The sound level in decibels is typically expressed as β = 10 log (I/I0), but since sound is a pressure wave, the sound level can be expressed in terms of a pressure difference. Intensity depends on the amplitude squared, so the expression is β = 20 log (P/P0), where P0 is the smallest pressure difference noticeable by the ear: P0 = 2.00·10-5 Pa. A hair dryer has a sound level of 79 dB, find the amplitude of the pressure wave generated by this hair dryer?
See 11-16-11, 6:30am post for solution.
To find the amplitude of the pressure wave generated by the hair dryer, we can use the formula β = 20 log (P/P0), where β is the sound level in decibels and P is the pressure difference.
Given the sound level of the hair dryer is 79 dB, we can plug this value into the formula as follows:
79 = 20 log (P/P0)
To solve for P, we need to isolate it. Let's start by dividing both sides of the equation by 20:
79/20 = log (P/P0)
Next, we need to convert the logarithmic equation to exponential form. Since logarithms are base 10, we can write it as:
10^(79/20) = P/P0
Now we can compute the left side of the equation:
10^(79/20) ≈ 10^3.95 ≈ 8908.45
Finally, multiply both sides of the equation by P0 to isolate P:
P = 8908.45 * P0
Given the value of P0 is 2.00 * 10^-5 Pa, we can substitute it into the equation:
P = 8908.45 * (2.00 * 10^-5) ≈ 0.1782 Pa
Therefore, the amplitude of the pressure wave generated by the hair dryer is approximately 0.1782 Pa.