I have an intuition problem here that need help with. I understand and have in front of me the typical "wave equation formula" $y (x, t) = A \sin(kx - \omega t + \phi )$. Sorry about the ugly art , I am not sure how to use the fonts yet. It does not look so pretty as the book. Also I don't know who to ask so I guessed who might know.
Does anyone know how to add or multiple waves together? I do not understand what happens to the amplitude A when this happens. Thank you!
When you add waves with different frequencies the result is no longer a sine wave so the simple notion of amplitude no longer applies.
When you add sine waves with the same frequency but possibly different phases you do get a sine wave so the amplitude is meaningful. You can compute the amplitude and phase of the combined wave by using formulas for the sine (or cosine) of sums of angles and the definition of arctangent. One illustration (in just a time variable) should show how this is done.
$$ y_1(t) = A\sin (t + \delta_1)\\ y_2(t) = B\sin (t + \delta_2) \\ y(t) = y_1(t)+y_2(t) = C\sin(t+\delta) $$ and you are asked to find $C$ and $\delta$ in terms of $A,B,\delta_1,\delta_2$.
Use addition formula for sines: $$ y_1(t) = A\sin t \cos \delta_1 + A \cos t \sin\delta_1 \\ y_2(t) = B\sin t \cos \delta_2 + B \cos t \sin\delta_2 \\ y(t) = (A\cos \delta_1+B\cos \delta_2)\sin t + (A\sin \delta_2+B\sin \delta_1)\cos t = C\sin t\cos \delta + C \cos t \sin \delta $$ Now you match coefficients of sine and cosine in the two expressions for $y(2)$: $$ C\sin\delta = A\sin \delta_1+B\sin \delta_2\\ C\cos \delta = A\cos \delta_2+B\cos \delta_1 $$ and here is the trick: Divide the two equations to get the tangent of $\delta$ on the left: $$ \tan\delta = \frac{A\sin \delta_1+B\sin \delta_2}{A\cos \delta_2+B\cos \delta_1} \\ \delta = \tan^{-1}\left(\frac{A\sin \delta_1+B\sin \delta_2}{A\cos \delta_3+B\cos \delta_2} \right) $$ That gives you $\delta$, and now you can find the combined amplitude $C$ from $$ C = \frac{A\sin \delta_1+B\sin \delta_2}{\sin\delta}$$ Two subtleties:
You may notice a $\sin\delta$ on the bottom of the fraction; what if $\delta$ comes out to zero? In that case the fraction will always be $0/0$ but if you yous the $C\cos\delta$ equation instead you will get a perfectly good determination of $\delta$.
And you might point out that $\tan^{-1}$ could be considered to have two possible values, $180^\circ$ apart. True, and then $\sin\delta$ would change signs. But this is fine because any sine wave could be described in two ways, one with a phase that is $180^\circ$ more than the other but with negative of the amplitude. Conventionally we adjust the phase such that the amplitude $C$ is positive.