Amplitude at successive wavefronts?

Question

Consider a spherical waves emanating from a point source initially having the amplitude is $A$. As he wave travels, forming wavefronts, will the amplitude of each point in all the secondary wavelets be the same? The new wavefront will surely have a larger size. So will each point in this wavefront also have the same initial amplitude $A$? If so, isn't it against the law of conservation of energy? Because, initially in a small wavefront less particles were oscillating with amplitude $A$ and after that many more particles are oscillating with the same amplitude $A$. So the energy is increasing as the wave is propagating. If we assume amplitude decreases at successive wavefronts, then won't the wave disappear after certain distance? I have thought about this a lot, but I am not able to figure it out. Can anybody shed some light on it in detail?

ANY HELP IS GREATLY APPRECIATED

Answer 1

The simple model of wavelets is adapted to take care of two things.

One is your idea that the amplitude of the wavelets must decrease with distance and the is done with a $\frac 1 r$ term where $r$ is the distance the wavelets has travelled. This ties in with the intensity (energy per second per unit area) falling off as $\frac{1}{r^2}$ as intensity is proportional to amplitude squared.

The other factor which is included is that the amplitude of the wave in a direction at an angle $\theta$ to the forward direction is proportional to $1+\cos \theta$. So the amplitude of the wavelets is not the same in all directions and zero in the reverse direction.

Answer 2

The conservation of energy is indeed a guiding principle for the way the amplitude changes for a spherical wave. Clearly it cannot stay the same, because then the energy would not be conserved; it would increase. So the amplitude must decrease, but it must decrease at such a rate that the energy remains constant as it propagates away from the source. So what one can do is to consider an amplitude that is a function of the radius $r$. Then we compute the intensity and integrate that over the surface of a sphere at an arbitrary radius, say $r=r_0$. The result is $$ {\rm energy} \sim 4\pi A(r_0)^2 r_0^2 , $$ where I assumed that the amplitude does not depend on the angles. So the integral only gives an overall constant factor, time $r_0^2$. We see that for the energy to remain constant, $A(r)^2$ must behave as $1/r^2$, which implies that the amplitude function must be $$ A(r) = \frac{A_0}{r} , $$ where $A_0$ is a constant that determines the total energy in the field. Note that, although the amplitude decreases with distance, it does not go to zero for any finite value of $r$.

As an aside, note that the amplitude would need to be infinite at $r=0$. This is artifact of assuming a point source. In practical situations, the source usually has a finite size.