Waves.

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:

Harmonic Plane Waves

Optics Series Lecture, Lecture – VIII.
“Harmonic Plane Waves”
This lecture was delivered on 13th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics honors students.

In our last lecture, lecture-VII we began by discussing what are electromagnetic waves. We also discussed in good detail what are harmonic waves. Harmonic waves are those waves whose wave-profile is either sine, cosine or in general both sine and cosine combined with each other. Shortly (after within a few lectures) we will discuss what is wave profile and how to transform a wave profile into a traveling wave.

A wave profile, wave form or wave shape is simply a time instant view of a more general moving wave. We also discussed what is a plane wave. We applied our harmonic plane waves to the interesting phenomena of interference between two plane waves that are in addition monochromatic that is have same wavelength. Such waves traveling in a homogeneous media do so at a  fixed frequency and as long as they are in free-space their speed remains unaltered at the sped of light value c = 3 × 108 m/s.

A plane wave is one traveling wave where the wave fronts are planar points with equal phases all over the plane. In that order a spherical wave front is the locus of uniform phase over spherical configuration and a cylindrical wave front would be a traveling wave where the locus of uniform phase is nothing but a cylindrical surface.

In one of the future lecture, shortly, we will discuss in much detail what are spherical waves. Cylindrical waves are left to the advanced and willing students to work out by themselves. If time permits sometime in the future we can fall back and make a case for cylindrical wave fronts as well. But no promise at this point.

Note that waves are simply motion of phase points as a function of space or location and time instant. A phase is nothing but the angular argument of the wave described in terms of harmonic functions.

Plane Waves.
Let us begin studying Plane waves in detail. Here are some of its features.
1. A plane wave is the simplest example of a 3-dimensional wave.
2. These are so called, because plane wave wave fronts are planar in shape.
A wave-front is a locus of points on which the phase of the wave is same. Its a surface of wave-disturbances which move together, at the same speed.
3. Optical devices are often tuned to produce plane waves.
This necessitates the study of plane waves as base examples, where more complicated features can be assigned when they become pertinent.
4. Wave fronts are always perpendicular to the direction of wave propagation.