What is progressive wave ?

Explanation

A jerk is given on a stretched string ast time t= 0 s. Assume that the wave pulse created during this disturbance moves along positive x direction with constant speed v as shown in Figure (a).

Represent the shape of the wave pulse, mathematically as y=y(x,0)=f(x) at time t= 0s. Assume that the shape of the wave pulse remains the same during the propogation. After some time t, the pulse moving towards the right and any point on it can be represented by x' (read it as x prime) as shown in Figure (b). Then,
`y(x,t) =f(x') = f(x-vt)`.
Similarly, if the wave pulse moves towards left with constant speed v, then y=f(x+vt). Both waves `y=f(x+vt)` and `y=f(x-vt)` will satisfy the following one dimensional differential equation known as the wave equation.
`(d^(2)y)/(dx^(2)) =1/v^(2)(d^(2)y)/(dt^(2))`
where the symbol `rho` represent partial derivative (read `(dy)/(dx)` as partial y by partial x). Not all the solutions satisfying this differential equation can represent waves, because any physical acceptable wave must take finite values for all values of x and t. But if the function represents a wave then it must satisfy the differential equation. Since, in one dimension (one independent variable), the partial derivative with respect to x is the same as total derivatives in coordinate x, we write So it can be written as
`(d^(2)y)/(dx^(2)) =1/v^(2) (d^(2)y)/(dt^(2))`