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महाराणा प्रताप ट्रेल सज्जनगढ़ उदयपुर

महाराणा प्रताप ट्रेल सज्जनगढ़ उदयपुर में इको ट्रेल 30 नवम्बर को राजस्थान वन विभाग उदयपुर डिविजन तथा WWF-India उदयपुर डिविजन के सानिध्य में महाराणा प्रताप ट्रेल सज्जनगढ़ उदयपुर में इको ट्रेल की गई, जिसमें WWF-India के स्टेट काॅर्डिनेटर श्रीमान अरूण सोनी तथा वन विभाग कीे ओर से डाॅ. सतीश कुमार शर्मा, सेवानिवृत्त अधिकारी मौजूद थे। मुझे भी इस इको ट्रेल में जाने का सुअवसर प्राप्त हुआ, जो गोरीला व्यू पाॅइंट से बड़ी-लेक व्यू पाॅइंट तक की गई इसमें मुझे विज्ञान की एक नई शाखा के बारे में पता चला, जिसे टट्टी विज्ञान कहा जाता है। सुनने में आपको थोड़ा अजीब लगेगा, मुझे भी सुनकर हैरानी हुई, परन्तु वास्तव में एक ऐसा भी विज्ञान है, जिसके बारे में डाॅ. सतीश शर्मा ने बड़े ही विस्तार पूर्वक बताया कि किस प्रकार वनों में जानवरों की टट्टी देखकर यह पता लगाया जा सकता है कि यहां कौनसा जानवर आया था। जानवरों की टट्टी कितनी पुरानी है, वह गीली है या सूखी है। इसी के आधार पर उस विशेष जंगल में कौन-कौनसे जानवर विचरण करते हैं, उसके बारे में वन विज्ञान के कर्मचारी पता लगा लेते हैं। जानवरों की टट्टी का विश्लेषण करके यह पता लगा...
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Phase velocity and Group velocity | Quantum mechanics

Phase velocity and Group velocity

Phase velocity (Wave velocity)

  • If a single monochromatic wave (wave of single frequency or wavelength) travels through a medium, then the velocity through which it propagate is wave velocity.
  • Equation of plane progressive wave propagating in x-direction
            y = a sin (ωt - kx)
  • Here a = amplitude, ω = angular frequency, and k = 2π / λ = wave vector or propagation constant
  • Since in wave motion, the particles of the medium vibrates about their mean position continuously, the velocity acquired by particle during this motion is particle velocity.
  • During this motion, the disturbance (crest or trough) also produced and they also move in the direction in which the wave propagate and also with the same velocity.
  • The velocity of the disturbance through which it advances through the medium is phase velocity (wave velocity).

  • Above equation gives the wave velocity. 
  • Thus the ratio of angular frequency to the propagation constant (wave vector) is known as wave velocity.
  • In equation  y = a sin (ωt - kx), (ωt - kx) is phase of wave motion
  • For the planes of constant phase or wavefronts
            (ωt - kx) = constant
  • Above expression represents the wave velocity.
  • Since the wave velocity is the velocity with which planes of constant phase advance through the medium, and above expression is obtained for such type of wave, so it is also known as phase velocity.
                

Group velocity 

  • In practical life we come across pulse or group of wave of slightly different frequencies. This group of waves are wave packets.
  • A wave packet refers to the case where two or more waves exist simultaneously.
  • The group velocity of a wave packet is the velocity with which the maximum amplitudes of the group advances in the medium.
  • This is the velocity with which the energy in the group is transmitted.
  • The group velocity is defined as the rate of change of angular velocity with respect to wave vector.
            
  • Let a wave packet is made up of two harmonic waves
  • Let on superposition, these waves form a wave packet, then resultant displacement of wave packet
                y = y1 + y2

  • The amplitude of this equation is modulated by

  • Here figure (a) and (b) shows the two waves of slightly different frequencies, and figure (c) shows the resultant wave of these two waves.

  • If △ω and △k are very small. 

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