Question

Three sources of sound $S_1,S_2,$ and $S_3$ of equal intensity are placed in a straight line with $S_1{S}_2=S_2{S}_3$ (figure below). At a point $P$, far away from the sources, the wave coming from $S_2$ is $120°$ ahead in the phase of that from $S_1$. Also, the wave coming from $S_3$ is $120°$ ahead of that from $S_2$. What would be the resultant intensity of sound at $P$?

Answer 1

Step 1: Given Data

Let the amplitude of the sound source from $S_1$ be $A_1$.

Let the amplitude of the sound source from $S_2$ be $A_2$.

Let the amplitude of the sound source from $S_3$ be $A_3$.

$A_2$ is $120°$ ahead of $A_1$.

$A_3$ is $120°$ ahead of $A_2$.

They have equal intensities, $I_1=I_2=I_3$

Step 2: Formula Used

Relationship between intensity and amplitude, $I\propto A^2$

Step 3: Phasor Diagram

Step 4: Calculate the Resultant Intensity

Since they have equal intensities,

$$\begin{gathered} I_1=I_2=I_3 \\ \Rightarrow {A_1}^2={A_2}^2={A_3}^2 \\ \Rightarrow A_1=A_2=A_3 \end{gathered}$$

Therefore, the intensity at point $P$,

$$\begin{gathered} I_r=I_1+I_2+I_3 \\ =A\left\{\cos \left(\theta -120°\right)\cos \theta +S\cos \left(\theta +120°\right)\right\} \\ =A\left\{\cos \theta +\cos \left(\theta -120°\right)+\cos \left(\theta +120°\right)\right\} \\ =A\left\{\cos \theta +2\cos \theta \cos \left(-120°\right)\right\} \\ =A\left(\cos \theta +2\cos \theta .\left(-\frac{1}{2}\right)\right) \\ =A\left(\cos \theta -\cos \theta \right)=0 \end{gathered}$$

Therefore, by vector method the resultant intensity at point $P=0$

Hence, the resultant intensity and point $P$ is zero.