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Related Concept Videos

Sound Waves: Interference00:53

Sound Waves: Interference

Sound waves can be modeled either as longitudinal waves, wherein the molecules of the medium oscillate around an equilibrium position, or as pressure waves. When two identical waves from the same source superimpose on each other, the combination of two crests or two troughs results in amplitude reinforcement known as constructive interference. If two identical waves, that are initially in phase, become out of phase because of different path lengths, the combination of crests with troughs...
Sound as Pressure Waves01:17

Sound as Pressure Waves

Sound waves, which are longitudinal waves, can be modeled as the displacement amplitude varying as a function of the spatial and temporal coordinates. As a column of the medium is displaced, its successive columns are also displaced. As the successive displacements differ relatively, a pressure difference with the surrounding pressure is created. The gauge pressure varies across the medium.
The pressure fluctuation depends on the difference in displacements between the successive points in the...
Continuous -time Fourier Transform01:11

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...
Interference: Path Lengths01:10

Interference: Path Lengths

Consider two sources of sound, that may or may not be in phase, emitting waves at a single frequency, and consider the frequencies to be the same.
Two special sources may be considered when they are in phase. This can be easily achieved by feeding the two sources from the same source. An example would be synchronizing the two speakers by feeding them with the same source, such as the sound waves produced by a tuning fork. This setup ensures that the two sources have the same frequency and are...
Interference and Superposition of Waves01:07

Interference and Superposition of Waves

When two waves of the same nature occur in the same region simultaneously, they result in interference. Interference of waves implies that the net effect of the waves is the sum of the individual waves' effects. However, it does not imply that the individual waves affect the propagation of other waves.
Interference occurs in mechanical waves, such as sound waves, waves on a string, and surface water waves. Mechanical waves correspond to the physical displacement of particles. Hence,...
The Entropy as a State Function01:14

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...

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Related Experiment Video

Updated: May 15, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

Joint entropy of continuously differentiable ultrasonic waveforms.

M S Hughes1, J E McCarthy, J N Marsh

  • 1Department of Medicine/Cardiology Division, Campus Box 8215, Washington University School of Medicine, 660 South Euclid Avenue, St. Louis, Missouri 63110-1093, USA. mshatctrain@gmail.com

The Journal of the Acoustical Society of America
|January 10, 2013
PubMed
Summary

This study extends joint entropy for continuous functions like ultrasound signals. The new method enhances materials characterization and nanoparticle detection sensitivity.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Biomedical Engineering

Background:

  • Traditional joint entropy applies to discrete random variables.
  • Continuous functions, like backscattered ultrasound, require advanced mathematical treatment.
  • Existing methods for materials characterization and contrast agent detection have limitations.

Purpose of the Study:

  • To extend the concept of joint entropy to continuous functions.
  • To develop a novel approach for analyzing ultrasound data.
  • To improve sensitivity in materials characterization and targeted nanoparticle detection.

Main Methods:

  • Extension of joint entropy to continuous functions using Schwartz distributions.
  • Application of a coarse-graining operation for imaging.
  • Removal of coarse-graining parameter via the ergodic theorem.

Main Results:

  • A novel expression for joint entropy of continuous functions was derived.
  • The method was applied to backscattered ultrasound data.
  • Enhanced sensitivity, up to double previous methods, was achieved for materials characterization and nanoparticle detection.

Conclusions:

  • The extended joint entropy concept is effective for analyzing continuous ultrasound data.
  • This technique offers superior sensitivity for materials characterization.
  • The method demonstrates significant utility in detecting targeted liquid nanoparticle ultrasonic contrast agents.