Travelling wave animation

Download scriptView on GitHubRun when the site was built.

A Gaussian pulse splitting into two halves that travel in opposite directions.

d’Alembert’s solution of the wave equation $u_{tt} = c^2 u_{xx}$ with $u(x, 0) = e^{-x^2}$ and $u_t(x, 0) = 0$ is

$$u(x, t) = \tfrac12\left[e^{-(x - ct)^2} + e^{-(x + ct)^2}\right].$$

Press play, or drag the slider. Plotly figures keep their frames, so animations work as well.

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Source code

travelling_wave.py

import numpy as np
import plotly.graph_objects as go

c = 1.0
x = np.linspace(-10.0, 10.0, 400)
times = np.linspace(0.0, 7.0, 36)


def u(t: float) -> np.ndarray:
    return 0.5 * (np.exp(-((x - c * t) ** 2)) + np.exp(-((x + c * t) ** 2)))


frames = [go.Frame(data=[go.Scatter(x=x, y=u(t))], name=f"{t:.1f}") for t in times]
steps = [
    {
        "label": f.name,
        "method": "animate",
        "args": [
            [f.name],
            {"mode": "immediate", "frame": {"duration": 0, "redraw": False}},
        ],
    }
    for f in frames
]
fig = go.Figure(
    data=[go.Scatter(x=x, y=u(0.0), fill="tozeroy", line={"width": 3})],
    frames=frames,
    layout={
        "title": "d'Alembert solution of the wave equation",
        "xaxis": {"title": "x", "range": [-10, 10]},
        "yaxis": {"title": "u", "range": [0, 1.05]},
        "margin": {"b": 130},
        "updatemenus": [
            {
                "type": "buttons",
                "showactive": False,
                "x": 0.0,
                "y": -0.3,
                "buttons": [
                    {
                        "label": "Play",
                        "method": "animate",
                        "args": [
                            None,
                            {
                                "frame": {"duration": 80, "redraw": False},
                                "fromcurrent": True,
                            },
                        ],
                    },
                    {
                        "label": "Pause",
                        "method": "animate",
                        "args": [
                            [None],
                            {"mode": "immediate", "frame": {"duration": 0}},
                        ],
                    },
                ],
            }
        ],
        "sliders": [
            {
                "steps": steps,
                "y": -0.3,
                "x": 0.15,
                "len": 0.85,
                "currentvalue": {"prefix": "t = "},
            }
        ],
    },
)
fig.show()