Problem 15 - University Physics with Modern Physics

Question

Why do circular water waves on the surface of a pond decrease in amplitude as they travel away from the source?

Step-by-Step Solution

Answer

Answer: The amplitude of circular water waves decreases as they travel away from the source because the energy of the waves is distributed over expanding circular wavefronts. As the wavefronts expand in radius, they cover a larger surface area, causing the energy density to decrease, which in turn leads to a decrease in amplitude.

Step 1: Define amplitude

Amplitude refers to the maximum displacement of a wave from its equilibrium position. It is a measure of the wave's energy, as a wave with a higher amplitude has more energy than a wave with a lower amplitude.

Step 2: Understand the circular wavefront

A circular wavefront is produced when a disturbance (e.g., a stone thrown into a pond) occurs at a single point on the surface of the water. The energy from this disturbance produces circular wavefronts, centered on the point of disturbance. As the wavefronts travel away from the source, they expand in a concentric and roughly circular pattern.

Step 3: Explain energy distribution in circular wavefront

The energy from the disturbance is spread evenly over the surface of each successive wavefront. As a wavefront expands, its surface area increases; specifically, the surface area of a wavefront is proportional to the square of its radius (A = 4πr^2). Consequently, at a greater distance from the source, the same amount of energy is distributed over a larger area, resulting in a reduced energy density.

Step 4: Relate energy density to amplitude

Since the amplitude of a wave is directly related to its energy, a decrease in energy density leads to a decrease in amplitude. As the wavefronts expand and travel away from the source, they encounter a larger area to cover. This causes the energy to be spread thinner, leading to a reduction in the amplitude of the waves.

Step 5: Conclusion

Circular water waves on the surface of a pond decrease in amplitude as they travel away from the source because the energy of the waves is distributed over expanding circular wavefronts. As the wavefronts expand in radius, they cover a larger surface area, causing the energy density to decrease, which in turn leads to a decrease in amplitude.