Optics Series Lecture, Lecture – XII and – XIII.
“Traveling waves, Differential wave equations, Particle and wave velocities.”
These lectures were delivered on 17th and 20th February 2017, in two lecture sessions of 1 and 1/2 hours each. The web version has been named “Waves.” and the lectures were delivered to Physics honors students.
In one of our earlier optics session lecture I had hinted at having waves defined by their pulse shape called as wave profile — or alternatively wave shape or wave form, and transcribing them into forms that represent actual wave motion. The later are then called as traveling or progressive waves. The former, the so called wave shape or wave profile are then time-snapshots of the full fledged time varying waves that we just called traveling waves.
Remember that stationary or standing waves are not wave profiles or any snapshots of a single traveling wave, they are rather the superposition of an advanced and a retarded wave — that is one traveling wave moving forward and another exactly shaped traveling wave moving in the reverse direction. We studied advanced and retarded waves, here.
We have also already dealt with traveling waves in much detail, eg, here and here. This lecture will justify what we have been espousing all along. Also in complex waves that are found in quantum mechanical theories, we have what are called as stationary states, these are like the time-snapshots of the quantum mechanical waves, represented through the energy of the system. Since the full energy or wave cycle is not necessarily contained in a given amount of time called as a time window, we have a corresponding uncertainty relation called an energy-time uncertainty relation.
But talking about an instant of time, a stationary state which represents the energy of the wave in that instant, are well defined states of energy and called as eigen-states. But what would happen if one takes a picture of a dynamic system? The fuzzed out region or so called “motion blur” might show up, because these time instants are not well defined eigen states rather superposition of random number of any of them, may be.
One dimensional Traveling Waves.
A traveling wave is a self sustaining oscillation of particles of a medium or oscillations of any physical quantity at different space-time points so that energy is transported across the medium when the oscillation propagates in the medium. There is no motion of the relevant medium in the ideal description of the wave. The oscillating particles move periodically about their equilibrium locations and in the case of physical quantities they take values around their equilibrium values. Examples of waves are mechanical waves: