"Derivation" of the Heisenberg Uncertainty Principle The question I outline below is my textbook's "derivation" of the Heisenberg Uncertainty Principle. The "derivation" my textbook uses involves 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
$$y(x,t) = A\sin(kx - wt)$$
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? (They have all 7 waves graphed, but not on top of each other. Did they do some mathematics or find this from the graph? Note, looking at this graph its hard to tell that all 7 waves are in phase at $x=\pm 12, \pm 24$, etc).
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 (looks like a beat graph) but did they determine this number from the graph or is it somehow related to the numbers given above?
Then it shows a plot of the amplitude of the waves ($y_0$) vs. $k$. It ``shows'' that the width at $y_0 = 1/2$ is $4\pi$. 
Just fyi, this is a physics textbook which goes on to say that $\Delta k \Delta x \sim 1$ (using the numbers above, $4\pi * 1/12 \approx 1$) and $\Delta w \Delta t \sim 1$ (by similar arguments). It then uses these as a basis to state the Heisenberg uncertainty principle.  
 A: I almost answered without reading the question (further than the title...by the way, my "would be" answer to it is this)
It is good that you cited from where you got that non-sense. Although, without reading the book I can already say that 12 is in the "middle" of 9 and 15 and I guess that that is the only the author wanted to point out.
The only meaning of "being in phase" can I come up with is that all the $ k_i x,\ i=9,\cdots 15$ are all equal modulo $2\pi$ which in that special case that $k_i=2\pi i,\ i=9,\cdots ,15$ is the largest common divisor of those integer, i.e. 1. They are all in phase $x$ any multiple of 1.
The "width $\Delta x$ of the group" makes no sense to me but you may look in signal theory.

The original motivation for my answer was to say that in the special case of the observable position $X$ and momentum $P$ and the Hilbert space $L^2(\mathbb{R}^3)$ of wave fonctions, some inequality from Fourier theory is used to proove the "Cauchy-Schwarz" inequality in the derivation of the Heisenberg uncertainty relation
