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Why equation of wave is written like $Asin(\omega t-kz+ \phi)$ , why $\omega t - kz+ \phi$ as domain of sine, how somebody came up with this kind of equation to represent a wave?

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  • $\begingroup$ desmos.com/calculator Type here the equation and play around with the parameters. $\endgroup$ – Superfast Jellyfish Jul 23 '20 at 11:09
  • $\begingroup$ I think if you start by understanding Simple Harmonic Motion and then come to this you will understand it. $\endgroup$ – Vamsi Krishna Jul 23 '20 at 12:34
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Waves are usually separated mathematically into two broad categories: standing and traveling. The equation you have written is for a traveling wave. This can be demonstrated as follows. Let's ignore the @ phase part of the equation. Then: $$y=Asin(wt-kz)=Asin(k(\frac{w}{k}t-z))$$ $\frac{w}{k}$ is the wave velocity,v (check the units). Now, let's say we want to the wave's phase to remain constant. For simplcity we'll say we want it to be 0. What has to happen for this to be true? $$vt-z=0$$ or $$z=vt$$ This tells us that if we want to follow a portion of the wave whose phase is 0, the z component has to increase at the speed v. Thus this point in phase (as are then of course all points of phase) is traveling to the right at the speed v. Thus a traveling wave.

A standing wave usually is written like this: $y=A(z)sinwt$. The shape of the wave is governed by A(z) and it basically sits "standing and waving" since sinwt runs from -1 to +1.

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For a sound wave, the air particles to perform Simple Harmonic Motion (SHM) about their mean position. The equation for displacement in SHM is generally represented as: $$x = x_o\sin(\omega t + \phi)$$ Where $x_o$ is the amplitude, $\omega$ is angular frequency of oscillation and $\phi$ is the phase difference.

Now, for a sound wave, the equation is: $$y = A\sin(\omega t - kx + \phi)$$

Now here $y$ is $y$ component of displacement of particle at $x$ position.

Other wave equations are also alike.

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