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    This study shows plane waves are the only coherent modes for statistically homogeneous fields when the spatial power spectrum is injective. Field randomness arises from spectral phase, not amplitude.

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    Area of Science:

    • Physics
    • Wave Phenomena
    • Signal Processing

    Background:

    • Spatially random fields are common in physics and engineering.
    • Understanding their decomposition is crucial for analysis.
    • Plane-wave decomposition is a key technique.

    Purpose of the Study:

    • To investigate the uniqueness of plane-wave decomposition for temporally deterministic, spatially random fields.
    • To determine the conditions under which plane waves represent the sole coherent modes.
    • To elucidate the origins of randomness in such fields.

    Main Methods:

    • Analysis of spatially ergodic and statistically homogeneous fields.
    • Examination of the relationship between spatial power spectrum and plane-wave decomposition.
    • Investigation of spectral amplitude and phase contributions.

    Main Results:

    • Plane waves are uniquely determined as the only coherent modes when the spatial power spectrum is injective.
    • Field randomness is attributed to the spatial spectral phase.
    • Spectral amplitude is deterministic, dictated by the spatial power spectrum.

    Conclusions:

    • The plane-wave decomposition of statistically homogeneous fields is unique under specific spectral conditions.
    • The findings clarify the roles of spectral amplitude and phase in defining field properties.
    • Results are translatable to the time domain, offering broader applicability.