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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What we have learnt
- The standard form of the one-dimensional wave equation is derived from physical principles governing wave motion.
- D'Alembert’s solution illustrates how waves travel without changing shape.
- Boundary and initial conditions significantly influence the behavior and solutions of wave equations.
Key Concepts
- -- OneDimensional Wave Equation
- A second-order linear partial differential equation that models the propagation of waves in a single spatial dimension.
- -- D'Alembert's Formula
- A formula representing the general solution of the wave equation, showing how waves propagate without changing shape.
- -- Boundary Conditions
- Constraints applied to the wave equation that determine the solution's behavior at the endpoints of the domain.
- -- Separation of Variables
- A mathematical method used to solve partial differential equations by separating the variables to reduce them into ordinary differential equations.
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