Question

To understand the decibel scale. The decibel scale is a logarithmic scale for measuring the sound intensity level. Because the decibel scale is logarithmic, it changes by an additive constant when the intensity as measured in W/m2 changes by a multiplicative factor. The number of decibels increases by 10 for a factor of 10 increase in intensity. The general formula for the sound intensity level, in decibels, corresponding to intensity I is

Answer 1

The question is incomplete. Here is the complete question.

To understand the decibel scale. The decibel scale is a logarithmic scale for measuring the sound intensity level. Because the decibel scale is logarithmic, it changes by an additive constant when the intensity when the intensity as measured in W/m² changes by a multiplicative factor. The number of decibels increase by 10 for a factor of 10 increase in intensity. The general formula for the sound intensity level, in decibels, corresponding to intensity I is

`\beta=10log((I)/(I_(0)) )dB`,

where
`I_(0)` is a reference intensity. for sound waves,
`I_(0)` is taken to be
`10^(-12) W/m^(2)`. Note that log refers to the logarithm to the base 10.

Part A: What is the sound intensity level β, in decibels, of a sound wave whose intensity is 10 times the reference intensity, i.e.
`I=10I_(0)`? Express the sound intensity numerically to the nearest integer.

Part B: What is the sound intensity level β, in decibels, of a sound wave whose intensity is 100 times the reference intensity, i.e.
`I=100I_(0)`? Express the sound intensity numerically to the nearest integer.

Part C: Calculate the change in decibels (
`\Delta \beta_(2),\Delta \beta_(4)` and
`\Delta \beta_(8)`) corresponding to f = 2, f = 4 and f = 8. Give your answer, separated by commas, to the nearest integer -- this will give an accuracy of 20%, which is good enough for sound.

Answer and Explanation: Using the formula for sound intensity level:

A)
`I=10I_(0)`

`\beta=10log((10I_(0))/(I_(0)) )`

`\beta=10log(10 )`

β = 10

The sound Intensity level with intensity 10x is 10dB.

B)
`I=100I_(0)`

`\beta=10log((100I_(0))/(I_(0)) )`

`\beta=10log(100)`

β = 20

With intensity 100x, level is 20dB.

C) To calculate the change, take the f to be the factor of increase:

For
`\Delta \beta_(2)`:

`I=2I_(0)`

`\beta=10log((2I_(0))/(I_(0)) )`

`\beta=10log(2)`

β = 3

For
`\Delta \beta_(4)`:

`I=4I_(0)`

`\beta=10log((4I_(0))/(I_(0)) )`

`\beta=10log(4)`

β = 6

For
`\Delta \beta_(8)`:

`I=8I_(0)`

`\beta=10log((8I_(0))/(I_(0)) )`

β = 9

Change is

`\Delta \beta_(2),\Delta \beta_(4)`,
`\Delta \beta_(8)` = 3,6,9 dB