.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/fields/electrodynamics/plot_em_cavity_modes.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_fields_electrodynamics_plot_em_cavity_modes.py: A standing TM cavity mode oscillating in place ==================================================== :func:`~physicskit.fields.electrodynamics.fdtd_2d_tmz` and :func:`~physicskit.fields.electrodynamics.fdtd_2d_tmz_evolve` integrate the same TMz-mode Maxwell curl equations as the other FDTD examples in this gallery, but already enforce ``Ez=0`` on all four grid edges every step -- exactly a perfectly-conducting (PEC) cavity wall, no absorbing boundary needed. A rectangular PEC cavity of size :math:`L_x\times L_y` supports discrete standing-wave modes .. math:: E_z(x,y) = \sin\!\left(\frac{m\pi x}{L_x}\right)\sin\!\left(\frac{n\pi y}{L_y}\right), \qquad \omega_{mn} = c_0\pi\sqrt{\left(\frac{m}{L_x}\right)^2 + \left(\frac{n}{L_y}\right)^2}, which already vanish on the boundary by construction. :func:`~physicskit.fields.electrodynamics.tmz_cavity_mode` builds this analytic :math:`TM_{mn}` mode shape directly; seeding the FDTD grid with it (here :math:`m=n=1`) launches a mode that oscillates in place at its analytic frequency :math:`\omega_{mn}`, rather than propagating outward the way the point-source and dipole-antenna demos do. .. GENERATED FROM PYTHON SOURCE LINES 25-31 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.fields import animate_field_2d, courant_limit_2d, fdtd_2d_tmz_evolve, tmz_cavity_mode .. GENERATED FROM PYTHON SOURCE LINES 32-34 The analytic TM_11 mode of a rectangular PEC cavity ------------------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 34-43 .. code-block:: Python Nx, Ny, dx, dy = 61, 41, 1e-3, 1e-3 m, n = 1, 1 Ez0, omega_mn = tmz_cavity_mode((Nx, Ny), dx, dy, m=m, n=n) Hx0 = np.zeros((Nx, Ny)) Hy0 = np.zeros((Nx, Ny)) eps_r, mu_r = np.ones((Nx, Ny)), np.ones((Nx, Ny)) dt = 0.4 * courant_limit_2d(dx, dy) .. GENERATED FROM PYTHON SOURCE LINES 44-46 Leapfrog the cavity forward and record snapshots ------------------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 46-51 .. code-block:: Python period = 2 * np.pi / omega_mn steps = int(1.2 * period / dt) frames, times = fdtd_2d_tmz_evolve(Ez0, Hx0, Hy0, eps_r, mu_r, steps=steps, dt=dt, dx=dx, dy=dy, snapshot_stride=max(steps // 60, 1)) .. GENERATED FROM PYTHON SOURCE LINES 52-56 The mode's spatial shape stays fixed -- it does not propagate anywhere, since it already vanishes on all four PEC walls -- and only its amplitude oscillates sinusoidally in time at the analytic frequency :math:`\omega_{mn} = c_0\pi\sqrt{(m/L_x)^2+(n/L_y)^2}`. .. GENERATED FROM PYTHON SOURCE LINES 56-64 .. code-block:: Python x = np.arange(Nx) * dx y = np.arange(Ny) * dy X, Y = np.meshgrid(x, y, indexing="ij") anim = animate_field_2d(X, Y, frames, times) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 65-69 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("cavity_mode.gif", writer="pillow", fps=20) .. GENERATED FROM PYTHON SOURCE LINES 69-76 .. code-block:: Python probe_i, probe_j = Nx // 3, Ny // 3 probe_series = frames[:, probe_i, probe_j] predicted = Ez0[probe_i, probe_j] * np.cos(omega_mn * times) print(f"TM_{m}{n} analytic frequency omega_mn = {omega_mn:.4e} rad/s") print(f"max deviation of the FDTD-evolved probe point from the analytic standing wave: {np.max(np.abs(probe_series - predicted)):.2e}") .. rst-class:: sphx-glr-script-out .. code-block:: none TM_11 analytic frequency omega_mn = 2.8298e+10 rad/s max deviation of the FDTD-evolved probe point from the analytic standing wave: 9.63e-03 .. GENERATED FROM PYTHON SOURCE LINES 77-84 The probe point's time trace against the analytic standing wave --------------------------------------------------------------------- The animation shows the mode's fixed spatial shape only oscillating in amplitude; plotting the FDTD-evolved probe point's value against the analytic :math:`\cos(\omega_{mn}t)` prediction directly, over the whole run, shows that oscillation tracking the exact frequency the cavity's boundary conditions predict. .. GENERATED FROM PYTHON SOURCE LINES 84-93 .. code-block:: Python fig2, ax2 = plt.subplots() ax2.plot(times, probe_series, "o", markersize=3, label="FDTD probe point") ax2.plot(times, predicted, "k--", label=r"$E_{z,0}\cos(\omega_{mn}t)$") ax2.set_xlabel("t") ax2.set_ylabel("Ez at probe point") ax2.set_title(f"TM_{m}{n} standing wave: FDTD probe vs. analytic frequency") ax2.legend() fig2.tight_layout() .. image-sg:: /api/gallery/fields/electrodynamics/images/sphx_glr_plot_em_cavity_modes_002.png :alt: TM_11 standing wave: FDTD probe vs. analytic frequency :srcset: /api/gallery/fields/electrodynamics/images/sphx_glr_plot_em_cavity_modes_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 4.954 seconds) .. _sphx_glr_download_api_gallery_fields_electrodynamics_plot_em_cavity_modes.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_em_cavity_modes.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_em_cavity_modes.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_em_cavity_modes.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_