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 \(120 .\). 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: The height of the ball would vary as a superposition of sinusoidal waves with varying amplitudes resulting from the sum of the three waves at each point in time. If a wave generator is moved to a different location, it could affect the resulting amplitude due to constructive or destructive interference, altering the waveform's height, frequency, or phase.

Step-by-step solution

Identifying the waves' wavelengths

To determine the wavelengths of the waves, we need the speed \(v\) of the water wave, which is \(5.0 \mathrm{~m/s}\), and the frequency \(f\) of each wave generator. The formula to find the wavelength \(\lambda\) is: \(\lambda = \dfrac{v}{f}\) Calculate the wavelengths for each wave generator: 1. Wave generator 1 frequency: \(2.00 \mathrm{~Hz}\), Wavelength: \(\lambda_1 = \dfrac{5.00}{2.00} = 2.50 \mathrm{~m}\). 2. Wave generator 2 frequency: \(3.00 \mathrm{~Hz}\), Wavelength: \(\lambda_2 = \dfrac{5.00}{3.00} = 1.67 \mathrm{~m}\). 3. Wave generator 3 frequency: \(4.00 \mathrm{~Hz}\), Wavelength: \(\lambda_3 = \dfrac{5.00}{4.00} = 1.25 \mathrm{~m}\).

Determining the phase shifts

Since the wave generators impart a phase shift of zero, we don't need to perform any calculations here for initial phase shifts.

Calculate the heights of the waves at the ball's location

At each point in time, the waves add up constructively or destructively, affecting the height of the ball. The height of each wave can be calculated using the sinusoidal wave equation: \(h(t) = A \cdot \sin(\omega t + \phi)\) Where \(h\) is the height, \(A\) is the amplitude, \(\omega\) is the angular frequency \((\omega=2\pi f)\), \(t\) is the time, and \(\phi\) is the phase shift. In this case, the amplitude is the same for all three waves, so: - Height of Wave 1: \(h_1(t) = A \cdot \sin(2\pi \cdot 2.00 \cdot t )\) - Height of Wave 2: \(h_2(t) = A \cdot \sin(2\pi \cdot 3.00 \cdot t )\) - Height of Wave 3: \(h_3(t) = A \cdot \sin(2\pi \cdot 4.00 \cdot t )\)

Calculating the total wave height as a function of time

To calculate the total wave height as a function of time, we need to sum the heights of each wave for each time unit, and subsequently plot the waveform \(h(t)\) from \(t=0\) to \(t=2.0\)s. \(h_{\text{total}}(t) = h_1(t) + h_2(t) + h_3(t)\)

Plot the height of the ball versus time

Using \(h_{\text{total}}(t)\), plot the height of the ball versus time for the given time interval. Since the sine function has a periodic nature and the frequencies are whole numbers, the waveform will look like a superposition of sinusoidal waves with varying amplitudes.

Discussing result if a wave generator were moved

If a wave generator is moved to a different location at the edge of the pool, it may change the time each wave takes to reach the center of the pool, and therefore affect the resulting amplitude due to the constructive or destructive interference between the waves. As a result, the waveform representing the height of the ball may change, showing different heights, or changes in frequency or phase, depending on the new placement.

Key concepts

Wave Frequency

Understanding wave frequency is essential when studying wave phenomena. Frequency, denoted as \( f \), represents how often the waves pass a given point in one second and is measured in hertz (Hz). Higher frequency means more waves per second and typically, higher energy. In the context of the exercise, wave generators create waves with frequencies of 2 Hz, 3 Hz, and 4 Hz. This difference in frequency means that the waves will interact with each other over time, leading to patterns of interference which can dramatically affect the resultant motion of a floating object, such as the small ball in the center of the pool.

Wavelength Calculation

Wavelength, designated by the Greek letter \( \lambda \), is the distance between two identical points on successive waves (e.g., crest to crest). You can calculate the wavelength if you know the wave's speed and frequency using the formula: \[ \lambda = \frac{v}{f} \]Where \( v \) is the speed of the wave and \( f \) is its frequency. In our scenario, with waves traveling at 5.0 m/s, the wavelengths for the generators' frequencies can be calculated easily. For instance, a wave with a frequency of 2.00 Hz has a wavelength of 2.50 m. Understanding how to calculate wavelength helps in visualizing how waves from different sources might interact with each other at various points in space.

Sinusoidal Wave Equation

The sinusoidal wave equation models the height of a wave over time and is an invaluable tool in understanding wave behavior. It's expressed as:\[ h(t) = A \cdot \sin(2 \pi f t + \phi) \]where \( h(t) \) is the height at any given time \( t \), \( A \) is the amplitude, \( f \) is the frequency, and \( \phi \) is the phase shift. This equation shows how a wave fluctuates around a central position, similar to how our ball's height changes over time as waves generated from different points interfere with each other. By applying this equation to each wave generator in the pool, we can predict the height profile of the ball over time.

Constructive and Destructive Interference

Interference is a critical concept when two or more waves meet. When the crests and troughs of interacting waves align, they amplify each other in constructive interference, resulting in a larger amplitude. Conversely, if a crest meets a trough, destructive interference occurs, and the waves can cancel each other out, leading to smaller or even zero amplitude. The interference pattern affects the total height of waves impacting the ball in the pool. Since the amplitude for all three generators is the same, their waves can constructively or destructively interfere depending on their phase relationship at the point of intersection, causing the ball to oscillate with varying intensity.

Phase Shift

A phase shift in wave motion indicates a change in the wave's cycle position relative to a reference point. In mathematical terms, it's the \( \phi \) in the sinusoidal wave equation and it affects where the wave starts its cycle. A phase shift of zero means no shift from the reference point. In our exercise, initially, all wave generators impart a phase shift of zero, starting their waves simultaneously at the same phase. If we moved one generator, it would introduce a phase shift due to the extra distance the wave needs to travel, changing how it interferes with the other waves at the center. In essence, this could lead to a complex interference pattern and alter the ball's motion on the water's surface.