Question

Find the intensity ratio of two sounds whose sound levels differ by \(1.00 \mathrm{~dB}\).

Short answer

The intensity ratio of the two sounds is 1.259.

Step-by-step solution

Understanding the Concept

The Decibel (dB) is a logarithmic unit used to express the ratio of two quantities, usually power or intensity. In the case of sound, it expresses the ratio between the intensity of the sound under consideration and a reference intensity, typically the lowest intensity sound that the average human ear can hear.

Formulate the Given Information and Apply the Formula

We know from the problem that the difference in sound levels is 1.00 dB. This means that \( dB_2 - dB_1 =1.00 \). Applying the formula \( dB =10 \log_{10}(\frac{I}{I_0}) \) separately for the two sounds we get \( dB_1 =10 \log_{10}(\frac{I_1}{I_0}) \) and \( dB_2 =10 \log_{10}(\frac{I_2}{I_0}) \). If we subtract these two equations we obtain that \( dB_2 - dB_1 =10 \log_{10}(\frac{I_2}{I_0}) - 10 \log_{10}(\frac{I_1}{I_0}) = 10 \log_{10}(\frac{I_2}{I_1}) \). We can therefore solve for \( \frac{I_2}{I_1} \) by substituting \( dB_2 - dB_1 =1.00 \) and using that \( 10 \log_{10}(x) \) is the inverse function of \( 10^x \). We therefore obtain \( \frac{I_2}{I_1} =10^{(dB_2 - dB_1) /10} =10^{1.00 /10} =1.259 \).

Interpret the Result

The intensity ratio \( \frac{I_2}{I_1} = 1.259 \) means that the second sound is roughly 1.259 times as intense as the first sound. This is the result when their decibel levels differ by 1.00 dB.

Key concepts

Decibel Scale

The decibel (dB) scale is a logarithmic unit that quantifies the ratio between two sound intensities. On the decibel scale, differences in sound level are measured based on a reference intensity which is usually the quietest sound a human ear can detect. This scale is widely used because it allows us to handle a large range of sound intensities in a manageable fashion.

Instead of dealing with large and cumbersome numbers, the decibel scale uses a relative measure. This means that each increase of 10 dB represents a sound that is ten times more intense. The equation that defines decibels for sound intensity is:

• \( dB = 10 \log_{10}(\frac{I}{I_0}) \),

where \( I \) is the intensity of the sound in question, and \( I_0 \) is the reference intensity.

Understanding this formula helps us see why a small change in dB represents a larger proportional change in intensity.

Logarithmic Measurement

Logarithmic measurement is fundamental in understanding the decibel scale. The human ear perceives sound intensity logarithmically, meaning equal ratios of sound intensities are perceived as equal differences in sound level. This is where the mathematics of logarithms becomes highly practical.

Using logarithmic measurement, we simplify the comparison of vastly different quantities. For sound, if two sound levels differ in decibels, we can express this in terms of intensity ratios. For instance, if the difference in intensity levels is 1 dB, this can be represented by the formula:

• \( dB_{difference} = 10 \log_{10}(\frac{I_2}{I_1}) \)

Logarithms reduce multiplicative differences to additive differences, making calculations manageable.

This logarithmic structure also explains why we can interpret a 1 dB difference as a 1.259-fold increase in intensity. The equation to find this intensity ratio is:

• \( \frac{I_2}{I_1} = 10^{(dB_{difference} /10)} = 10^{(1.00 /10)} \).

Logarithms in sound intensity measurement help us transition from intuitive human perception to precise mathematical representation.

Intensity Ratio

The intensity ratio is a way to quantify how much more powerful one sound is compared to another. When sound levels differ, it isn't just about subtracting one number from another; it's about finding how many times more intense one is in terms of power.

For example, a 1 dB difference might sound minor, but in terms of intensity, it means one sound has about 1.259 times the power of the other. This is a vital concept to grasp because it shifts focus from raw numbers to understanding their physical implications.

Using the equation for intensity ratio:

• \( \frac{I_2}{I_1} = 10^{(dB_{difference} /10)} \)

You can calculate how sound levels compare relative to each other. For the example of a 1 dB difference, the result is \( 10^{(1.00 /10)} = 1.259 \). This means not only louder but over 25% more intense, which can be significant in contexts like engine noise levels, concert sound checks, or even everyday listening experiences. Understanding intensity ratios helps explain why some sounds seem disproportionately louder despite small decibel changes.