Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion

Area of Science:

• Acoustics and Wave Physics
• Theoretical Physics

Background:

• Causality imposes fundamental constraints on system responses in both time and frequency domains.
• The Kramers-Kronig (K-K) relations mathematically connect the real and imaginary components of a system's frequency-domain response.
• In ultrasonics, K-K relations are crucial for linking signal attenuation and dispersion phenomena.

Purpose of the Study:

• To review integral and differential forms of frequency-domain Kramers-Kronig relations applicable to theoretical acoustics and experimental measurements.
• To examine methods for implementing K-K relations with finite-bandwidth data, including data extrapolation and integration limit restriction.
• To assess the accuracy of K-K predictions based on system behavior and integral truncation.

Main Methods:

• Review of integral and differential Kramers-Kronig relations.
• Analysis of finite-bandwidth data implementation techniques: extrapolation and integration limit restriction.
• Evaluation of K-K prediction accuracy for frequency power-law attenuation and resonant scatterer media.

Main Results:

• Demonstrated accurate prediction of ultrasonic attenuation and dispersion using K-K relations for specific media types.
• Investigated the impact of finite bandwidth and integral truncation on K-K relation accuracy.
• Reviewed time-causal relations as time-domain counterparts to frequency-domain K-K relations.

Conclusions:

• Kramers-Kronig and time-causal relations are valuable tools for analyzing acoustic measurements and models.
• Understanding causality-imposed restrictions enhances the interpretation of ultrasonic wave propagation.
• The study provides insights into applying K-K relations for accurate characterization of acoustic media.