An inequality for longitudinal and transverse wave attenuation coefficients

Area of Science:

• * Physics
• * Materials Science
• * Acoustics

Background:

• * Total absorption, the net energy flux from a region per cycle, must be non-negative for passive systems.
• * This principle applies to isotropic linearly viscoelastic materials undergoing time-harmonic motion.
• * Existing research has not explicitly defined constraints on absorption coefficients for these materials.

Purpose of the Study:

• * To derive and present a novel inequality constraining the absorption coefficients of longitudinal and transverse waves.
• * To establish a relationship between wave attenuation and wave speed based on the passivity condition.
• * To provide a consistency check for experimental ultrasonic measurements in viscoelastic materials.

Main Methods:

• * Theoretical derivation based on the definition of total absorption and passivity.
• * Analysis of non-dimensional absorption coefficients (γ_L, γ_T) for small values.
• * Formulation of an inequality involving attenuation coefficients (α_L, α_T) and wave speeds (c_L, c_T).

Main Results:

• * A new inequality is derived: α_L/α_T≥4c_T³/3c_L³ for small absorption coefficients.
• * This inequality provides a relative bound between wave speed and attenuation.
• * Most, but not all, previously reported ultrasonic measurement data satisfy this positive absorption test.

Conclusions:

• * The derived inequality offers a fundamental constraint for viscoelastic materials, ensuring physical consistency.
• * It serves as a valuable tool for validating experimental data in acoustics and materials science.
• * The findings highlight the importance of passivity conditions in wave propagation studies.