1. Relate intensity and amplitude

For harmonic waves of the same frequency, intensity is proportional to the square of amplitude:

$I \propto A^2$

If each wave has intensity $I_0$, let the amplitude of each wave be $a$. Then:

$I_0 \propto a^2$

2. Write the three waves in phasor form

Since all have same frequency, we can add their amplitudes vectorially using phase angles:

• Wave 1: phase $0$ $\Rightarrow a e^{i0}$
• Wave 2: phase $\frac{\pi}{4}$ $\Rightarrow a e^{i\pi/4}$
• Wave 3: phase $-\frac{\pi}{4}$ $\Rightarrow a e^{-i\pi/4}$

Resultant phasor amplitude:

$A_R = a\left(e^{i0}+e^{i\pi/4}+e^{-i\pi/4}\right)$

Using

$e^{i\theta}+e^{-i\theta}=2\cos\theta$

we get:

$A_R = a\left(1+2\cos\frac{\pi}{4}\right)$

Since

$\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}$

so

$A_R = a\left(1+\sqrt{2}\right)$

3. Find resultant intensity

Intensity is proportional to square of amplitude, so:

$I_R = I_0(1+\sqrt{2})^2$

Now,

$(1+\sqrt{2})^2 = 1+2+2\sqrt{2}=3+2\sqrt{2}$

Numerically,

$3+2\sqrt{2} \approx 3+2(1.414)=3+2.828=5.828$

Thus,

$I_R \approx 5.8 I_0$

4. Match with options

The closest option is:

$\boxed{5.8 I_0}$

So, Option A is correct.