Consider a stretched string which lies along x-axis when in equilibrium. During wave motion each particle described by its equilibrium position x, is displaced by distance y (say) in transverse direction. The value of y depends on which particle of the string is being disturbed (that is, on x) and also on time t at which it is observed.
The y is the function of x and t i.e., y = f(x,t). This function is known as wave function and if it is known in the form of f(x,t), it gives complete information regarding the wave motion.
During the wave propagation of a sinusoidal wave left to right along the string each particle of the wave undergoes simple harmonic motion, but successive particles are excited at later times in comparison to the particle at origin.
Let the displacement of the particle at the origin (x = 0) be given by,
y = A sin ωt
The time required for the disturbance to travel from x = 0 to some point x to the right of the origin is given by x/v, where v is the speed of the wave. The motion of the particle at x = 0 at earlier times (t-x/v). Thus, displacement of point x at time t is obtained simply by replacing 't' by (t - x/v) and, we obtain,
y(x,t) = A sin ω(t - x/v)
= A sin 2πv(t - x/v)
= A sin 2π/Tv (v × t - x)
= A sin 2π/λ (vt - x) ...(i)
It can also be written as
y = A sin 2π(t/T - x/λ) ...(ii)
Equ. (i) and (ii) are the equations of the progressive waves.
