Practice Summary (11.4) - One-Dimensional Wave Equation - Mathematics - iii (Differential Calculus)

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6 sample questions

Question 1.
What is the standard form of the one-dimensional wave equation?
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It involves second derivatives with respect to both time and space.
Question 2.
What variables are represented in the wave equation?
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Look for the terms that involve movement in time and space.
3Question 3.
What does the wave equation describe?
- Thermodynamics
- Wave propagation
- Electromagnetism
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Think about the physical processes involved in different media.
4Question 4.
True or False: D'Alembert's solution allows for wave shapes to change over time.
- True
- False
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Recall how traveling waves behave.
Question 5.
A string fixed at both ends vibrates according to the equation $\frac{\partial^2 u}{\partial t^2} = 16 \frac{\partial^2 u}{\partial x^2}$ . If the initial conditions are $u(x,0) = 0$ and $\frac{\partial u}{\partial t}(x,0) = 10 \sin(\frac{\pi x}{L})$ , find the solution for $u(x,t)$ .
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Apply separation of variables and use Fourier series expansions.
Question 6.
Consider a wave traveling in a string modeled by the equation $\frac{\partial^2 u}{\partial t^2} = 25 \frac{\partial^2 u}{\partial x^2}$ . If the free end at x = L has an amplitude of 5 cm, derive the expressions for displacement and motion at x over time.
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Identify the wave parameters and relate them back to the wave equation.

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