Sound waves, simple harmonic motion, and wave equations Why is the equation $y=\sin (\omega t-kx)$ considered as a wave equation, whereas $y=\sin\omega t$ not considered a wave equation? I am really confused about it and its application.
 A: First, $y$ cannot be a wave equation, it can be a wave function.
The function $y=\sin(\omega t-kx)$ is a wave function simply because satisfies the wave equation,
$$\frac{\partial^2y}{\partial t^2}=v^2\frac{\partial^2y}{\partial x^2},$$
with $v^2=\omega^2/k^2$, whereas $y=\sin(\omega t)$ does not. Just check it. The latter however satisfies the simple harmonic motion equation
$$\frac{d^2y}{dt^2}+\omega^2y=0,$$
whereas the former does not.
A: The difference is in the kind of physical system each solution describes. 


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*$y = A \sin(\omega t + \phi)$


This equation describes the displacement $y$ of a point with respect to its equilibrium position $y = 0$, under the effect of a force of the form $F = -m\omega^2 y$. It is important to highlight the fact that this is a single moving point


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*$y = A \sin(\omega t - k x)$


This represents the motion of continuos medium. In this case, each position of the medium is labeled with the coordinate $x$. You can see the resemblance with the first case, in fact, if you look just one single point of the medium, say the one located at position $x = x_0$, then the equation of motion of this element is $y = A\sin(\omega t + \phi)$, with $\phi = k x_0 = {\rm cst}$, which is exactly the same description as before, meaning that each element of the medium behaves as a simple harmonic oscillator
A: Simply: a wave describes a periodic motion in space and time. Your equation must have both kinds of components (space - with the $kx$ term; and time - with the $\omega t$ term).
The equation for a point exhibiting simple harmonic motion does not show the wave moving in space - in other words, there is no propagation of the wave form.
