Question

Suppose that the sound level of a conversation is initially at an angry 71 dB and then drops to a soothing 54 dB. Assuming that the frequency of the sound is 504 Hz, determine the (a) initial and (b) final sound intensities and the (c) initial and (d) final sound wave amplitudes. Assume the speed of sound is 346 m/s and the air density is 1.21 kg/m3.

Answer 1

Step-by-step explanation:

In the decibel scale , intensity of sound changes logarithmically as follows

`10log(I)/(I_0) =` Value in decibel scale , the value of I₀ = 10⁻¹² W /m².

Putting the values

`10log(I)/(10^(-12)) = 71`

`log(I)/(10^(-12)) = 7.1`

`(I)/(10^(-12)) = 10^(7.1)`

`I= 10^(-4.9)` W/m²

Similarly for 54 dB sound intensity can be given as follows

I = 10⁻¹² x
`10^(5.4)`

`I= 10^(-6.6 )` W / m²

For intensity of sound the relation is as follows

I = 2π²υ²A²ρc where υ is frequency , A is amplitude , ρ is density of air and c is velocity of sound .

Putting the given values for 71 dB

`I= 10^(-4.9)` = 2π² x 504²xA²x 1.21 x 346

A² = 60.03 x 10⁻¹⁶

A = 7.74 x 10⁻⁸ m

For 54 dB sound

`10^(-6.6)` = 2π² x 504²xA²x 1.21 x 346

A² = 1.1978 x 10⁻¹⁶

A = 1.1 x 10⁻⁸ m