# How to understand motion of waves through functions of two variables - time and distance? [closed]

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

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.

• 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. – G. Smith Oct 20 '19 at 17:18

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 $$v=\lambda\nu$$.
• It is in the SE page you cited. $\omega=2\pi/T$ and $k=2\pi/\lambda$. – Lawrence B. Crowell Oct 22 '19 at 0:33