Three sound sources S_1 , S_2 , and S_3 of equal intensity are...

Let the intensity of each individual sound source be $I_0$, and the corresponding amplitude be $A_0$.

The wave arriving from $S_2$ leads the wave from $S_1$ by $120^\circ$, and the wave from $S_3$ leads the wave from $S_2$ by $120^\circ$. The equations of the three waves arriving at point $P$ can be written as:

$y_1 = A_0 \sin(\omega t)$

$y_2 = A_0 \sin(\omega t + 120^\circ)$

$y_3 = A_0 \sin(\omega t + 240^\circ)$

The resultant displacement at point $P$ is given by the principle of superposition:

$y = y_1 + y_2 + y_3$

$y = A_0 [\sin(\omega t) + \sin(\omega t + 120^\circ) + \sin(\omega t + 240^\circ)]$

Using the trigonometric identity $\sin C + \sin D = 2 \sin$\left(\dfrac{C+D}{2}\right) \cos$\left(\dfrac{C-D}{2}\right)$ for the last two terms:

$\sin(\omega t + 120^\circ) + \sin(\omega t + 240^\circ) = 2 \sin(\omega t + 180^\circ) \cos(-60^\circ)$

Since $\sin(\omega t + 180^\circ) = -\sin(\omega t)$ and $\cos(-60^\circ) = \dfrac{1}{2}$:

$\sin(\omega t + 120^\circ) + \sin(\omega t + 240^\circ) = 2 (-\sin(\omega t)) \left(\dfrac{1}{2}\right) = -\sin(\omega t)$

Substituting this back into the equation for $y$:

$y = A_0 [\sin(\omega t) - \sin(\omega t)] = 0$

Since the resultant amplitude is zero, the resultant intensity at point $P$ is also zero.