Question

A small ball floats in the center of a circular pool that has a radius of \(5.00 \mathrm{~m}\). Three wave generators are placed at the edge of the pool, separated by \(120 .^{\circ}\). The first wave generator operates at a frequency of \(2.00 \mathrm{~Hz}\). The second wave generator operates at a frequency of \(3.00 \mathrm{~Hz}\). The third wave generator operates at a frequency of \(4.00 \mathrm{~Hz}\). If the speed of each water wave is \(5.00 \mathrm{~m} / \mathrm{s}\), and the amplitude of the waves is the same, sketch the height of the ball as a function of time from \(t=0\) to \(t=2.00 \mathrm{~s}\), assuming that the water surface is at zero height. Assume that all the wave generators impart a phase shift of zero. How would your answer change if one of the wave generators was moved to a different location at the edge of the pool?

Short answer

Answer: If one of the wave generators is moved to a different location at the edge of the pool, the time it takes for the wave to reach the center of the pool would change, causing a change in the phase shift φ for that wave. This would lead to a different combined height function and would ultimately affect the height of the ball at each point in time. The changes would be visible in the sketch of the height of the ball as a function of time.

Step-by-step solution

Calculate the time each wave takes to reach the center of the pool

As the pool has a radius of 5m and the speed of water waves is 5m/s, we can calculate the time it takes for each wave to travel from its origin to the center of the pool using the formula: time = distance / speed. All waves will take the same time since they all travel the same distance with the same speed: time = 5m / 5m/s = 1s

Find the height of each wave at each point in time

To find the height of each wave, we can use the formula for wave displacement: y = A * sin(2π * f * t + φ) where A is the amplitude, f is the frequency, t is the time, and φ is the phase shift. Since the amplitude and phase shift for all the waves remain the same, we calculate the height for each wave generated by the respective wave generator: Wave 1 height = A * sin(2π * 2Hz * t) Wave 2 height = A * sin(2π * 3Hz * t) Wave 3 height = A * sin(2π * 4Hz * t)

Find the total height of the ball at each point in time.

To find the total height of the ball, we need to add the heights of all three waves. We know the height of each wave as a function of time: Total height = Wave 1 height + Wave 2 height + Wave 3 height = A * sin(2π * 2Hz * t) + A * sin(2π * 3Hz * t) + A * sin(2π * 4Hz * t)

Sketch the height of the ball as a function of time.

To sketch the height of the ball as a function of time, we simply need to plot the total height function from t=0 to t=2s. Due to limitations in text form, we recommend using graphing software to visualize the function.

Discuss how the answer changes if one wave generator is moved.

If one of the wave generators is moved to a different location at the edge of the pool, the time it takes for the wave to reach the center of the pool would change. Consequently, the phase shift φ for that wave would also change, leading to a different combined height function. This difference would be visible in the sketch of the height of the ball as a function of time.

Key concepts

Wave Superposition Principle

The wave superposition principle is a fundamental concept in physics that describes the behavior of interacting waves. It states that when two or more waves traverse the same space, their displacements add together at every point. This principle applies to all types of waves, including sound waves, light waves, and, as in our exercise, water waves.

Imagine throwing two stones into a calm pond. Ripples from where each stone entered the water spread out, and where they meet, the height of the water is the sum of the ripples created by each stone. Similarly, in the textbook exercise, the waves generated by the three wave generators in the pool meet in the center, where the ball floats. Their individual effects on the ball's height are additive, giving a resultant wave that oscillates with a combined amplitude and phase. By calculating the height of each wave at any point in time and summing them (Step 3), we apply the wave superposition principle to predict the total effect on the ball.

Wave Frequency

Wave frequency is the number of complete wave cycles that pass a given point in one second, measured in hertz (Hz). The cycle of a wave includes one full crest and one full trough, so a frequency of 2Hz means the wave oscillates two times per second.

In the outlined exercise, the three wave generators produce waves of differing frequencies—2Hz, 3Hz, and 4Hz—which affects the ball's motion. Each wave's frequency determines how rapidly the ball bobs up and down due to that wave. When sketching the ball's height as a function of time (Step 4), the combined effect of these frequencies results in a complex motion. This is because the wave with the highest frequency will cause the ball to oscillate more rapidly than the others, leading to a pattern that repeats less frequently than any of the individual waves alone.

Phase Shift

A phase shift in wave motion occurs when there is a displacement in the wave's cycles relative to a given reference point. It's often measured in degrees, where 360° corresponds to one full wave cycle. In the context of the pool and wave generators, if there were no movement of any wave generator, no phase shift would be present; the waves would reach the center of the pool simultaneously, starting their cycles at the exact same time—if we assume an initial phase shift of zero.

However, if we were to move one of the wave generators (Step 5), the timing of when its wave reaches the center would change, creating a delay or advancement in the wave cycle, also known as phase shift, relative to waves from the other wave generators. This shift would change the interference pattern and therefore the amplitude of the resultant wave experienced by the ball. This means the ball's motion in the pool would alter, and the sketch from Step 4 would differ.

Standing Waves

Standing waves, also known as stationary waves, occur when two waves of the same frequency and amplitude interfere with each other while traveling in opposite directions. They are characterized by nodes, where the wave displacement is always zero, and antinodes, where the displacement reaches a maximum.

While standing waves are not explicitly described in our pool exercise, understanding them can enhance comprehension of wave interactions. If the waves from our generators were reflected back with the same amplitude and frequency, and at an angle that allows them to superimpose, we could see the formation of standing waves at certain points in the pool. These would not contribute to the ball's motion because at the nodes there is no vertical displacement. However, if the ball were located at an antinode, it would experience the maximum vertical displacement.