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$$ s(x,t)= A \sin(\frac{2\pi}{T}t-\frac{2\pi}{\lambda}x) $$

Basically I would love to get some plausible and thorough explanation of plotting these two independent variables in order to satisfy the harmonic motion of waves.

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    $\begingroup$ You can plot $s$ against $x$ for several values of $t$. You can plot $s$ against $t$ for several values of $x$. You can make a 3D plot of $s$ above a $t$-$x$ plane. I recommend the first one. $\endgroup$ – G. Smith Oct 20 '19 at 17:18
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Assume you are moving with the wave. The amplitude of the wave is then constant in that frame. The wave has a constant phase $$ \theta=\frac{2\pi}{T}t-\frac{2\pi}{\lambda}x, $$ so that $d\theta/dt=0$ which leads to $$ 0=\frac{2\pi}{T}-\frac{2\pi}{\lambda}\frac{dx}{dt}. $$ This means the velocity is $v=\lambda/T$ and for frequency $\nu=1/T$ we get the standard formula $v=\lambda\nu$.

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