Two Speakers - Sound Maximum and Sound Minimum Problem

davichi

Hi, I am having difficulty solving the following problem:

1. The problem statement, all variables and given/known data

Two loudspeakers 5.0 m apart are playing the same frequency. If you stand 13.0 m in front of the plane of the speakers, centered between them, you hear a sound of maximum intensity. As you walk parallel to the plane of the speakers, staying 13.0 m in front of them, you first hear a minimum of sound intensity when you are directly in front of one of the speakers.

What is the frequency of the sound? Assume a sound speed of 340 m/s.


2. Relevant equations

Sound Maximum:
L1 - L2 = n[tex]\lambda[/tex]

Sound Minimum:
L1' - L2 = (n+[tex]\frac{1}{2}[/tex])[tex]\lambda[/tex]

Frequency:
f = [tex]\frac{v}{\lambda}[/tex]

3. The attempt at a solution



Sound Maximum:
L1 - L2 = n[tex]\lambda[/tex]

L2 = 13.0 m
L1 = [tex]\sqrt{13.0^{2}+2.50^{2}}[/tex] = 13.23820229

L[tex]_{1}[/tex] - L[tex]_{2}[/tex] = n[tex]\lambda[/tex]
13.23820229 - 13 = n[tex]\lambda[/tex]
n[tex]\lambda[/tex] = 0.23820229

Sound Minimum
L1' - L2 =(n + [tex]\frac{1}{2}[/tex])[tex]\lambda[/tex]

L2 = 13.0 m
L1' = [tex]\sqrt{13.0^{2}+5.0^{2}}[/tex] = 13.92838828

Sub in n[tex]\lambda[/tex]= 0.23820229:

L1' - L2 = (n + [tex]\frac{1}{2}[/tex])[tex]\lambda[/tex]
13.92838828 - 13 = n[tex]\lambda[/tex] + [tex]\lambda[/tex]/2
[tex]\lambda[/tex]/2 = 0.92838828 - 0.23820229
[tex]\lambda[/tex] = 1.380371974

Sub in [tex]\lambda[/tex] = 1.380371974:
f = [tex]\frac{v}{\lambda}[/tex]
f = [tex]\frac{340}{1.380371974}[/tex]
f = 246.3104195 Hz

I'm not sure if my approach is wrong or if I'm interpreting the question incorrectly. Any help would be greatly appreciated!

Thanks.

rl.bhat

In the central position the two speakers are at equal distance. So the path difference is zero. In between the first and the second position, there is neither a maximum nor a minimum. So at the second position ( l1' - l2) = λ/2.

davichi

Ooh.. no wonder. Thank you very much!