# Optics Series Lecture, Lecture – X.

“Harmonic Spherical Waves” This lecture was delivered on 16th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics honors students.

In our lecture ( lecture-VIII ) we worked out the form of plane harmonic traveling waves. Note that soon we will barge into the concept of wave profile and 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 Den. 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 region that move out with equal phase are the wave fronts in this case, spherical in shape, called as spherical wave fronts.

We evidently need to describe the spherical wave fronts in spherical polar coordinate system, owing to the spherical symmetry in problems of 3-dimensional propagation of light waves.

Let us recall that the Laplacian in spherical polar coordinate system is given as:  $\inline&space;\dpi{150}&space;\bg_red&space;\fn_jvn&space;\tiny&space;\boxed{\nabla&space;^2&space;=&space;\frac{1}{r^2}\frac{\partial&space;}{\partial&space;r}\Big&space;(r^2&space;\frac{\partial&space;}{\partial&space;r}&space;\Big)&space;+\frac{1}{r^2&space;sin\,&space;\theta}\frac{\partial&space;}{\partial&space;\theta}\Big&space;(sin&space;\,\theta&space;\,\frac{\partial&space;}{\partial&space;\theta}&space;\Big)+&space;\frac{1}{r^2&space;sin&space;^2&space;\,\theta}\frac{\partial&space;^2}{\partial&space;\phi&space;^2}&space;}$ where r, θ, φ are given by: x = r sin θ cos φ , y = r sin θ sin φ , 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. Our wave-function, the solution to our wave equation which we call the spherical traveling wave now looks like: $\inline&space;\dpi{150}&space;\bg_red&space;\fn_jvn&space;\psi&space;(\vec&space;{r})=\psi&space;(r,&space;\theta,&space;\phi)=\psi&space;(r)$ with $\inline&space;\dpi{200}&space;\bg_red&space;\fn_jvn&space;\tiny&space;\boxed{\nabla&space;^2&space;=&space;\frac{1}{r^2}\frac{\partial&space;}{\partial&space;r}\Big&space;(r^2&space;\frac{\partial&space;}{\partial&space;r}&space;\Big)}$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 those Spherical System identities — that you should take as home work, look like. $\inline&space;\bg_red&space;\fn_jvn&space;\boxed{\nabla&space;^2&space;=&space;\frac{\partial&space;^2&space;}{\partial&space;x^2}&space;+&space;\frac{\partial&space;^2&space;}{\partial&space;y^2}&space;+&space;\frac{\partial&space;^2&space;}{\partial&space;z^2}&space;}$$\inline&space;\bg_red&space;\fn_jvn&space;\boxed{\frac{\partial&space;\psi&space;}{\partial&space;x}&space;=&space;\frac{\partial&space;\psi&space;}{\partial&space;r}&space;\times&space;\frac{\partial&space;r&space;}{\partial&space;x}&space;;\,and\,for\,&space;y&space;\,and\,&space;z.}$$\inline&space;\dpi{120}&space;\bg_red&space;\fn_jvn&space;\boxed{\nabla&space;^2&space;\psi&space;(r)&space;=&space;\frac{\partial&space;^2&space;\psi&space;}{\partial&space;r^2}&space;+&space;\frac{2}{r}\frac{\partial&space;\psi&space;}{\partial&space;r}}$ and also:$\inline&space;\dpi{200}&space;\bg_red&space;\fn_jvn&space;\tiny&space;\boxed{\nabla&space;^2&space;\psi&space;=&space;\frac{1}{r}\frac{\partial&space;^2}{\partial&space;r^2}\Big&space;(r\psi&space;\Big)}$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 will devote one full lecture towards this important idea, with a historical as well as technical perspective. (In-fact wile I am typing the current lecture today, on 16-3-2017 this lecture “Electromagnetic Nature of Light — A brief history of light” was delivered to both honors and elective students, its my 2nd favorite lecture so far, not second-most, but second, the other being, Application of matrix method to thick lens, lecture-VI. This will be available soon, perhaps tomorrow, I will try, my students are still copying the lecture, so I don’t have the copy, else instead of the present lecture that would have found its place here, today — so look forward to have it. ) So without much fuss, lets write the differential wave equation in our spherically symmetric time-progress mode — again for the time being check first few sections of lecture-VII. We will also have a full lecture devoted to explaining how to obtain the wave equation and related concept of traveling wave. Wave Equation in 3-dimension: $\inline&space;\dpi{200}&space;\bg_red&space;\fn_jvn&space;\tiny&space;\boxed{\nabla&space;^2&space;\psi&space;=&space;\frac{1}{v^2}\frac{\partial&space;^2&space;\psi}{\partial&space;t^2}}$. For spherical symmetry this assumes the form — by employing the spherical symmetric form of Laplacian stated above: $\inline&space;\dpi{200}&space;\bg_red&space;\fn_jvn&space;\tiny&space;\boxed{\frac{1}{r}\frac{\partial&space;^2}{\partial&space;r^2}\Big&space;(r\psi&space;\Big)=\frac{1}{v^2}\frac{\partial&space;^2&space;\psi}{\partial&space;t^2}}$.

We readily see from the above “spherically symmetric differential wave equation” that the wave function — or solution to the same, the wave equation, is where ψ  varies only in radial direction ψ = ψ(r) and not in any angular manner θ or φ. Thus this wave function, the solution, being a traveling or progressive wave has a general functional form, f(r – vt), which we will see in one of our future lectures, soon. Thus we have: ψ (r, t) = f(r – vt)/r. Such a wave is a spherical wave which moves outward –unhindered by choice-less indecisiveness, that is radially from origin. We can also envisage the exact opposite. Instead of originating in a radial outward manner Mr Wave can decide to fall backwards, converging towards center, the so called backward or retarded wave as opposed to the diverging, outward, forward or advanced wave. It has a similar form, except the the representing function is different in general. So, 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 differential wave equation is the harmonic traveling wave. This can be given as a cosine wave or a more general exponential wave-profile, moving in time.$\inline&space;\dpi{120}&space;\bg_red&space;\fn_cs&space;\boxed{&space;\psi&space;(r,&space;t)&space;=&space;\frac{A}{r}\,cos\,k(r\mp&space;vt)}$$\inline&space;\dpi{120}&space;\bg_red&space;\fn_cs&space;\boxed{&space;\psi&space;(r,&space;t)&space;=&space;\frac{A}{r}\,e^{i\,k(r\mp&space;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 surfaces of constant phase is given by kr = constant. This we saw in lecture-VIII. The amplitude of spherical waves depends upon r, the radial distance. r-1 is called as the attenuation factor. The plane wave amplitude was seen in lecture-VIII to be constant but spherical wave amplitude is inversely proportional to radial distance.

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