Question

Loudness of sound coming from a point source at distance R is 50dB. If the distance is doubled then the new loudness value will become?

Answer 1

Given:

Loudness of sound coming from a point source at distance R is 50dB.

To find:

New loudness value at doubled distance.

Calculation:

Loudness level at any point is measured by the intensity of the sound wave at that point. Intensity of sound from any source at a distance d is dependent as follows:

$\boxed{ \sf{intensity \: \propto \: \dfrac{1}{ {d}^{2} } }}$

Let k be a constant:

Loudness level at distance r is 50 dB.

$\rm{ \therefore \: 50 \: = \dfrac{k}{ {r}^{2} } }$

Let loudness at distance 2r be L:

$\rm{ \therefore \:L \: = \dfrac{k}{ {(2r)}^{2} } = \dfrac{k}{4 {r}^{2} } }$

Dividing the 2 Equations:

$\rm{ \therefore \: \dfrac{L}{50} = \dfrac{1}{4} }$

$\rm{ = > \: L = \dfrac{50}{4} }$

$\rm{ = > \: L = 12.5 \: dB }$

So, final answer is:

$\boxed{ \red{ \large{\sf{ \: L = 12.5 \: dB }}}}$

Answer 2

Answer:

given = distance { R} = 50 dB

to find =  2R =?

solution =    

                             intensity  ∝ 1/ d²

k will be consider as constant .

so,  

loudness at R  

   50 = k /  r² ----------------------------- {1}

loudness when R become twice /double

L = loudness

so,

L = K / (2r )²

L = k / 4r²---------------------------{2}  

divide the equation 2 and 1 we get

L /50 =  1/4

L  = 50/ 4

L = 12.5 dB