Practice Final Form Of D'alembert’s Solution (with Initial Conditions) (14.5)

Practice: Final Form of D'Alembert’s Solution (with initial conditions)

Lesson 14.5D’Alembert’s Solution of Wave Equation

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6 sample questions

Question 1.
What is D'Alembert's solution?
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Think about how waves behave with given initial states.
Question 2.
What do the terms $\phi(x)$ and $\psi(x)$ represent?
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What do we predict when we start with a wave?
3Question 3.
What is the general form of D'Alembert's solution?
- u(x,t) = f(x + ct)
- u(x,t) = f(x + ct) + g(x - ct)
- u(x,t) = c²u(x,t)
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Think about how waves can part ways.
4Question 4.
True or False: D'Alembert's solution applies only in the scenario of non-linear wave forms.
- True
- False
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Recall the types of equations we discussed.
Question 5.
Given an initial displacement of $u(x,0) = e^{-x^2}$ and an initial velocity $\partial u / \partial t(x,0) = e^{-x^2}$ , derive the complete wave solution using D'Alembert's formula.
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Recognize how the exponential function behaves when traveling across the wave.
Question 6.
Analyze how increasing the wave speed $c$ affects the propagation of $u(x,0) = ext{sin}(x)$ and $\partial u / \partial t(x,0) = 0$ .
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Think about a race—how does speed make a difference?

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