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 ]

====================================================

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)

====================================================

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 ..

===================================================

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.

 

[kyle8921]

Last edited by a moderator: May 3, 2017

Answers and Replies

- Apr 2, 2007

- #2

Astronuc
Staff Emeritus
Science Advisor

19,138

2,450

I think the problem is in the use of the intensity formula. The intensity should be the same at both observers.

So one should be able to equate the two intensities and determine the value of r.

You have the correct equation, but one must remember that the intensity is a measure of power per unit area, and the power must be the same at both observers.

One can also use the inverse-square law for sound, which is essentially the same as the intensity formula.

P/4πr^2 = P/4π(110-r)^2

The power (P) should be the same at both observers.

- Apr 3, 2007

- #3

HallsofIvy
Science Advisor
Homework Helper

41,833

963

Astronuc is right. Your equation, ( 1 x 10^-4 )*4*pi*r^-2 = Intesity2*4*pi*[(110-r)^-2], is correct but you need to solve for r.

( 1 x 10^-4 )*4*pi*r^-2 = Intesity2*4*pi*[(110-r)^-2] can be rewritten as
( 1 x 10^-4 )= Intesity2*[(110-r)^-2]/r^-2. Since 110-r= (110-r)^-1, that can be rewritten as
( 1 x 10^-4 )= Intesity2*[(110-r)/r]^2.

Now, (110-r)/r= 110/r- 1 and so [(110-r)/r]^2= 110^2/r^2- 2(110/r)+ 1. Replace [(110-r)/r]^2 by that in the equation above and you have a quadratic equation in r^2.

Related Threads on Decibel levels homework

Hot Threads

Equivalent resistor of a simple circuit
Engineering Hydrostatic Pressure Sluice Gate - Civil Engineering
Comp Sci Keypad Translation Help
Engineering Simulate a step-down chopper circuit using LTSpice

Recent Insights

Insights How to Solve Second-Order Partial Derivatives - Comments
Insights Is Pressure A Source