Young experiment: square of classical real wave function

Question

I can't understand why the sum of two real waves result in a time dependent wave, but not so for the complex waves. In details, I can't get this passage on p.38-39 in A.C. Phillips, Introduction to Quantum Mechanics:

$$\tag{3.9}\Psi ~=~ A\cos (kR_1 - \omega t) + A\cos (kR_2 - \omega t)$$

$$\Downarrow$$

$$\tag{3.10}\Psi^2 ~=~ 2A^2\cos^2\left(\frac{k(R_1-R_2)}{2}\right)\cos^2\omega t.$$

This is driving me crazy!

Answer 1

Yes, OP is right. The book contains an error. The formula (3.10) for the classical real wave$^1$ should have been

$$\tag{3.10'}\Psi^2 ~=~ 4A^2\cos^2\left(\frac{k(R_1-R_2)}{2}\right)\cos^2\left(\omega t-\frac{k(R_1+R_2)}{2}\right).$$

Note however, that the qualitative physical conclusions that Phillips draws from eq. (3.10) in subsequent paragraphs remain valid.

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$^1$ Note that Phillips is here (on p.38-39) for pedagogical reasons considering a real classical wave (3.9) rather than the correct quantum mechanical wave function.