AIIMS 2006 Physics Wave Propagation MCQ Question

When a stone is thrown into still water, it creates circular waves that propagate outward from the point of disturbance. The amplitude of these waves is influenced by how energy is distributed as the waves move away from the source.

Correct Answer: C) $r^2$

The amplitude $A$ of waves generated in this manner decreases with distance $r$ from the center of the disturbance due to the conservation of energy. As the wave travels outward, the energy spreads over a larger area. The area $A$ of a circle is given by:

$$A = \pi r^2$$

As the wave propagates, the energy per unit area (intensity) decreases. The intensity $I$ of a wave is proportional to the square of the amplitude $A$, expressed mathematically as:

$$I \propto A^2$$

As the intensity is distributed over the area of the circular wave, we can express this relationship as:

$$I \propto \frac{E}{A} \propto \frac{E}{\pi r^2}$$

Where $E$ represents energy. Since intensity is inversely proportional to the area, the amplitude must decrease as the distance from the source increases. Specifically, if we assume the initial amplitude at the source is $A_0$, the amplitude $A$ at a distance $r$ can be modeled as:

$$A(r) \propto \frac{A_0}{r}$$

Since the energy is distributed over a larger area as it travels, the amplitude actually decreases proportional to $\frac{1}{r}$ for wave propagation in two dimensions (circular waves). Therefore, since energy (and thus amplitude) decreases with distance, we should expect:

$$A(r) \propto r^{-1}$$

However, this doesn't match with option C directly. The key point here is understanding that the amplitude is proportional to the inverse square root of the distance, which is consistent with how waves propagate in a two-dimensional medium. Thus, the amplitude decreases as:

$$A(r) \propto r^{-3/2}$$

This leads us to the conclusion that the relationships described in the options might not have direct physical implications here.

Clarifying Why Other Options Are Incorrect:

• Option A: $r^{1/2}$ – This suggests that the amplitude increases as we move away from the source, which contradicts the principle that energy spreads over a larger area, causing the amplitude to decrease.

• Option B: $r^{1}$ – Similar to option A, this implies a linear increase in amplitude with distance, which is incorrect as the energy dissipates with distance.

• Option D: $r^{-3/2}$ – This option aligns with the idea that amplitude decreases with distance due to energy dispersion; however, it incorrectly states the exponent. The correct relationship for two-dimensional waves is indeed a decrease proportional to $r^{-1}$.

Conclusion

Thus, the correct understanding is that the amplitude decreases as $r^{-1}$ for circular waves originating from a point source. The rationale behind the choice of $r^2$ as the correct option is based on the incorrect interpretation of the amplitude relationship. Despite the confusion, the real physics of wave propagation highlights how energy disperses, resulting in a decrease in amplitude as the radial distance increases.