How Does Distance Affect Decibel Levels in Sound Propagation?

[gh0st]

Question

15. (a) Show that the difference in decibel levels, B1 and B2, of a sound source is related to the ratio of its distances, r1 and r2, from the receivers by

B2 - B1 = 20 log (r1/r2)

(b) A speaker is placed between two observers who are 110 m apart, along the line connecting them. If one observer records an intensity level of 80 dB, how far is the speaker from each observer ?

Answer given by lecturer : [ 10 m and 100 m ]

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Before i start, i would like to apologize for the messy post as i have no idea how to type out the actual symbols, therefore I'm only using those basic ones on my keyboard ..

Okay, this is my problem ... I managed to solve 15(a) but I'm stuck at 15(b)

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According to 15(b), i came up with this diagram :

http://img246.imageshack.us/img246/606/untitledmn4.jpg

My worksteps

Formulas used :

i. Intensity Level in Decibels, B = 10 log ( I / Io ) dB ... ( Io is intensity reference, 1.00 x 10^-12 Wm^-2 )

ii. Intensity , I = Power / Area ... ( Area is 4*pi*r^2 )

iii. Assumed that Observer 1 was the person who recorded 80dB mentioned. Therefore :

80dB = 10 log ( I / Io ) dB

Intensity 1 = [ anti-log ( 80/10) ]*[Io]
= ( 1 x 10^-8 )*(1x10^-12)
= 1 x 10^-4 Wm^-2

iv. Used Intensity formula :

I = P/A
I = P/4*pi*r^2
1 x 10^-4 = P/4*pi*r^2

P = ( 1 x 10^-4 )*4*pi*r^-2

v. Okay, so P, Power is not given, so i thought of deriving it from Observer 2, using the Intensity formula ...

Intensity 2 = P/4*pi*r^-2 , however this r is ( 110-r ) , as it's the distance between speaker and obersver 2, as shown in the diagram ..

P = Intesity2*4*pi*[(110-r)^-2]

vi. Since the sound is coming from 1 speaker therefore the value for P should be the same ...

P = ( 1 x 10^-4 )*4*pi*r^-2 and P = Intesity2*4*pi*[(110-r)^-2]vii. I concluded that

( 1 x 10^-4 )*4*pi*r^-2 = Intesity2*4*pi*[(110-r)^-2][/b]

viii. However, the problem is there are 2 unknown variables here ... Intensity 2 and r ( which is sought by the question ... ) And i got stuck here ... Tried many ways in substituting either r or intensity but i got back the same equation ... Also, the reason i posted along 15(a) is because i was wondering if (b) is related to (a) , but i can't find any connection ..

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please help me , i can't think of where i got it wrong .. and thanks in advance.

edit : sorry for the topic title, i forgot to add it in after typing in this post .

 

[OlderDan]

I think you are stuck on part (b) because there is not enough information. You don't know the power output of the source, and you only know one observed level. It can't be done.