Expression of a progressive wave

Equation of a progressive wave

We are well aware of  a progressive wave, that a progressive wave continuously move in forward direction in a given medium without change in form. Since it is a progressive wave we need to understand this wave in form of time and displacement. As the progressive wave move in forward direction it is periodic in time as well as space hence it is a function of time and distance.

Progressive wave

In above figure the wave shown is transverse wave as it is periodic in space therefore position of the particle of the medium is described by fixes value of x. The displacement of a particle about its mean position is in Y direction. General equation of displacement of a particle from mean position is given by ,

Y = A sinθ

* For a progressive wave moving in right direction, the equation for sinusoidal wave is given by

Y(x,t) = A sin ( Kx -ωt+Φ)----(1)

As K = 2π/λ   and  ω =2π/T----(2)

from (1) and (2)

Y(x,t) = A sin [(2π/λ)x - (2π/T)t+Φ] 

At a particular instant say t = t₀ , 

 y (x, t₀ ) = a sin (kx - ωt₀  + φ) 

 y (x, t₀ ) = a sin (kx + constant ) 

 Thus the shape of the wave at t =t₀, as a function of x is a sine wave. 

Also, at a fixed location x = x₀  ,

 y (x₀ ,t) = a sin (kx₀  -ωt + φ) 

y (x₀ ,t) = a sin (constant - ωt) 

Hence the displacement y, at x = x₀  varies as a sine function. 

* For a progressive wave moving in left direction, the equation for sinusoidal wave is given by

Y(x,t) = A sin ( Kx + ωt+Φ)----(3)

As K = 2π/λ   and  ω =2π/T----(4)

from (3) and (4)

Y(x,t) = A sin [(2π/λ)x + (2π/T)t+Φ] 

Note : (kx - ωt + φ) is the argument of the sinusoidal wave and is the phase of the particle at x at time t.

_{│<<<Progressive wave│   │Superposition of wave>>>│}    

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