3.1 General equation of wave motion

2.1 Transverse waves
2.2 Longitudinal waves
2.3 Differences between transverse and longitudinal waves

3.1 General equation of wave motion
3.2 Wave function
3.3 Equation of a plane progressive harmonic wave
3.4 Important relations

4.1 Speed of different types of wave

5.1 Power $(P)$
5.2 Intensity $(I)$

Wave is the transmission of disturbance with a certain velocity.

Suppose the disturbance is propagating along positive $X$-axis with a velocity $v$.

The disturbance which is a function of $x$ and $t$ can be represented as, $$y = f(x,t)$$ The above equation is also known as wave equation.

Consider an infinite length of the string lying along $x$-axis. Suppose that a single linearly polarised pulse travels in the direction of increasing value of $x$.

Let at time $t=0$, a stationary observer standing at origin $O$ observes, $y$ be the transverse displacement of the string, then the wave equation can be written as, $$y = f(x,t)$$ The above equation states that the transverse displacement $y$ depends on propagation distance $(x)$ and time $(t)$.

In an ideal case, the pulse travels along the string with constant velocity while maintaining its shape.

Let an observer is moving with the same velocity at which the pulse moves along the string. The observer would observe a stationary pulse having a shape described by function, $$y' = f(x')\quad ...(i)$$

Relation between the two frame of reference can be given by, $$\begin{equation} \begin{aligned} x = x' + vt\quad ...(ii) \\ y = y'\quad ...(iii) \\\end{aligned} \end{equation} $$

From equation $(i),(ii)$ & $(iii)$ we can write, $$y = f(x - vt)$$

Similarly, if the wave propagates along the negative $X$-axis then, $$y = f(x + vt)$$

Then the general equation of wave can be written as, $$y = f(x \pm vt)\quad {\text{or}}\quad y = f(ax \pm bt)$$

As shown in the figure, a pulse propagates along positive $X$-axis with a velocity $v$ along a tensioned string.

At time $t_1$ & $t_2$, the peak of the pulse is at co-ordinate $x_1$ & $x_2$ respectively.

Mathematically, $$\begin{equation} \begin{aligned} {\text{velocity}}(v) = \frac{{{\text{Distance travelled}}}}{{{\text{Time taken}}}} \\ v = \frac{{{x_2} - {x_1}}}{{{t_2} - {t_1}}} \\\end{aligned} \end{equation} $$

We know, $$\begin{equation} \begin{aligned} {x_2} - {x_1} = \lambda \\ {t_2} - {t_1} = T \\\end{aligned} \end{equation} $$ So, $$v = \frac{\lambda }{T}$$

Alternatively, $${\text{velocity}}(v) = \left| {\frac{{{\text{Coefficient of }}t}}{{{\text{Coefficient of }}x}}} \right| = \left| {\frac{b}{a}} \right|$$