# Amplitude at successive wavefronts?

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

• @JohnForkosh but my textbook says >energy carried by a wave depends on the amplitude of the wave< (NCERT textbook physics class-12 part -2 chapter -wave optics and example question2)
– JM97
Commented Feb 22, 2016 at 4:56
– JM97
Commented Feb 22, 2016 at 5:22

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.

• If the amplitude decreases so will the wave disappears after certain distance?
– JM97
Commented Feb 22, 2016 at 7:38
• Consider this case in two dimensions then the circular wavefront near the source will have less number of particles oscillating than the wavefront which is away from the source so will the energy still conserved? 2. So In case of wavelets if we add all the wavelets arising from a wavefront which has particles oscillating with amplitude A , then in the direction of propagation the particles in wavelets will have amplitude 2A( since proportional to 1+cos0 ) so how is energy conserved here?
– JM97
Commented Feb 22, 2016 at 7:44
• Your comment about the oscillations of particles is a good one. As the energy is transported further out it is ditrbuted over more and more particles so each particle has less - a smller amplitude of oscillation. Instead of "disappears after certain distance" substitute "will get smaller and smaller" and indeed that is what happens. My answer above was related to 3D waves. Commented Feb 22, 2016 at 7:50
• The angular relationship works because other particles have a smaller amplitude so overall no energy is lost or gained. Commented Feb 22, 2016 at 7:54
• How do you justify energy conservation in wavelets formation?
– JM97
Commented Feb 22, 2016 at 9:00

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.