Sound wave

Overview

A small-amplitude sound wave propagates at 45° across a doubly-periodic box through a fluid at rest. The perturbation is an acoustic eigenmode — density, internal energy, and velocity are all perturbed simultaneously according to the linearized relativistic Euler equations, and the wave propagates at the relativistic sound speed \(c_s\) with no background velocity. The analytic solution is known at all times, making this a test of the code's ability to correctly propagate compressive waves and maintain the proper phase speed over one full wave period.

Setup

The domain is the unit square \([0,1]\times[0,1]\) in Minkowski coordinates with periodic boundaries on all four sides. The background state is a fluid at rest with \(\rho_0 = 1\), \(u_0 = 1\), \(\tilde{u}^i = 0\), and adiabatic index \(\Gamma = 4/3\). The sound speed of the background is

\[ c_s = \sqrt{\frac{\Gamma p_0}{\rho_0 h_0}}, \qquad p_0 = (\Gamma-1)u_0,\qquad h_0 = 1 + \frac{u_0 + p_0}{\rho_0}. \]

The initial perturbations to all fluid variables are set to the left-propagating acoustic eigenmode,

\[ \delta\rho = A\,\sqrt{\tfrac{63}{187}},\quad \delta u = A\,\sqrt{\tfrac{112}{187}},\quad \delta\tilde{u}^x = A\,\sqrt{\tfrac{12}{187}}\cos(\pi/4),\quad \delta\tilde{u}^y = A\,\sqrt{\tfrac{12}{187}}\sin(\pi/4),\quad \mathbf{B} = 0, \]

where \(A = 10^{-4}\), superimposed as a cosine wave propagating at 45°,

\[ q(x,y,t=0) = q_0 + \delta q\,\cos(k_1 x + k_2 y), \]

with \(k_1 = k_2 = 2\pi\) (one full wavelength along each axis). The analytic solution at time \(t\) is

\[ q(x,y,t) = q_0 + \delta q\,\cos\!\bigl(k_1(x - v_p^x\, t) + k_2(y - v_p^y\, t)\bigr), \]

where \(v_p^x = v_p^y = c_s/\sqrt{2}\) are the phase velocity components. The final time \(t_f \approx 1.62\) corresponds to exactly one full wave period.

Parameters

Relevant compile-time parameters are:

Parameter	Default	Notes
N1TOT, N2TOT	256	Grid resolution; change for convergence study
METRIC	MINKOWSKI
RECONSTRUCTION	LINEAR
X{1,2}{L,R}_BOUND	PERIODIC

Output and convergence

The video below shows \(\rho\) over one full wave period at the default \(256\times256\) resolution.

A plotting script for individual dumps is provided at prob/sound_wave/plot_sound_wave.py.

Because the analytic solution is known at all times, the L1 error is computed for all four perturbed primitives (\(\rho\), \(u\), \(\tilde{u}^x\), \(\tilde{u}^y\)) at the final dump.

Below is the convergence plot with LINEAR reconstruction; the expected slope is \(L_1 \propto N^{-2}\).