Moving Sound Field Model & Supersonic Effects

• The moving sound field model characterizes the spatial and temporal evolution of acoustic fields generated by dynamic sources, incorporating both classical and quantum interference effects.
• It explains how supersonic motion results in dual retarded emissions that create hyperboloidal interference patterns and nonlocal field superpositions.
• The quantization approach uses operator splitting and S-matrix factorization to link classical source trajectories with Poisson-distributed phonon emission statistics.

A moving sound field model describes the spatial and temporal evolution of acoustic fields generated by dynamic or mobile sources, with a focus on accurate physical, mathematical, and statistical characterization. Modern moving sound field models extend well beyond simplified Doppler-shifted solutions, encompassing quantum and classical interference effects for supersonic sources, nonstationary and nonisotropic reverberation, microphone-trajectory-aware reconstruction, quantifiable phonon statistics, and explicit analytical solutions for source-field-observer geometry. Such models are fundamental in acoustical physics, room acoustics, computational audio, advanced simulation, and modern signal processing methodologies.

1. Supersonic Moving Source Model and the Double Image Effect

The foundational moving sound field model for a supersonic source—designated as a "tachyonic" acoustic emitter—considers a point-like source traveling along the x-axis in a fluid at rest with speed $v > c$ (the speed of sound) (Arias et al., 2011). The source is described by the current density:

$$Q(t, \mathbf{x}) = q(t) \delta(x-vt) \delta(y) \delta(z)$$

This configuration produces, for a fixed observer, two distinct retarded emission events that simultaneously contribute to the observed sound field. The field solution to the inhomogeneous wave equation,

$\square \psi(t,\mathbf{x}) = Q(t,\mathbf{x}),$

gives rise to two physically meaningful retarded distances,

$$R = \frac{(v/c)(x-vt) \pm \sqrt{(x-vt)^2 + (1-v^2/c^2)(y^2 + z^2)}}{1-v^2/c^2}$$

(Both $R_+$ and $R_-$ are allowed for $v > c$.) This results in the "double image" phenomenon, where the sound field at a given observation time is a superposition of contributions from two spacelike-separated source locations. The resulting spatial and temporal field pattern cannot be reduced to a single causal emission and fundamentally requires treatment of non-causal, space-like source trajectories.

2. Classical Sound Radiation and Interference Pattern

For a harmonic source modulation $q(t) = q_0 e^{-i\omega t}$, the moving sound field exhibits a prominent interference signature:

$$\psi(t,\mathbf{x}) = \frac{q_0 e^{-i\omega t}}{4\pi R_1} \left( e^{i\omega R_+/c} + e^{i\omega R_-/c} \right) = \frac{q_0 e^{-i\omega (t-R_+/c)}}{4\pi R_1} \left[1 + e^{-2i\omega R_1 / (c(M^2-1))} \right]$$

where $R_1 = \sqrt{(vt-x)^2 - (M^2-1)(y^2+z^2)}$ and $M = v/c$.

The phase term,

$\Delta\varphi = \frac{2\omega R_1}{c(M^2-1)},$

dictates spatially varying constructive ($\Delta\varphi = 2\pi n$) and destructive ($\Delta\varphi = (2n+1)\pi$) interference, leading to a pattern of hyperboloidal nodal surfaces in three-dimensional space. This structure is a direct result of the nonlocal double emission inherent to supersonic motion and is illustrated in the referenced figures as intensity-modulated spatial field patterns.

3. Quantization: Acoustic Field as Quantum Scalar Field and Phonon Emission Statistics

Quantization of the classical field is performed in the interaction picture by expressing the field operator as $\psi = \psi^{(+)} + \psi^{(-)}$ and introducing the S-matrix:

$$S = \exp\left\{ -i\int d^4x \psi_{in}(x) Q(x) \right\}$$

which, after normal ordering, separates into factors acting on the annihilation and creation operators, as well as a second-order term encoding the vacuum fluctuations. The generated phonons (quantized sound quanta) due to the interaction with the supersonic classical source obey a Poisson distribution:

$P(N) = \frac{e^{-\nu} \nu^N}{N!}$

with

$\nu = 2\int d\tilde{k} |\tilde{Q}(k)|^2$

and

$\tilde{Q}(k) = \int d^4x Q(x) e^{-ik \cdot x}$

Notably, the factor of $2$ in $\nu$ arises from the two spatially distinct emission points of the tachyonic source—a contrast to the single emission event in the subsonic regime. This quantization asserts the direct relation between the multiple-emission geometry and the emergent statistics of phonon production.

4. Mathematical Formalism and Analytical Results

The moving sound field model integrates several key analytic results:

• The distance-time relationship incorporating superluminal ($v>c$) trajectories and their analytic continuation.
• The inhomogeneous d'Alembertian for the scalar field driven by a moving delta-function source.
• The explicit formulae for interference patterns arising from the phase difference between the two emission points.
• The S-matrix factorization for the quantum description, facilitating calculation of transition amplitudes and phonon emission probabilities.

The complete moving field is formalized through the Green's function techniques and explicit evaluation of integrals involving classical and quantum components: $$R = \frac{(v/c)(x-vt) \pm \sqrt{(x-vt)^2 + (1-v^2/c^2)(y^2 + z^2)}}{1-v^2/c^2}$$

$P(N) = \frac{e^{-\nu} \nu^N}{N!}$

5. Physical Interpretation: Nonlocality, Causal Structure, and Analogies

The model exemplifies how super-causal (space-like) source trajectories violate the naive expectation of single-event emission fields and give rise to nonlocal field superpositions. The "double image" echoes the tachyonic double appearance from field-theoretic tachyon analogs. The interference pattern’s hyperboloidal geometry is intrinsically tied to the Mach number, with the spatial pattern a direct acoustic signature of supersonic motion in analogy to Cerenkov-Mach cone effects but with richer structure due to the quantum-classical correspondence.

The quantized picture not only reproduces classical interference but explicitly relates it to discrete, statistically independent phonon emissions arising from the spacetime geometry of the moving source.

6. Implications and Broader Context

This moving sound field model provides:

• A framework for interpreting nonlocal features and interference in supersonic and tachyonic analog systems, relevant for analog models of field theory and "Mach cones".
• A template for quantizing acoustic perturbations driven by deterministic classical sources, with clearly calculable statistics, useful in phonon generation and detection problems.
• Methods and insights applicable to lab-based analog gravity experiments and the paper of quantum field effects in moving media.
• An explicit demonstration of how classical source trajectory geometry gets imprinted both in macroscopic wave patterns and in microscopic (phonon) statistics.

The methodology and mathematical structure presented in this model is foundational for current and future studies of supersonic and dynamically modulated acoustic fields, as well as for generalizing to other media and quantum analog systems.