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Solution The formula derived in lecture is valid for a system with damping since the kinetic and potential energies of the string only depend on the displacement u x t and its derivatives
The equation that governs this setup is the so called one dimensional wave equation y tt a 2 y xx nonumber for some constant a 0 The intuition is similar to the heat equation replacing velocity with acceleration the acceleration at a specific point is proportional to the second derivative of the shape of the string
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Graphing Linear Equations Practice Worksheet
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Graphing Linear Equations Practice Worksheet
The wave equation is a linear second order partial differential equation which describes the propagation of oscillations at a fixed speed in some quantity y y A solution to the wave equation in two dimensions propagating over a fixed region 1
The Wave Equation For a one dimensional wave equation with a fixed length the function u x t describes the position of a string at a specific x and t value This leads to the classical wave equation dfrac partial 2 u partial x 2 dfrac 1 v 2 cdot dfrac partial 2 u partial t 2 label W1
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Equation ref 16 4 is known as a simple harmonic wave function A wave function is any function such that f x t f x v t Later in this chapter we will see that it is a solution to the linear wave equation
The Vibrating String We motivate the study of the wave equation by considering its application to the vibrations of a string such as a violin string tightly stretched in equilibrium along the x axis in the xu plane and tied to the points 0 0 and L 0 Figure 12 2 1 Figure 12 2 1 A stretched string
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The equation that governs this setup is the so called one dimensional wave equation y tt a 2 y xx nonumber for some constant a 0 The intuition is similar to the heat equation replacing velocity with acceleration the acceleration at a specific point is proportional to the second derivative of the shape of the string
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https://www.wtamu.edu/~dcraig/PHYS4330/m201.pdf
The Harmonic Wave Function A harmonic wave function is sinusoidal in functional form and for a one dimensional wave may be expressed as either 21 4 A sin x vt 0 1 or 21 4
The equation that governs this setup is the so called one dimensional wave equation y tt a 2 y xx nonumber for some constant a 0 The intuition is similar to the heat equation replacing velocity with acceleration the acceleration at a specific point is proportional to the second derivative of the shape of the string
The Harmonic Wave Function A harmonic wave function is sinusoidal in functional form and for a one dimensional wave may be expressed as either 21 4 A sin x vt 0 1 or 21 4
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