Question

A ping-pong ball is floating in the middle of a lake and waves begin to propagate on the surface. Can you think of a situation in which the ball remains stationary? Can you think of a situation involving a single wave on the lake in which the ball remains stationary?

Short answer

What conditions are required for this to occur? Answer: A ping-pong ball can remain stationary on a lake's surface with waves propagating on it if there are multiple waves of equal amplitude coming from different directions, and their interference patterns create a point of destructive interference where the ball is floating. In a situation with just a single wave, it's unlikely that the ball would remain stationary, as there is no other wave to create destructive interference.

Step-by-step solution

Consider wave interference patterns

When multiple waves interact, they can combine to form constructive and destructive interference patterns. Constructive interference occurs when the peaks and troughs of two or more waves align, causing a larger wave. Destructive interference happens when the peak of one wave aligns with the trough of another wave, resulting in a smaller amplitude or even cancellation.

Situation with multiple waves

In a situation where multiple waves of equal amplitude are propagating on the lake from different directions, it's possible for the ping-pong ball to remain stationary. This can happen if the ball is located at a point where destructive interference occurs. At this point, the waves cancel each other out, and there is no net movement or disturbance in the water, allowing the ball to stay in place.

Situation with a single wave

In a scenario involving just a single wave, it's challenging to create a situation where the ping-pong ball remains stationary. A single wave will cause the water to move up and down, which will cause the ping-pong ball to move accordingly. Although the displacement of the water might be minimal at some points, it's unlikely that the ball would remain entirely stationary throughout the wave's progression as there is no other wave to create destructive interference. In conclusion, the ping-pong ball can remain stationary when multiple waves of equal amplitude propagate on the lake's surface from different directions, and their interference patterns create a point of destructive interference where the ball is floating. However, in a situation involving a single wave on the lake, it's unlikely that the ball remains stationary, as there is no other wave to create destructive interference.

Key concepts

Destructive Interference

Destructive interference is a fascinating phenomenon occurring when two or more waves come together in such a way that they cancel each other out. Imagine two waves on a lake moving towards each other. One wave has a crest (the highest point of the wave) arriving at the same location and at the same time as a trough (the lowest point of the wave) of another wave. Here, the height of the crest is precisely negated by the depth of the trough, creating a still or flattened surface.

This concept explains the stationary ping-pong ball exercise. If the ball is positioned at a point of destructive interference, the upward force from one wave is countered exactly by the downward force from another. No net force means no motion – the ball stays put! In everyday life, this principle is used in noise-cancelling headphones, where sound waves produced by the headphones destructively interfere with ambient noise, reducing what you hear.

Constructive Interference

In contrast to destructive interference, constructive interference amplifies wave interaction. When two waves coincide perfectly in phase – meaning their crests and troughs match up – they combine to form a wave with greater amplitude. Hence, the wave's overall height increases.

You could think of constructive interference as a team effort where two waves join forces to make a 'superwave'. If the ping-pong ball were at a point of constructive interference, this 'superwave' would move the ball dramatically. In applications like laser physics, constructive interference is important for amplifying light waves to produce high-intensity beams.

Wave Propagation

Wave propagation refers to the movement and transmission of waves through a medium. In our lake example, waves travel across the water, transferring energy from one place to another. This energy causes the ping-pong ball to move as well.

Wave propagation isn't limited to water; it occurs in a variety of mediums, including air (sound waves), space (light waves), and even solid ground (earthquakes). Understanding how waves propagate allows us to predict and explain the motion of objects influenced by those waves, whether it’s the bouncing of a ping-pong ball or the ripples from a pebble tossed into a pond.

Amplitude

Amplitude is a measure of a wave's strength or intensity – specifically, the height of its crests or the depth of its troughs. Higher amplitude means a more powerful wave, capable of exerting more force on an object, like our floating ping-pong ball. If a wave on the lake had a large amplitude, the ball would bob up and down with greater intensity.

Amplitude is not only important in physical waves; it is also crucial in sound (loudness), light (brightness), and even quantum mechanics (probability of finding a particle). Understanding amplitude helps us to comprehend the energy and potential impact of different kinds of waves and their influence on the environment around them.