# Multiple Waves all in phase (Wave packets)

Suppose there are seven waves of slightly different wavelengths and amplitudes and we superimpose them (textbook is talking about wave packets). The wavelengths range from $\lambda _9 = 1/9$ to $\lambda _{15} = 1/15$. Their wavenumbers ($k = 2\pi / \lambda$) ranges from $k_9 = 18\pi$ to $k_{15} = 30\pi$. Note, the waves are of the form

$y(x,t) = Asin(kx - wt)$

The waves are all in phase at $x = 0$ and again at $x = \pm 12, \pm 24$ etc. My question is the last line. How does my textbook (from which I copied what they wrote) know that they are all in phase at $x = \pm 12$ etc. ?

If you can do this in simple terms that would be great (i.e., no fourier transform math since I have yet to learn about it). Is there some rule to know when $n$ number of waves are in phase?

Second question, my textbook goes on to say that the width of the group $\Delta x$ of superposition is just a big larger than 1/12. There's a graph of the superposition but did they determine this number from the graph or is it somehow related to the numbers given above?

Just fyi, this is a physics textbook which goes on to say that $\Delta k \Delta x \sim 1$ and $\Delta w \Delta t \sim 1$. It then uses these as a basis to state the Heisenberg uncertainty principle.