Mathematics - iii (Differential Calculus) - Vol 2 | 11. One-Dimensional Wave Equation by Abraham | Learn Smarter
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11. One-Dimensional Wave Equation

The one-dimensional wave equation describes the propagation of wave phenomena in a medium along a single spatial dimension. Deriving from Newton's laws, this second-order linear partial differential equation showcases critical aspects including boundary and initial conditions necessary for unique solutions. D'Alembert's formula provides a general solution, while the method of separation of variables aids in solving complex problems involving fixed and free boundary conditions.

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Sections

  • 11

    Partial Differential Equations

    The one-dimensional wave equation describes the propagation of waves through a medium along a single spatial dimension.

    Learning Practice

  • 11.1.1

    Derivation Of The One-Dimensional Wave Equation

    This section presents the derivation of the one-dimensional wave equation, highlighting its significance in describing wave phenomena in various media.

    Learning Practice

  • 11.1.2

    General Solution Of The One-Dimensional Wave Equation

    This section presents D'Alembert's formula as the general solution to the one-dimensional wave equation, explaining how it describes wave propagation.

    Learning Practice

  • 11.1.3

    Initial And Boundary Value Problems (Ibvp)

    This section discusses Initial and Boundary Value Problems essential for solving the one-dimensional wave equation.

    Learning Practice

  • 11.1.4

    Boundary Conditions (Bcs)

    This section introduces boundary conditions in relation to the one-dimensional wave equation, outlining fixed, free, and mixed conditions.

    Learning Practice

  • 11.1.5

    Method Of Separation Of Variables

    The Method of Separation of Variables allows us to solve the one-dimensional wave equation by breaking it into two ordinary differential equations.

    Learning Practice

  • 11.1.7

    Worked Example

    This section presents a worked example of solving the one-dimensional wave equation with specific initial and boundary conditions.

    Learning Practice

  • 11.2

    One-Dimensional Wave Equation

    The one-dimensional wave equation is a key second-order linear partial differential equation that models wave propagation in a single spatial dimension.

    Learning Practice

  • 11.3

    Introduction

    This section introduces the one-dimensional wave equation, a fundamental second-order linear partial differential equation that models wave propagation in various mediums.

    Learning Practice

  • 11.4

    Summary

    The one-dimensional wave equation models wave propagation and is derived from Newton's laws, with significant applications in physics and engineering.

    Learning Practice

References

Unit_2_ch11.pdf

Class Notes

Memorization

What we have learnt

  • The standard form of the on...
  • D'Alembert’s solution illus...
  • Boundary and initial condit...

Final Test

Revision Tests