) is the initial distance. This particular formula reveals why a logarithmic scale is used: It allows for a wide range of sound intensities to be condensed into a smaller range of numbers, facilitating easier comprehension and comparison of sound levels.

Sound Intensity and Distance

Sound intensity, which represents the power per unit area, diminishes as the distance from the sound source increases. This is due to the fact that sound spreads out in spherical waves, and as the distance from the source increases, these waves cover a larger area.

This phenomenon is encapsulated in the concept known as the inverse square law, which states that the intensity of sound is inversely proportional to the square of the distance from the source. Consequently, when you double the distance (\( d_{f} = 2d_{n} \) in our example), the intensity drops, not to half, but to one quarter of its original level.

It's important to understand that decibels measure a ratio. So, when we are talking about the decibel level decrease due to an increase in distance, we're looking at the ratio of the sound intensity at two different points. This relational aspect is where logarithmic calculation becomes essential as it succinctly represents the rate at which the sound diminishes with distance.

Mathematical Application in Acoustics

The study of sound, known as acoustics, involves a great deal of mathematical application. Acousticians use math to describe sound behavior in different environments, to design buildings with better acoustics, and to solve problems related to noise pollution.

Acoustics relies heavily on concepts such as frequency, wavelength, and amplitude, all of which can be quantified and modeled mathematically. Besides the inverse square law, mentioned in the context of intensity and distance, mathematical models can also take into account the absorption of sound by the atmosphere, which can differ depending on the frequency and the medium through which the sound is traveling.

Real-world applications might include determining the optimal location for speakers in a concert hall or calculating the noise reduction needed in a factory to meet safety standards. The formula for decibel decrease due to distance change is just one example of how math is used to quantify and manage the properties of sound in real and complex environments.