We use the method of partial derivative to determine the wave equation. We have two similar quantities \(k = 2\pi/\lambda\) and \(\omega = 2 \pi /T\), both have similar expressions. You know the wave speed is \(v = \lambda f = \lambda/T\), and \(\omega / k\) is also equal to \(\lambda /T\), so you can find,
\[v=\frac{\omega }{k} \tag{6} \label{6}\]
First you know the wave equation for the wave traveling in positive x-direction is
\[y=A\cos (kx-\omega t)\]
Differentiating the above equation with respect to \(t\) keeping \(x\) constant we get the velocity of the particle \(v_\text{y}\) at \(x\).
\[\frac{\partial y}{\partial t}={{v}_{y}}=A \omega \sin (kx-\omega t) \tag{7} \label{7}\]
Note that the magnitude of velocity \(v_\text{y}\) is not the wave speed. Here the velocity of the particle \(v_\text{y}\) is the velocity when the particle moves up and down or back and forth in simple harmonic motion as the wave disturbance reaches the particle. You can find the acceleration \(a_y\) of the particle at position \(x\) by again differentiating Eq. \eqref{7} with respect to \(t\) keeping \(x\) constant:
\[{{a}_{y}}=\frac{\partial {{v}_{y}}}{\partial t}=\frac{{{\partial }^{2}}y}{\partial {{t}^{2}}}=-A{{\omega }^{2}}\cos (kx-\omega t) \tag{8} \label{8}\]
Now differentiating equation \(y=A\cos (kx-\omega t)\) with respect to position \(x\) keeping \(t\) constant, you get the slope of the curve at position \(x\):
\[\frac{\partial y}{\partial x}=-Ak \sin (kx-\omega t) \tag{9} \label{9}\]
Again differentiating Eq. \eqref{9} with respect to \(x\) keeping \(t\) constant, you'll get the curvature of the curve at position \(x\):
\[\frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}=-A{{k}^{2}}\cos (kx-\omega t) \tag{10} \label{10}\]
Now using Eqs. \eqref{8} and \eqref{10}, and you know \(v = \omega/k\) from Eq. \eqref{6}, you can get a form:
\[\begin{align*} \frac{{{\partial }^{2}}y/\partial {{t}^{2}}}{{{\partial }^{2}}y/d{{x}^{2}}}&=\frac{{{\omega }^{2}}}{{{k}^{2}}} \\ \text{or,}\quad \frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}&=\frac{1}{{{v}^{2}}}\frac{{{\partial }^{2}}y}{\partial {{t}^{2}}} \tag{11} \label{11} \end{align*}\]
The above equation Eq. \eqref{11} is called linear wave equation which gives total description of wave motion. This equation is obtained for a special case of wave called simple harmonic wave but it is equally true for other periodic or non-periodic waves. This is one of the most important equations of physics. You will get the same wave equation for the wave traveling in negative x-direction.
