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Related Concept Videos

Vector Forms of Green’s Theorem01:26

Vector Forms of Green’s Theorem

The study of fluid motion often involves understanding how local rotational behavior relates to global circulation. In the context of a pond with pollutants, direct measurement of water movement along an irregular shoreline can be impractical. Green’s Theorem in vector form provides an alternative by relating the circulation around a closed boundary to properties of the flow within the enclosed region.Measurements of water velocity at different points define a continuous vector field that...
Green’s Theorem01:27

Green’s Theorem

Green’s Theorem establishes a relationship between a line integral around a closed plane curve and a double integral over the region enclosed by that curve. It applies to a vector field F(x, y) = 〈P(x, y), Q(x, y)〉, where P and Q have continuous first partial derivatives on an open set containing the region.Let C be a positively oriented, simple, closed, piecewise smooth curve, and let R be the plane region bounded by C. Green’s Theorem states that\begin{equation*}\oint_C P\,dx+Q\,dy =\iint_R...
Extended Versions of Green’s Theorem01:27

Extended Versions of Green’s Theorem

Green’s Theorem connects the circulation of a vector field around a closed curve with the behavior of the field across the region enclosed by that curve. It provides a way to replace a line integral around a boundary with a double integral over the interior region, making it especially useful in plane geometry, fluid flow, and vector calculus.Although Green’s Theorem is often introduced using simple regions without gaps, it can also be applied to regions made from several simple parts. This...
Uniform Depth Channel Flow: Problem Solving01:18

Uniform Depth Channel Flow: Problem Solving

To calculate the flow rate for a trapezoidal channel, first, identify the bottom width, side slope, and flow depth of the channel. The cross-sectional area (A) corresponding to the depth of flow (y), channel bottom width (B), and side slope (θ) is determined by:Next, calculate the wetted perimeter, which includes the bottom width and the sloped side lengths in contact with the water. Using the values of the cross-sectional area and the wetted perimeter, determine the hydraulic radius by...
Energy Line and Hydraulic Gradient Line01:27

Energy Line and Hydraulic Gradient Line

Based on Bernoulli's equation, the energy line (EL) and hydraulic grade line (HGL) provide graphical representations of energy distribution in a fluid flow system. For steady, incompressible, inviscid flows, Bernoulli's equation is expressed as:
Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

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Updated: Jun 27, 2026

Investigating the Relationship between Sea Surface Chlorophyll and Major Features of the South China Sea with Satellite Information
10:28

Investigating the Relationship between Sea Surface Chlorophyll and Major Features of the South China Sea with Satellite Information

Published on: June 13, 2020

Extracting the local Green's function on a horizontal array from ambient ocean noise.

S E Fried, W A Kuperman, Karim G Sabra

    The Journal of the Acoustical Society of America
    |December 10, 2008
    PubMed
    Summary
    This summary is machine-generated.

    Researchers used ocean ambient noise to approximate the time domain Green's function (TDGF) and analyze seismic wave travel times. This method accurately reveals environmental details, including the critical angle at the water-sediment interface.

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    Area of Science:

    • Ocean acoustics
    • Seismic signal processing
    • Environmental monitoring

    Background:

    • Passive acoustic monitoring leverages ambient noise.
    • Seismic wave propagation is key to understanding subsurface environments.
    • Accurate environmental characterization is crucial for geophysical studies.

    Purpose of the Study:

    • To approximate the time domain Green's function (TDGF) using only ambient ocean noise.
    • To determine optimal time windows for noise correlation analysis.
    • To extract detailed environmental information from acoustic data.

    Main Methods:

    • Utilized ocean ambient noise recordings from a bottom hydrophone array.
    • Calculated the time domain Green's function (TDGF) from noise correlations.
    • Compared noise correlation functions over increasing time windows to identify optimal durations.
    • Validated results against computer simulations of environmental noise responses.

    Main Results:

    • Successfully approximated the local TDGF using ambient noise.
    • Identified an optimal time window for noise correlation analysis by comparing function strength with residual fluctuations.
    • Demonstrated that the resulting time series accurately approximates environmental noise responses.
    • Obtained accurate environmental details, specifically the critical angle at the water-sediment interface.

    Conclusions:

    • Ocean ambient noise is a viable source for approximating the TDGF.
    • The methodology provides a robust way to extract seismic travel times and environmental parameters.
    • This passive acoustic approach offers a cost-effective and non-invasive method for oceanographic and geophysical research.