Question

Plot a wave packet. The function. $$ f(x, t)=e^{-(x-3 t)^{2}} \sin (3 \pi(x-t)) $$ describes for a fixed value of \(t\) a wave localized in space. Make a program that visualizes this function as a function of \(x\) on the interval \([-4,4]\) when \(t=0\). Name of program file: plot_wavepacket.py.

Short answer

Create a program using matplotlib to plot the wave packet for \(t=0\) on \([-4, 4]\).

Step-by-step solution

Understanding the Function

The given wave packet is represented by the function:\[f(x, t) = e^{-(x-3t)^2} \sin(3\pi(x-t))\]This is a product of a Gaussian function and a sinusoidal wave. When \(t = 0\), the function simplifies to:\[f(x, 0) = e^{-x^2} \sin(3\pi x)\]

Setting up the Environment

Open your text editor or IDE and start a new Python file named `plot_wavepacket.py`. Ensure you have the `matplotlib` library available for plotting; if not, install it using `pip install matplotlib`.

Import Necessary Libraries

In your Python file, start by importing the required libraries: ```python import numpy as np import matplotlib.pyplot as plt ```

Define the Function

Create a Python function to represent the wave packet: ```python def wave_packet(x): return np.exp(-x**2) * np.sin(3 * np.pi * x) ```

Generate Data Points

Define an array of \(x\) values over the interval \([-4, 4]\):```pythonx_values = np.linspace(-4, 4, 400)```

Compute Function Values

Compute the corresponding \(f(x, 0)\) values:```pythony_values = wave_packet(x_values)```

Plot the Function

Use `matplotlib` to visualize the wave packet: ```python plt.plot(x_values, y_values, label="Wave Packet at t=0") plt.xlabel("x") plt.ylabel("f(x, 0)") plt.title("Wave Packet Visualization at t=0") plt.legend() plt.grid(True) plt.show() ```

Finalizing and Running the Program

Save the file `plot_wavepacket.py`. Run the program using the Python interpreter to generate and display the plot of the wave packet for \(t=0\) over the specified interval.

Key concepts

wave packet visualization

Visualizing a wave packet helps in understanding how waves behave mathematically over space and time. A wave packet is essentially a combination of two mathematical constructs: a Gaussian function, which gives a localized shape, and a sinusoidal function, which describes wave-like oscillations. In our function, the Gaussian part is represented by \(e^{-(x-3t)^2}\), focusing the wave around certain points as time \(t\) changes. The sinusoidal part \(\sin(3\pi(x-t))\) makes the wave oscillate, detailing its periodic nature.
To visualize this wave packet when \(t=0\), the function becomes \(f(x, 0) = e^{-x^2} \sin(3\pi x)\), showing a wave mainly centered around \(x=0\), decreasing in amplitude as \(x\) moves away from zero. Both the shape and oscillations are distinct, making visualization essential for deeper insights. When plotted, you can see how the wave packet forms peaks and troughs, representing different wave positions at a single time point.

matplotlib plot

Plotting images of mathematical functions is a breeze with Python's `matplotlib`, a versatile plotting library. It helps turn complex equations into visual representations. For our wave packet, we leverage `matplotlib` to craft a detailed visual graph. This process starts by setting up your environment and making sure `matplotlib` is installed. Using `pip install matplotlib` can help if it's not already installed.
In your Python script, importing `matplotlib.pyplot` as `plt` is crucial. This alias allows you to use `plt` as a shorthand to invoke various plotting commands. The command `plt.plot()` lets you pass your data for visualization, and here, it shows our wave packet's behavior when \(t=0\). Subsequently, adding labels, titles, and grid lines through methods like `plt.xlabel()` and `plt.title()` enrich the plot, making the mathematical function easy to interpret.
Lastly, invoking `plt.show()` renders the plot, opening a window with a clear visual of the wave packet function, helping us see the peaks and oscillations clearly described by the mathematics.

function plotting

Plotting a function involves defining a mathematical model and representing it graphically over a specified domain. In our case, the function \(f(x, t)\) is plotted over the interval \([-4, 4]\). To begin, a set of data points that represent this interval must be generated using `numpy`. The `np.linspace()` function is ideal, as it creates an array of equally spaced \(x\) values, providing a smooth transition across the plot domain.
Next, a Python-defined function calculates the wave packet’s value for each \(x\), specifically \(f(x, 0)\) in this case. When defining this function, `numpy` operations like `np.exp()` and `np.sin()` are essential, as they handle mathematical operations efficiently over arrays.

• Defining the function within Python ensures it can be reused and verified easily.
• Generating \(y\) values from \(x\) values let you prepare the data set needed for plotting.

Once you have both \(x\) and \(y\) values, plotting the function using `matplotlib` becomes straightforward, providing a clear and easy-to-read visual of the wave packet. This is crucial for students and professionals who want to analyze the characteristics and phenomena described by the mathematical function.