Wave Formula

✦Equation of a Harmonic Wave:

         Harmonic wave are generated by sources that execute simple harmonic motion.

    A Harmonic wave travelling along the positive (+ve) direction of x-axis is represented by 

       y=A sin (ωt–kx) 

         =A sin {2π(t/T–x/λ)}

         =A sin {2π/λ(vt–x)}

   

        Where, 

 ☞ y = displacement of the particle of          the medium at a location X at time t

☞ A = amplitude of the wave 

☞ λ = wavelength 

☞ T = time period 

☞ v = velocity of the wave in the 

      medium, Ʋλ

☞ ⍵ = angular frequency, 2π/T

☞ k = angular wave number, 2π/λ

           

            If the wave is travelling along the negative (–ve) direction of x-axis then

        y = A sin (⍵t + Kx)  

Differential equation of wave motion:

   d^2y/dx^2 = 1/v^2 × d^2/dt^2

Relation between wave velocity and particle velocity: 

 y = A sin (⍵t – kx) .....(i) 

   

Particle velocity, 

        Vₚ = dy/dt = A⍵ cos(ωt–kx) ......(ii) 

Slope of displacement curve,

         dy/dx = –Ak cos (ωt–kx) ......(iii) 

When we divide equation (ii) by equation (iii), we get the following 

              Vₚ = –v. dy/dx

☞ λ = Ʋ2π/ω = ƲT.

☞ k = 2π/λ

        = 2πv/Ʋ

        = ω/Ʋ.

           

Relation between phase difference, path difference and time difference :

☞Phase difference of 2π radian is equivalent to a path difference λ and a time difference of period T. 

                                   

☞Phase difference = (2π/λ)× path difference

                               Φ = (2π/λ)×x 

                                                  x= (λ/2π)×Φ

☞Phase difference =(2π/T)× time difference

                               Φ = (2π/T) ×t 

                                                    t = (T/2π)×Φ

☞Time difference = (T/λ)×path difference 

                                t = (T/λ)×x 

                                               x= (λ/T)×T