To find the amplitude of the resultant wave for the given electric field components, we use the formula for the resultant amplitude of two combining waves. These components are:

$ E_1 = E_0 \sin \omega t $

$ E_2 = E_0 \sin \left(\omega t + \frac{\pi}{3}\right) $

The general formula for the amplitude $ E $ of two superimposed waves $ E_1 $ and $ E_2 $ with a phase difference is:

$ E = \sqrt{E_1^2 + E_2^2 + 2E_1E_2 \cos\phi} $

where $ \phi $ is the phase difference between the two waves. In this scenario, $ \phi = \frac{\pi}{3} $.

Substituting the given expressions and phase difference into the formula, we get:

$ E = \sqrt{(E_0)^2 + (E_0)^2 + 2(E_0)(E_0)\cos\left(\frac{\pi}{3}\right)} $

Given that $\cos \left( \frac{\pi}{3} \right) = \frac{1}{2}$, this simplifies to:

$ E = \sqrt{2E_0^2 + E_0^2} = \sqrt{3}E_0 $

Thus, the amplitude of the resultant wave is $\sqrt{3}E_0 \approx 1.73E_0$.