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Usually the cases (that I have seen) with wave functions involve trigonometric functions like sine or cosine (for example, $y(x,t)=A\sin(kx-\omega t+\phi)$), but not about gaussian wave pulses.

So, let's suppose a case with two gaussian pulses like $y_1(x,t)=A_1e^{-(k(x-v_1t))^2}$ and $y_2(x,t)=A_2e^{-k(x+v_2t)}$, where $A_2=1.7A_1$ and $v_1=1.6v_2 \implies v_2=(\frac{5}{8})v_1$. Trying to analyze the resulting wave ─ that is, $$y(x,t)=y_1 + y_2=A_1e^{-(k(x-v_1t))^2}+1.7A_1e^{-k(x+(\frac{5}{8})v_1t)}, $$ I got confused about the amplitude of this wave with maximum overlap and find the time when this event happens, I tried to approach it using maxima and minima criterion of critical points (derivating the resulting function with respect to $x$ and $t$: $f(x,t)=y(x,t); D(x,t)=[f_{xx} f_{tt}-(f_{xt})^2]$) but the results were extensive and complex.

Therefore, my question is: According $y_1$ and $y_2$ (mentioned previously), what would be the maximum amplitude of the resulting wave when the two wave pulses overlap each other and what time this occurs?

I would like to understand the cases with gaussian wave pulses and resulting waves of their behaviour because this is and interesting case of waves.

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closed as off-topic by sammy gerbil, JMac, Mitchell, Rob Jeffries, Daniel Griscom Dec 27 '17 at 13:23

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  • $\begingroup$ I'm confused about what it is you're actually asking. What do you mean by "amplitude of this wave with maximum overlap"? $\endgroup$ – Emilio Pisanty Nov 28 '17 at 19:59
  • $\begingroup$ Thanks for your corrections, excuse if I didn't write correctly the question, this kind of waves represents something new for my understanding. $\endgroup$ – ht1204 Nov 29 '17 at 4:38
  • $\begingroup$ Think about what happens when $x$ and $t$ are both zero. What is the amplitude there? Can it get any larger than that? $\endgroup$ – Chris Nov 29 '17 at 4:39
  • $\begingroup$ $y_2$ doesn't appear to be gaussian $\endgroup$ – Kyle Kanos Nov 29 '17 at 11:12
  • $\begingroup$ Hi, I'm starting to approach this topic, about $y_2$ I purpose an exponential not elevated to 2 and I thought it was gaussian. $\endgroup$ – ht1204 Nov 29 '17 at 22:33