Practice Applying Initial Conditions (14.4) - D’Alembert’s Solution of Wave Equation

Practice: Applying Initial Conditions

Lesson 14.4D’Alembert’s Solution of Wave Equation

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10 questions

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6 sample questions

Question 1.
What are the two initial conditions typically defined for wave problems?
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Think about what describes the wave at the start.
Question 2.
What does $u(x, 0) = \phi(x)$ signify?
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What component of the wave might this represent?
3Question 3.
What are the initial conditions for the wave equation?
- Initial displacement and initial acceleration
- Initial displacement and initial velocity
- Initial position and acceleration
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Think about what you need to set the state of the wave.
4Question 4.
True or False: The solution for the wave equation can be determined without initial conditions.
- True
- False
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Consider how we define the state of a wave.
Question 5.
Given the wave equation with initial conditions $u(x, 0) = e^{-x}$ and $\frac{\partial u}{\partial t}(x, 0) = 2e^{-x}$ , find the expressions for $f(x)$ and $g(x)$ .
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Think about how to apply the conditions to formulate the two functions.
Question 6.
Apply D'Alembert's Solution for $u(x, 0) = cos(x)$ and $\psi(x) = \sin(x)$ and find the resultant wave function.
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Use the properties of the cosine function when finding specific solutions.

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