Problem 54 - Pearson Physics

Question

Imagine waking up to two different alarm clocks, one \(20 \mathrm{~dB}\) louder than the other. How many times louder does the "loud" alarm sound to your ears?

Step-by-Step Solution

Answer

The loud alarm sounds 100 times louder than the quieter one.

Step 1: Convert Decibels to Linear Scale

The decibel scale is logarithmic, which means that an increase of 20 dB in intensity level corresponds to a multiplication factor of the intensity. The formula to convert from decibels to the linear scale is \(I = 10^{(L/10)}\), where \(L\) is the level in decibels. Thus, 20 dB represents \(10^{(20/10)} = 100\).

Step 2: Determine the Relative Loudness

Since a 0 dB sound level is the reference threshold, the loudness of the quieter alarm is \(10^{0/10} = 1\). Meanwhile, the louder alarm is 100 times the intensity of this original level, which means it is 100 times louder than the quieter alarm to our ears.

Key Concepts

Decibel Scale

The Decibel Scale is a way to measure sound intensity in a manner that reflects human perception. It is logarithmic rather than linear, meaning each step on this scale represents a multiplicative change, not an additive one. Decibels (dB) become a practical tool because our ears perceive sound intensities logarithmically. In simple terms, an increase of 10 dB means sound intensity is perceived to be twice as loud. So, an increase of 20 dB corresponds to a sound perceived as approximately four times louder. The decibel scale starts at 0 dB, which is roughly the threshold of human hearing, and extends upwards, sometimes beyond 130 dB, the threshold of pain.

Logarithmic Scale

The Logarithmic Scale is crucial in understanding the behavior of the Decibel Scale. Unlike a linear scale, where equal intervals represent equal additions, a logarithmic scale represents equal intervals as equal multiplications. For instance, moving from 10 dB to 20 dB on the decibel scale does not double the sound intensity; it increases it by a factor of 10. Essentially, every 10 dB increase represents a tenfold increase in sound intensity. This is why converting decibels to a linear scale requires the use of a mathematical formula:

• \( I = 10^{(L/10)} \), where \( L \) is the level in decibels.

Understanding this concept helps us realize why sounds like alarms or music can seem much louder than we might expect from just a small increase in decibel reading.

Relative Loudness

Relative Loudness helps us compare different sound levels and understand their perception. When you encounter sounds with differing decibel levels, you can determine how much louder one is compared to another. For example, a sound that is 20 dB louder than another isn't just twice as loud but instead can be perceived as significantly more intense due to its position on the logarithmic scale. In practical terms, using an example from the exercise:

• If one alarm is 0 dB (the quietest sound you can hear), and another is 20 dB above that, the louder alarm is not just twice as loud—it's 100 times more intense in terms of sound wave energy.

This understanding underlines how our perception of sound doesn’t increase linearly with decibels, but rather exponentially.