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Convolution: Math, Graphics, and Discrete Signals01:24

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In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
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Convolution formulations for non-negative intensity.

Earl G Williams1

  • 1Naval Research Laboratory, Code 7106, Acoustics Division, 4555 Overlook Avenue, Washington, DC 20375, USA. earl.williams@nrl.navy.mil

The Journal of the Acoustical Society of America
|August 10, 2013
PubMed
Summary
This summary is machine-generated.

New formulas reveal sound-producing areas on vibrating structures using near-field acoustics. This method quantifies acoustic radiation and improves source localization, offering fast computations without Fourier transforms.

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Area of Science:

  • Acoustics
  • Structural Dynamics
  • Wave Physics

Background:

  • Near-field acoustic holography is crucial for identifying sound sources.
  • Existing methods for active intensity calculation have limitations in resolution and applicability.
  • Outgoing-only intensity techniques offer potential for improved source identification.

Purpose of the Study:

  • To derive novel spatial convolution formulas for active normal intensity in planar coordinates.
  • To extend the outgoing-only intensity technique for near-field acoustic measurements.
  • To quantify sound-radiating regions of vibrating structures and explore source localization.

Main Methods:

  • Derivation of spatial convolution formulas for active normal intensity using pressure or normal velocity near-field holograms.
  • Application of the derived formulas to measured data from a point-driven, unbaffled rectangular plate.
  • Development and application of hybrid-intensity formulas using a different spatial convolution operator.

Main Results:

  • Successfully constructed positive-only (outward) intensity distributions, clearly revealing sound-producing regions.
  • Demonstrated that spatial resolution is limited to a half-wavelength.
  • The velocity formula yielded classical active intensity, while the pressure formula provided a novel hybrid intensity useful for source localization.
  • Computations were performed rapidly in real space, avoiding Fourier transforms.

Conclusions:

  • The derived spatial convolution formulas effectively identify sound-radiating areas on vibrating structures.
  • The novel hybrid-intensity formulas offer potential for enhanced source localization.
  • The method provides a computationally efficient approach for near-field acoustic analysis.