Wave

✦ Wave Motion:

        Wave motion is a type of motion in which the disturbance travels from one point of the medium to another but the particles of the medium do not travel form one point to another. 

For the propagation of wave, medium must have inertia and elasticity. These two properties of medium decide the speed of wave. 

✦Amplitude:

               The maximum displacement of any particle from its mean position is called amplitude (a) of the wave.

 

✦Period:

        The time taken (T) for any particle to complete one vibration is called period of the wave. 

✦Frequency:

     The number of vibrations per second by a particle is called frequency (n) of the waves where

 n = 1/T. The frequency can also be said to be the number of waves passing per second across any point of the medium.

✦Wavelength:

         The distance between two consecutive particles of the medium which are in the same phase or which differ in phase by 2π (radians) is called wavelength (λ) of the wave.

✦ Types of waves

    There are Two types of waves 

1) Mechanical wave :  These waves require material for their propagation. 

For example : Sound waves, etc. 

     There are two types of Mechanical waves

i) Transverse wave: The particles of medium oscillate in a direction perpendicular to the direction of wave propagation. 

ii) Longitudinal waves: The particles of medium oscillate about their mean position along the direction of wave propagation. 

2) Non mechanical waves (Electromagnetic wave) : This waves do not require any material medium for their propagation. 

For example : X-ray,  Light waves etc. 

✦ 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:

   d²y/dx² = 1/v² × d²/dt²

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

✦ Speed of Sound in a gas

Newton's formula

Laplace's correction 

  

Did you know 🤔🤔

• For a wave 

          v = fλ

• The velocity of sound in air 

• Particle velocity is given by vp = dx/dt. It changes with time. The wave velocity is the velocity with which disturbance travel in the medium and is given by vw = ω/k.

• When a wave reflects from denser medium, the phase change is  π and when the wave reflects from rarer medium, the phase change is zero. 

• In a tuning fork, the wave produced in the prongs is transverse whereas in the stem is longitudinal. 

• A medium in which the speed of wave is independent of the frequency of the waves is called non-dispersive. 

    For example Air is non-dispersive.                  medium for the sound waves. 

• Transverse waves can propagate in the medium with shear modulus of elasticity.  

                                                                                                                                    More....