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Energy conservation is an important principle in the study of waves, especially when analyzing how they propagate through a medium. Imagine you drop a pebble into a still pond; the energy transferred to the water from this initial disturbance causes waves to ripple outward. This energy must be conserved, meaning that it doesn't magically increase or disappear as the wave spreads—it just gets distributed over an expanding area.

As suggested in the exercise, we consider the energy carried by one moving ripple. Since energy is conserved, the total amount of energy present in the initial disturbance is the same as the energy distributed across the wave fronts as they expand. Mathematically, if the power of the wave remains constant, the intensity of the wave will change in response to the increasing area it has to cover. Therefore, we understand that despite the energy being constant, the larger the area, the more diluted the energy becomes, which affects the amplitude of the wave.

The concept of circular wave expansion can be visualized as the ripples on a pond after the aforementioned pebble is dropped. These waves travel outward in concentric circles, each representing a wave front. The area of each subsequent circle increases as the wave moves further from the origin.

Calculating the area of these circles with the formula \(A = \pi r^2\) helps us understand that as the radius \(r\) gets larger, the area increases by the square of the radius. It implies that, at a larger distance, the same amount of wave energy spreads over a much greater area, effectively reducing the energy per unit area available to exert influence, which in practical terms, can be experienced as a decrease in the wave's amplitude.

The intensity of a wave is a measure of the power per unit area that is transmitted by the wave. It's a concept that can be somewhat counterintuitive since it involves the amount of energy passing through a unit area, not just the total energy in the wave.

By knowing that intensity is power divided by area (\(I = P/A\)), and considering that wave energy (and therefore power) is conserved, any increase in area results in a decrease in intensity. Intuitively, as a wave front expands outward, each point on the wave front has less energy passing through it per unit of time because the same amount of energy is now spread over a larger area.

Thus, connecting the dots, since the square of the amplitude is directly proportional to intensity (\(I \propto A^2\)), a reduction in intensity with increased distance from the source means that the amplitude must decrease correspondingly. When the amplitude's dependence on radius is calculated for a circular wave, it confirms the relationship \(A \propto 1/\sqrt{r}\), providing a mathematical explanation for our observations of wave behavior.