harmonic waves


Optics Series Lecture, Lecture – XII and – XIII

“Traveling waves, Differential wave equations, Particle and wave velocities.”

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.

Spherical harmonic waves

In our lecture — VIII, we worked out the form of plane harmonic traveling waves. Note that soon we will have to a. address the concept of wave profile and b. how to convert a wave profile into its corresponding time-dependent or traveling form.

But before we do that here is yet another general form of a traveling wave which we often meet in the physicists work-place. The traveling spherical wave fronts. Let us work out its details.

Spherical Waves
When a stone is dropped in water it sends out circular waves. Similarly a sphere or a glob of matter that oscillates inside of a water body would send out 3 – dimensional waves or ripples.

Sources of light wave, which we will study in great detail, in this course, to fulfill our insatiable hunger for understanding the nature of optical phenomena similarly send out oscillations which propagate radially and uniformly in all directions. These are the spherical waves and the points or regions that move out with equal phase are the wave fronts in this case, spherical in shape and known as spherical wave fronts.

We obviously need to describe the spherical wave fronts in spherical polar coordinate system, due to the spherical symmetry of problems of 3 – dimensional propagation of light waves.

Let us recall that the Laplacian in spherical polar coordinate system is given as:

Harmonic plane waves

Optics Series Lecture, Lecture – VIII.

“Harmonic Plane Waves”

This lecture ( 1.5 hours ) was delivered on 13th February 2017 to 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 a combination of both sine and cosine. You can learn more about what is wave profile and how to transform a wave profile into a traveling wave in the following lecture.

A wave profile, wave form or wave shape is simply a spatial snapshot view of a more general moving wave, at a suitable time. We have also discussed what is a plane wave. We studied the interesting phenomena of interference between two plane waves in the context of our harmonic plane waves. We assumed that our harmonic plane waves are also monochromatic waves, that is they have the same same wavelength.

These waves traveling in a homogeneous media have a fixed frequency and as long as they are in free – space their speed remains unaltered at the speed of light value c = 3 × 108 m/s.

A plane wave is a traveling wave where the wave fronts are planar points with equal phases anywhere on the plane. In a similar manner a spherical wave front is the locus of uniform phase over a 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 lecture we have discussed in much detail what are spherical waves. Cylindrical waves have been 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 I make no promises at this point.

Note that the waves are simply a motion of the phase points and therefore waves are a function of the space ( or location ) and time instant by which they are described. A phase is nothing but the angular argument of the wave described in terms of harmonic functions. Since harmonic functions ( another name for sinusoidal functions ) are periodic functions and trigonometric functions have a certain periodicity in terms of angular values, a phase represents the angular value ( or argument ) of these functions.

Plane Waves
Let us begin studying the plane waves in detail. Here are some of its features.

1. plane wave: a plane wave is the simplest example of a 3-dimensional wave.

2. why named so: plane waves are so called, because plane wave have wave fronts that 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. simplest example: 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.