Question

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

Short answer

Question: Briefly explain why the amplitude of circular water waves on the surface of a pond decreases as the waves travel away from the source. Answer: The amplitude of circular water waves decreases as they travel away from the source due to the conservation of energy and the spreading of wave energy across larger areas as the waves propagate outwards. As the waves move outwards, their energy is spread over larger circles, and since the energy is proportional to the square of the amplitude, the amplitude must decrease for the energy to remain constant.

Step-by-step solution

Understanding the wave energy and conservation

The energy of a wave is proportional to the square of its amplitude. As per the conservation of energy, the total energy in the wave remains constant as it moves through the medium (in this case, water). So, when the same amount of energy is spread across a larger area, the amplitude of the wave decreases. We will now illustrate this concept using the circular waves on a pond.

Spreading of wave energy in circular waves

The circular waves on a pond move outwards from the source (e.g., a pebble dropped in the water) in concentric circles. The energy of these waves spreads over larger and larger circles as they propagate. The area A of a circle is given by the formula A = pi * r^2, where r is the radius of the circle. As the radius increases, the area of the circle also increases, which means the wave energy has to spread across a larger area.

Relationship between wave energy and amplitude

As mentioned earlier, the energy E of a wave is proportional to the square of its amplitude A, i.e., E ∝ A^2. This means that if the energy of the wave spreads over an increasing area, the amplitude of the wave must decrease in order for the energy to remain constant.

Decrease in amplitude of circular water waves

As the circular water waves move away from the source, their energy spreads over larger and larger circles. Because the energy of a wave is conserved, and the energy is proportional to the square of the amplitude, the amplitude of the wave must decrease as it travels away from the source. In conclusion, the amplitude of circular water waves on the surface of a pond decreases as the waves travel away from the source due to the conservation of energy and the spreading of wave energy across larger areas as the waves propagate outwards.

Key concepts

Wave Energy Conservation

When we observe the serene, rippling surface of a pond after a pebble is tossed into it, we witness the principles of wave energy conservation in action. The concept of energy conservation states that energy cannot be created or destroyed; it can only be transformed or transferred from one form to another. In the context of water waves, the kinetic energy transferred to the water by the pebble must remain constant as the waves move away from the source. To understand how this works, consider that the energy of a wave is closely related to its amplitude—the height of the wave.
As the wave travels, energy conservation dictates that the total energy remains unaltered, but this energy is now distributed over a larger area. Given the conservation of energy, if the amplitude remained the same while the area increased, the wave would have to gain energy, which would violate the conservation principle. Therefore, a decrease in wave amplitude is necessary for the conservation law to hold true as the wave energy spreads outwards over progressively larger circular areas.

Wave Amplitude

The amplitude of a wave is a measure of its 'strength' or the height of its crest above the undisturbed surface of the water. Conceptually, the higher the amplitude, the more forceful the wave is perceived to be. In technically precise terms, the amplitude of a wave is intimately linked to the energy carried by the wave: the larger the amplitude, the more energy the wave carries. This relationship is quadratic, meaning that if you double the amplitude, the energy does not just double—it increases by a factor of four.
However, as the wave travels and the radius of the circular pattern increases, the same energy must be spread over a larger area. The increase in area follows a square law, much like the energy-amplitude relationship, which naturally leads to the amplitude diminishing as the square of the radius increases. This diminishing of amplitude is a visual cue that the energy is being conserved—it's not lost but spread out over a growing circumference.

Energy Propagation in Waves

Energy propagation in waves is the process through which wave energy travels through a medium, such as water. It's a mesmerizing sight—the energy from the pebble's fall propagates in all directions in the form of circular waves. It is important to note that it's the energy that is moving, not the water itself. The water particles merely move up and down, transferring energy to neighboring particles, creating a flow of energy away from the source.
The speed at which this energy travels is known as the wave speed and is determined by the properties of the medium. While the energy moves outward, something rather counterintuitive happens: the amount of motion or disturbance (amplitude) seen at any point on the water's surface decreases. This occurs because the energy must now 'do more work' and 'cover more ground,' figuratively speaking, as the wave extends over a larger area.

Circular Wave Behavior

Circular wave behavior is particularly visible on the surface of still water when disturbed. The waves radiate outwards in a pattern that appears to be simple, but the physics behind it is quite intricate. These waves maintain a circular shape because each point along the wavefront is equidistant from the source, much like the expanding rings you see after a pebble's splash. This uniformity in the wavefront's propagation direction maintains the circular symmetry.
The behavior of circular water waves showcases a fundamental characteristic of waves: as they travel outward, their energy does not dissipate randomly but rather is systematically distributed along increasing circumferences. Due to this patterned spread, the energy remains spread out evenly across the wavefront, ensuring that every point on a given circular wavefront has the same amount of energy. In short, the circular behavior ensures an equitable distribution of wave energy, which aligns with the observations we see—a gradual but consistent decrease in amplitude with increasing distance from the source.