Question

As the drawing shows, one microphone is located at the origin, and a second microphone is located on the +y axis. The microphones are separated by a distance of D = 1.76 m. A source of sound is located on the +x axis, its distances from microphones 1 and 2 being L1 and L2, respectively. The speed of sound is 343 m/s. The sound reaches microphone 1 first, and then, 1.60 ms later, it reaches microphone 2. Find the distances (in m) (a) L1 and (b) L2.

Answer 1

From the figure with the help of geometry we can write
L₂² = L₁²+ (1.76)² -----------------(1)
Now time taken by sound to reach the microphone 1 , say
t₁ = Distance /speed = L₁/V
where V is speed of sound = 341 m/s
t₁ = L₁ /341 ------------------(2)
Similarly time taken by sound to reach the microphone 2
t₂ = L₂/341 -------------(3)
now sound reaches to the microphone 2 after 1.6 ms that of microphone 1, therefore
t₂ = t₁ + (1.6*10^-3)
using equation 2
t₂ = (L₁/341) +(1.6*10^-3) ---------------(4)
hence from 3 and 4
L₂/341 = (L₁/341) +(1.6*10^-3)
L₂ - L₁ = 341*1.6*10^-3 = 0.5456 m
L₂ = L₁ + 0.5456
Squaring both side
L₂² = ( L₁ + 0.5456 )²
L₂²= L₁² + 0.2977 + 1.0912L₁-----------(5)
Now using equation 1
L₁²+ (1.76)² =  L₁² + 0.2977 + 1.0912L₁
3.0976 = 0.2977 + 1.0912L₁
L₁ = 2.799 /1.0912
L₁ = 2.566 m
And
L₂ = L₁ + 0.5456 = 2.566 +0.5456
L₂ = 3.112 m