Spherical harmonic waves

Optics Series – lecture — X
This lecture ( 1.5 hrs ) was delivered on 16 – 02 – 2017 to honors students

( all optics series lectures ) go to other available optics lectures

“Harmonic spherical Waves”


In 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.

( lecture-VIII: harmonic plane waves ) reading harmonic plane wave will help you with this lecture

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:

\boxed{\nabla ^2 = \frac{1}{r^2}\frac{\partial }{\partial r}\Big (r^2 \frac{\partial }{\partial r} \Big) +\frac{1}{r^2 sin\, \theta}\frac{\partial }{\partial \theta}\Big (sin \,\theta \,\frac{\partial }{\partial \theta} \Big)+ \frac{1}{r^2 sin ^2 \,\theta}\frac{\partial ^2}{\partial \phi ^2} }

where r, θ and φ are related to x, y and z by: x = r sin θ cos φ, y = r sin θ sin φ  and z = r cos θ. We shall also note that waves that are spherically symmetric do not depend upon angular dimensions, in other words they are independent of θ and φ.

spherical system

Spherical polar coordinates: azimuth angle ( φ ) zenith angle ( θ ) and vector magnitude variable (radial component, here r) Photo Credit: mdashf.org

Our wave – function, the solution to the wave equation which we call the spherical traveling wave now looks like: \psi \,(\,\vec {r}\,)=\psi \,(\,r, \,\theta, \,\phi\,)=\psi \,(\,r\,) with \nabla ^2 = \frac{1}{r^2}\frac{\partial }{\partial r}\Big (r^2 \frac{\partial }{\partial r} \Big)

To learn more about wave equation and the concepts of waves in physics read lecture — XII and XIII. If you are already comfortable with what we are discussing currently, you can skip the same and continue reading the following, and read the linked lecture sometime later in your convenience.

I must also advise that you might need to read lectures — VII and VIII, in conjunction with the current lecture ( preferably before the current lecture: lecture VIII first and then lecture — VII), for an efficient grasp of the subject matter at hand. I am therefore also linking these two lectures below.

( lecture — XII and XIII: Waves ) read about waves and wave equation, it will help with current lecture

( lecture — VII: Interference of two plane harmonic waves ) read about interference, it will help you with current lecture

( lecture — VIII: Harmonic Plane Waves ) read harmonic plane waves, it will help you with current lecture

The following results can be shown eg by using Cartesian coordinate system. I proffer it to you to try them as home work assignments. Here is how the Cartesian system Laplacian and the spherical system identities — that you should take as home work look like.

\nabla ^2 = \frac{\partial ^2 }{\partial x^2} + \frac{\partial ^2 }{\partial y^2} + \frac{\partial ^2 }{\partial z^2}

\frac{\partial \psi }{\partial x} = \frac{\partial \psi }{\partial r} \times \frac{\partial r }{\partial x}, \,\,\,\,\,\,\,\, \frac{\partial \psi }{\partial y} = \frac{\partial \psi }{\partial r} \times \frac{\partial r }{\partial y}, \,\,\,\,\,\,\,\,\frac{\partial \psi }{\partial z} = \frac{\partial \psi }{\partial r} \times \frac{\partial r }{\partial z}

\boxed{\nabla ^2 \psi (r) = \frac{\partial ^2 \psi }{\partial r^2} + \frac{2}{r}\frac{\partial \psi }{\partial r}} and also: \boxed{\nabla ^2 \psi = \frac{1}{r}\frac{\partial ^2}{\partial r^2}\Big (r\psi \Big)}

When we discussed “interference of two plane harmonic waves” in lecture — VII we introduced the concept of light waves as transverse waves of oscillating electromagnetic fields that satisfy differential equations known as wave equations. Please have a look there of the first few sections where the ideas have been given.

We will continue to delve into the deeper aspects of these electromagnetic and transverse nature of light time and again. In-fact we have devoted one full lecture towards this important idea, with a historical as well as technical perspective.

So without much fuss, lets write the differential wave equation in our spherically symmetric time – progressive mode. The wave equation in 3 – dimension is given by: \nabla ^2 \psi = \frac{1}{v^2}\frac{\partial ^2 \psi}{\partial t^2}. For spherical symmetry this assumes the form — by employing the spherical symmetric form of Laplacian stated above: \frac{1}{r}\frac{\partial ^2}{\partial r^2} (r\psi )=\frac{1}{v^2}\frac{\partial ^2 \psi}{\partial t^2}.

We readily see from the above “spherically symmetric differential wave equation” that the wave function — or solution to the wave equation is where ψ  varies only in radial direction i.e. ψ = ψ ( r ) and not in any angular variable θ or φ. Thus this wave function or the solution being a traveling or progressive wave has a general functional form of f ( r – vt ).

There is a lecture which discusses this in detail. This has been linked earlier but for convenience linked again.

Thus we have the forward wave: ψ ( r, t ) = f ( r – vt ) / r .

Such a wave is a spherical wave which moves outward radially from origin or source of the wave. We can also envisage the exact opposite. Instead of originating in a radial outward manner the wave can decide to travel backwards converging towards the center.

This is the so called backward or retarded wave as opposed to the diverging outward, forward or advanced wave. It has a similar form to the advance or forward wave except the representing function is different in general.

The backward wave looks like: ψ ( r, t ) = g ( r – vt ) / r.

The more general solution that accommodates both advanced and retarded wave components has a form: ψ ( r, t ) = C1 f ( r – vt ) / r + C2 g ( r – vt ) / r

Harmonic spherical waves

A specific solution to the above general solution to the differential wave equation is the harmonic traveling wave. This can be given as a cosine wave or with a more general exponential wave – profile, moving in time.

\boxed{ \psi\, (\,r,\, t\,) = \frac{A}{r}\,\cos\,k\,(\,r\mp vt\,)} and \boxed{ \psi\, (\,r,\, t\,) = \frac{A}{r}\,e^{\,i\,k\,(\,r\,\mp\, vt\,)}}.

A is called as the source strength of the wave. Any instant of time represents a snapshot of a cluster of concentric spheres.  The wave – front or the surfaces of constant phase is given by kr = constant. This we already saw in lecture — VIII.

The amplitude of spherical waves depends upon r, the radial distance. -1 ( inverse of the radial distance ) is called as the attenuation factor. The plane wave amplitude was seen in lecture — VIII to be constant but in spherical wave the amplitude is inversely proportional to radial distance.

A spherical harmonic wave front with radial distance. Its amplitude is dropped exponentially. This results in a decaying amplitude for the overall wave which is also harmonic or sinusoidal in nature.

Spherical harmonic wave: A spherical harmonic wave front varies with radial distance in a manner shown in the figure. Its amplitude drops exponentially. This results in a decaying amplitude for the overall wave which has additional harmonic or sinusoidal properties. Photo Credit: mdashf.org

 



Categories: Courses I developed, interference, light, optics, Physics, Teaching, waves

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  1. Electromagnetic Nature of Light — A brief history of light « Invariance Publishing House
  2. Waves. « Invariance Publishing House
  3. Harmonic plane waves – M Dash Foundation

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