Question

Two small loudspeakers emit sound waves of different frequencies equally in all directions. Speaker A has an output of 1.30 mW, and speaker B has an output of 1.60 mW.

Two speakers A and B are represented as points which serve as the centers of concentric circles representing sound waves emanating from the speakers. The concentric circles about speaker A are spaced more closely together than the concentric circles about speaker B. The points lie along a horizontal line, and speaker A is 5.00 m to the left of speaker B. A vertical line is drawn that intersects the horizontal line at a distance of 3.00 m from the left speaker and 2.00 m from the right speaker. Point C lies along this vertical line at a distance of 4.00 m above the point of intersection.

(a) Determine the sound level (in decibels) at point C in the figure if only speaker A emits sound. (Enter your answer to at least one decimal place.)
dB

(b) Determine the sound level (in decibels) at point C in the figure if only speaker B emits sound. (Enter your answer to at least one decimal place.)
dB

(c) Determine the sound level (in decibels) at point C in the figure if both speakers emit sound. (Enter your answer to at least the nearest dB.)
dB

Answer 1

Given,

Output of A, P= 1.3 mW

Output of B, P= 1.6 mW

d(OA)= 3 m. d(OC) = 4 m

d(OB) = 2 m

We can find distance of R(ac) and R(bc) can be determined by using Pythagoras theorem.

R(ac) = √[(OA)²+(OC)²] = √[9+16] = 5m

R(bc) = √[(OB)²+(OC)²] =√[16+4] = 4.47 m

a) sound level at point C due to only A

Here intensity, I = P/4πR(ac)² =

I = 1.3 * 10^-3 /( 4 * 3.142 * 25 )

= 4.14 *10^-6 W/m²

Now sound level is given by,

dB = 10 log (I /Io)  

Io = 10^-12 W/m²

dB = 10 log ( 4.14 * 10^-6 / 10^-12)

= 10 log ( 4.14 * 10 ^6 )

dB = 66.17 dB

b) sound level at point C due to only B

Intensity at B is, I = P/4π R(bc)²

I = 1.6 * 10^ -3 / (4 * 3.142 * 4.47²)

I = 6.37 * 10^-6 W/m²

Sound level is given by,

dB = 10 log (I / Io)

= 10 log ( 6.37 * 10^-6 / 10^-12 )

= 68.04 dB

Ans dB = 68.04 dB

c) sound level at point C due to both speakers A and B

Here total intensity at C is

I(total) = (4.14 + 6.37 ) * 10^-6 W/m²

= 10.51 * 10^-6 W/m²

So sound level at point C is given by

dB = 10 log ( I(total) / Io)

= 10 log ( 10.51 * 10^-6 / 10^-12)

= 10 log ( 10.51* 10^6)

dB = 70.21dB

Ans = 70.21 dB