Problem 11 - College Physics

Question

that made by the explosion of Krakatoa on August \(26-27,1888\). According to barometric measurements, the sound had a decibel level of \(180 \mathrm{~dB}\) at a distance of \(161 \mathrm{~km}\). Assuming the intensity falls off as the inverse of the distance squared, what was the decibel level on Rodriguez Island, \(4800 \mathrm{~km}\) away?

Step-by-Step Solution

Answer

The decibel level on Rodriguez Island would be obtained by subtracting the decrease in dB (\(ΔL\)) from the original decibel level of 180 dB. The result will be \(L_2\), the decibel level at Rodriguez Island.

Step 1: Identify constants and variables

In the problem, we are given: Initial decibel level \(L_1 = 180 dB\), Initial distance \(d_1 = 161 km\), and Final distance \(d_2 = 4800 km\) to calculate the final decibel level \(L_2\). Remember that Decibel level \(L\) is calculated using the formula: \(L = 10 × \log(I / I_0)\) where \(I\) is the Intensity and \(I_0\) is the reference Intensity (threshold of hearing).

Step 2: Find the ratio of the intensity levels

As we know, intensity is inversely proportional to the square of the distance. Therefore, we can create the following equation: \((I_1 / I_2) = (d_2 / d_1 )^2\).

Step 3: Find the difference in decibels

By using the formula \(L = 10 × \log(I / I_0)\) we find the difference in decibels to be \(ΔL = 10 \log(I_1 / I_2)\). We substitute the ratio we found in Step 2 into this equation. Once we find \(ΔL\) we can find the decibel level at Rodriguez Island by subtracting from the original decibel level. So, \(L_2 = L_1 - ΔL\).

Key Concepts

Decibel Level

The concept of the Decibel Level is essential in understanding sound intensity. The decibel (dB) is a logarithmic unit used to express the ratio of one value of a power or field quantity to another. When it comes to sound, decibels measure the intensity of a sound relative to a reference level, which is generally the threshold of hearing (usually taken as intensity \( I_0 = 10^{-12} \text{ W/m}^2 \) ). The decibel formula is given by \[ L = 10 \times \log\left( \frac{I}{I_0} \right) \] where \( L \) is the sound level in decibels, \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity. Because decibels operate on a logarithmic scale, each increase of 10 dB represents a tenfold increase in intensity.When we talk about sound reaching different locations, the decibel level changes depending on the distance from the source. In the problem about Krakatoa, the sound intensity level at Rodriguez Island changes based on how far it traveled from the source, affecting the observed decibel level at different points.

Inverse Square Law

The Inverse Square Law describes how the intensity of sound decreases as the distance from the source increases. This principle is not only applicable to sound but to any phenomena that radiate from a point source, such as light.According to the Inverse Square Law, the intensity \( I \) of sound is inversely proportional to the square of the distance \( d \) from the source:\[ I \propto \frac{1}{d^2} \]For instance, if you double the distance from the sound source, the intensity of sound becomes one-fourth as strong. This is because the same amount of energy spreads out over a larger area.In the exercise regarding the explosion of Krakatoa, this law explains how the intensity of the sound decreases as people move further away from the explosion. The distance ratios deduced: \[ \left( \frac{I_1}{I_2} \right) = \left( \frac{d_2}{d_1} \right)^2 \]help compute changes in sound intensity, hence altering decibel levels observed at different distances.

Acoustics

Acoustics is the branch of physics that deals with the study of sound. It encompasses the production, transmission, and effects of sound waves. Understanding acoustics is crucial for solving problems related to sound intensity and decibel levels, as seen in the given exercise on the Krakatoa explosion. Sound waves are longitudinal waves that travel through mediums like air, water, or solids. The properties of sound, like speed and intensity, can vary depending on these mediums. Factors that influence acoustics include:

• Material properties like density and elasticity
• Environmental conditions such as temperature and humidity
• Geometric considerations like distance and space configuration

In practical applications and problems, acoustic principles help determine how sound behaves over distances and under various conditions. By comprehending acoustics, one can better predict how sound intensity and decibel levels will change as sound moves from its source to different locations, as demonstrated in the Krakatoa exercise.