Four harmonic waves of equal frequencies and equal intensities $I_{0}$ have phase angles $0, \pi / 3$, $2 \pi / 3$ and $\pi$. When they are superposed, the intensity of the resulting wave is $n I_{0}$. The value of $n$ is

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Waves & Sound

Four harmonic waves of equal frequencies and equal intensities

I_{0}

have phase angles

0, π /3

2 π /3

and

π

. When they are superposed, the intensity of the resulting wave is

n I_{0}

. The value of

n

Answer: 3

Solution

Let the amplitude of each individual wave be

A

. Using phasor addition, the four waves with phases

0, π /3, 2 π /3, π

can be represented as vectors.

The waves at phases

0

and

π

cancel each other (opposite phases).

The waves at phases

π /3

and

2 π /3

add vectorially. Their resultant amplitude is:

A_{res} = 2 A cos (\frac{π}{6}) = 2 A \cdot \frac{3}{2} = 3 A

Since intensity

I \propto A^{2}

I = (3 A)^{2} = 3 A^{2} = 3 I_{0}

Therefore,

n = 3

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Integer answer

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