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, 2021 In this chapter, partial differential equations that govern many featured wave propagation phenomena are discussed. We start with modelling the process of string vibration through a partial differential equation, known as the wave equation, or the equation for string vibration. Then the concepts of initial and boundary conditions are introduced.
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On the wave equation with a potential
Communications in Partial Differential Equations, 1999We prove Strichartz estimates for wave equations with a ...
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2001
In this chapter we study the wave equation $$u_{tt} = \Delta u$$ on an open subset \(\Omega\) of \(\mathbb{R}^n.\)
Frank Neubrander +3 more
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In this chapter we study the wave equation $$u_{tt} = \Delta u$$ on an open subset \(\Omega\) of \(\mathbb{R}^n.\)
Frank Neubrander +3 more
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2014
Waves occur in various forms such as longitudinal acoustic waves in a medium such as air, transverse waves in a string, waves in a membrane and as electromagnetic radiation. All these are governed by the wave equation with the dependent variable being pressure or displacement or electric field etc.
S.P. Venkateshan, Prasanna Swaminathan
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Waves occur in various forms such as longitudinal acoustic waves in a medium such as air, transverse waves in a string, waves in a membrane and as electromagnetic radiation. All these are governed by the wave equation with the dependent variable being pressure or displacement or electric field etc.
S.P. Venkateshan, Prasanna Swaminathan
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The Leading Edge, 1987
The foundation of seismology is the theory of wave motion, a complicated concept that is still — after centuries of experiments and speculations by many of the very greatest scientists — an area of active research in many disciplines. Even simple forms of wave motion are difficult to describe verbally; but, ironically, the simplest type of wave is ...
Dean Clark, Enders A. Robinson
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The foundation of seismology is the theory of wave motion, a complicated concept that is still — after centuries of experiments and speculations by many of the very greatest scientists — an area of active research in many disciplines. Even simple forms of wave motion are difficult to describe verbally; but, ironically, the simplest type of wave is ...
Dean Clark, Enders A. Robinson
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1971
(a) Let (a1) be an ordinary linear differential equation with constant coefficients.
Giovanni Prouse, Luigi Amerio
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(a) Let (a1) be an ordinary linear differential equation with constant coefficients.
Giovanni Prouse, Luigi Amerio
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The Laplace Equation and Wave Equation
1996In this chapter we introduce the central linear partial differential equations of the second order, the Laplace equation $$\Delta u = f$$ (0.1) and the wave equation $$\left (\frac{{\partial }^{2}} {\partial {t}^{2}} -\Delta\right )u = f.$$ (0.2) For flat Euclidean space \(\mathbb{R}^{n}\), the Laplace operator is defined by ...
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2020
In Chaps. 29 and 30, you have seen how Fourier and Laplace transforms help us solve Poisson’s equation and the diffusion equation. We turn our attention now to another very important equation of mathematical physics, the wave equation. As you might expect, this is a well-studied topic with a vast literature.
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In Chaps. 29 and 30, you have seen how Fourier and Laplace transforms help us solve Poisson’s equation and the diffusion equation. We turn our attention now to another very important equation of mathematical physics, the wave equation. As you might expect, this is a well-studied topic with a vast literature.
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1986
In this chapter we introduce the wave equation, formulate initial-boundary value problems, and prove uniqueness and existence via the spectral theorem. To get more information about the behaviour of the solutions, we have to study the spectrum of the underlying operators more carefully, which will be done in Chapter 4.
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In this chapter we introduce the wave equation, formulate initial-boundary value problems, and prove uniqueness and existence via the spectral theorem. To get more information about the behaviour of the solutions, we have to study the spectrum of the underlying operators more carefully, which will be done in Chapter 4.
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2019
In the previous chapter, the concept of sound was introduced and a simple example of a physical system that can produce sound was given. More specifically, we saw how a metal bar struck by a hammer exhibits a displacement over time that is sinusoidal in nature.
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In the previous chapter, the concept of sound was introduced and a simple example of a physical system that can produce sound was given. More specifically, we saw how a metal bar struck by a hammer exhibits a displacement over time that is sinusoidal in nature.
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