Add Klein-Gordon operator

examples/klein_gordon.py

@@ -0,0 +1,71 @@
+"""Klein-Gordon wave equation example.
+
+Solves the Klein-Gordon equation in the frequency domain using the
+`klein_gordon` operator provided by j-Wave.
+
+The Klein-Gordon equation for a massive scalar field reads::
+
+    (nabla^2 + omega^2/c^2 - m^2) u = f
+
+where *m* is the field mass.  Setting *m = 0* recovers the ordinary
+Helmholtz equation.
+
+This example sets up a 2-D domain with a point source, solves the
+equation for two mass values (massless and massive), and plots the
+resulting pressure fields side by side.
+"""
+
+import jax.numpy as jnp
+import matplotlib.pyplot as plt
+from jaxdf.discretization import FourierSeries
+
+from jwave.acoustics import klein_gordon
+from jwave.geometry import Domain, Medium
+
+# ── Domain setup ──────────────────────────────────────────────────────
+N = (128, 128)
+dx = (0.1e-3, 0.1e-3)  # 0.1 mm grid spacing
+domain = Domain(N, dx)
+
+# Homogeneous medium (water-like)
+sound_speed = 1500.0  # m/s
+medium = Medium(domain=domain, sound_speed=sound_speed, pml_size=15)
+
+# Angular frequency corresponding to 1 MHz
+f0 = 1e6
+omega = 2 * jnp.pi * f0
+
+# ── Source term ───────────────────────────────────────────────────────
+src = jnp.zeros(N)
+src = src.at[N[0] // 2, N[1] // 2].set(1.0)
+source = FourierSeries(src, domain)
+
+# ── Solve for two mass values ─────────────────────────────────────────
+params = klein_gordon.default_params(source, medium, omega=omega)
+
+# Massless (equivalent to Helmholtz)
+kg_massless = klein_gordon(source, medium, omega=omega, mass=0.0, params=params)
+
+# Massive (m = k/2, where k = omega / c)
+k0 = omega / sound_speed
+mass = k0 / 2
+kg_massive = klein_gordon(source, medium, omega=omega, mass=mass, params=params)
+
+# ── Visualisation ─────────────────────────────────────────────────────
+fig, axes = plt.subplots(1, 2, figsize=(10, 4))
+
+for ax, field, label in zip(
+    axes,
+    [kg_massless, kg_massive],
+    ["Helmholtz (m = 0)", f"Klein-Gordon (m = k/2)"],
+):
+    img = jnp.abs(field.on_grid[..., 0])
+    ax.imshow(img.T, cmap="inferno", origin="lower")
+    ax.set_title(label)
+    ax.set_xlabel("x [grid points]")
+    ax.set_ylabel("y [grid points]")
+
+plt.tight_layout()
+plt.savefig("klein_gordon_example.png", dpi=150)
+plt.show()
+print("Done.")

jwave/acoustics/__init__.py

@@ -17,6 +17,7 @@
 from .conversion import db2neper
 from .operators import (
   helmholtz,
+  klein_gordon,
   laplacian_with_pml,
   scale_source_helmholtz,
   wavevector,

jwave/acoustics/operators.py

@@ -300,6 +300,43 @@ def helmholtz(u: OnGrid, medium: Medium, *, omega=1.0, params=None) -> OnGrid:
     return L + k
 
 
+def klein_gordon_init_params(u: Field,
+                             medium: Medium,
+                             omega,
+                             *args,
+                             **kwargs):
+    return helmholtz.default_params(u, medium, omega)
+
+
+@operator(init_params=klein_gordon_init_params)
+def klein_gordon(u: Field,
+                 medium: Medium,
+                 *,
+                 omega=1.0,
+                 mass=0.0,
+                 params=None) -> Field:
+    r"""Klein-Gordon operator: :math:`(\nabla^2 + k^2 - m^2)u`.
+
+    Extends the Helmholtz operator with a mass term.  The Klein-Gordon
+    equation arises in relativistic wave mechanics and describes massive
+    scalar fields.  Setting ``mass=0`` recovers the Helmholtz operator
+    exactly.
+
+    Args:
+      u (Field): Complex field.
+      medium (Medium): Medium object.
+      omega (float): Angular frequency.
+      mass (float): Mass parameter :math:`m`.
+      params (None, optional): Parameters for the underlying Helmholtz
+          operator.
+
+    Returns:
+      Field: Klein-Gordon operator applied to ``u``.
+    """
+    h = helmholtz(u, medium, omega=omega, params=params)
+    return h - mass**2 * u
+
+
 def scale_source_helmholtz(source: Field, medium: Medium) -> Field:
     if isinstance(medium.sound_speed, Field):
         min_sos = functional(medium.sound_speed)(jnp.amin)