(Solved): 1. [20 p] In this question we'll check the applicability and limitations of the complex representa ...

1. [20 p] In this question we'll check the applicability and limitations of the complex representation for calculating a few physical quantities. Consider the complex representation of waves \( \tilde{f}= \) \( A e^{i\left(\varphi_{1}-\omega t\right)} \) and \( \tilde{g}=B e^{i\left(\varphi_{2}-\omega t\right)} \) (assume \( A \) and \( B \) are real), where \( f=\operatorname{Re}[\tilde{f}] \) and \( g=\operatorname{Re}[\tilde{g}] \) are the physical fields. a) Is it true that \( \partial f / \partial t=\operatorname{Re}[\partial \tilde{f} / \partial t] ? \) b) Is it true that \( f \cdot g=\operatorname{Re}[\tilde{f} \cdot \tilde{g}] \) ? c) Show that \( \langle f \cdot g\rangle=\operatorname{Re}\left[\tilde{f} \cdot \tilde{g}^{*}\right] / 2 \). d) Use (c) to show that \( \left\langle f^{2}\right\rangle=|\tilde{f}|^{2} / 2 \).