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Related Concept Videos

Equations of Wave Motion01:02

Equations of Wave Motion

Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered.
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Types of Responses of Series RLC Circuits01:11

Types of Responses of Series RLC Circuits

A second-order differential equation characterizes a source-free series RLC circuit, marking its distinct mathematical representation. The complete solution of this equation is a blend of two unique solutions, each linked to the circuit's roots expressed in terms of the damping factor and resonant frequency.
Partial Fractions01:28

Partial Fractions

A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...
RLC Circuit as a Damped Oscillator01:30

RLC Circuit as a Damped Oscillator

An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
Consider a series RLC circuit. Here, the presence of resistance in the circuit leads to energy loss due to joule heating in the resistance. Therefore, the total electromagnetic energy in the circuit is no longer constant and decreases with time. Since the magnitude of charge, current, and potential difference continuously decreases, their oscillations are said to be damped. This is...

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Related Experiment Video

Updated: May 15, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Stochastic solution to a time-fractional attenuated wave equation.

Mark M Meerschaert1, Peter Straka, Yuzhen Zhou

  • 1Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, mcubed@stt.msu.edu.

Nonlinear Dynamics
|December 22, 2012
PubMed
Summary

This study introduces a random walk model to interpret fractional derivative terms in a power law wave equation, enabling tractable modeling of wave propagation with power law attenuation.

Related Experiment Videos

Last Updated: May 15, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Area of Science:

  • Physics
  • Applied Mathematics
  • Biomedical Engineering

Background:

  • Wave propagation phenomena often exhibit complex nonlinear dynamics.
  • Power law attenuation is a common characteristic in various physical systems, including wave propagation.
  • Fractional calculus offers advanced tools for modeling such complex behaviors.

Purpose of the Study:

  • To develop a random walk model for understanding fractional derivative terms in the power law wave equation.
  • To provide a physical interpretation for these fractional derivative terms.
  • To present a new strictly causal solution to the fractional wave equation and discuss its application in medical ultrasound.

Main Methods:

  • Development of a random walk model to elucidate fractional derivative terms.
  • Derivation of an explicit analytical solution for the fractional wave equation.
  • Formulation of a new strictly causal solution.

Main Results:

  • The random walk model successfully explains the fractional derivative terms in the power law wave equation.
  • A novel strictly causal solution to the fractional wave equation was derived.
  • The study demonstrates the applicability of the model to medical ultrasound wave propagation.

Conclusions:

  • Fractional calculus provides a powerful framework for modeling wave propagation with power law attenuation.
  • The random walk model offers valuable physical insights into fractional wave equations.
  • The new causal solution has potential applications in medical imaging and diagnostics.