Traveling and Standing Waves:

(a) “A standing wave is the sum of two traveling waves propagating at the same speed in opposite directions”. Prove analytically that this statement is true or untrue by explicitly adding the waves ${{\psi }_{1}}={{A}_{1}}\sin \left( kx-\omega t \right)$ and ${{\psi }_{2}}={{A}_{2}}\sin \left( -kx-\omega t \right)$. You could look at an animated function to try some possibilities, but you also need an analytical proof.

(b) “A traveling wave is the sum of standing waves”. Demonstrate analytically the truth, falsehood, or partial truth of this statement.