General wave equation:
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$$ \frac{\partial^2 y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2} $$
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Sinusodial waves can be thought of as the real component of a rotating phasor.
Note that this can be derived by taking the relations between neighbouring elements to form partial derivatives, as the number of infinitesimal elements tends to infinity.
This can be solved as a superposition of two progressive waves moving in opposite directions:
$$ y=f(x-vt)+g(x+vt) $$
$$ y(x,t)=A\cos(kx-\omega t+\phi), \quad v_p =\frac{\omega}{k}, \ v_g = \frac{\mathrm d \omega}{\mathrm d k} $$
$k$ is the rate of change of phase in space, while $\omega$ is the rate of change of phase in time.
The phase velocity describes how fast the phase of the sinusodial wave changes, while the group velocity describes how fast an entire wavepacket moves. They are equal for a linear wave equation. The function $\omega(k)$ is known as a dispersion relation.
Potential energy can be derived by treating the string as a slightly elastic rope (if the string were completely inelastic, it would be impossible to create waves!) , so the string stretches a little and stores energy
$$ \frac{dK}{dx}=\frac{1}{2}\mu\dot y^2 \qquad \frac{dV}{dx}=\frac{1}{2}Ty'^2 $$
A standing wave is a solution of the wave equation of the form
$$ y(x,t)=f(x)\cos \omega t $$
Where it appears that energy is not transferred.
A finite version of a sinusodial wave is a wavepacket. Suppose we superimpose two waves of wave numbers $k \pm \Delta k$. The wavefunction is
$$ y=2e^{i(kx-\omega t)}\cos(\Delta k\, x-\Delta \omega\, t) $$
This looks like a sinusodial wave with a envelope dependent on the range of the wave numbers.