Practice Introduction (11.3) - One-Dimensional Wave Equation - Mathematics - iii (Differential Calculus) - Vol 2

Practice: Introduction

Lesson 11.3One-Dimensional Wave Equation

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8 questions

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6 sample questions

Question 1.
What does the wave equation model?
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Think of examples of waves.
Question 2.
What is the standard form of the wave equation?
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Recall the variables involved.
3Question 3.
What is the standard form of the one-dimensional wave equation?
- $$\\frac{\\partial^2 u}{\\partial t^2} = c^2 \\frac{\\partial^2 u}{\\partial x^2}$$
- $$\\frac{\\partial u}{\\partial t} = c \\frac{\\partial^2 u}{\\partial x^2}$$
- $$u = c^2 t^2 + x^2$$
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Look for the equation that involves second derivatives.
4Question 4.
True or False: The wave equation can only be applied to sound waves.
- True
- False
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Consider other examples of wave phenomena.
Question 5.
Given a vibrating string fixed at both ends, write the general solution using Fourier series. Consider the fixed conditions at $u(0,t)=0$ and $u(L,t)=0$ .
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Think about how to incorporate boundary conditions into the Fourier expansion.
Question 6.
Solve the one-dimensional wave equation with initial conditions $u(x,0) = sin(x)$ and $u_t(x,0) = 0$ on the interval $[0,\pi]$ with Dirichlet conditions.
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Remember to apply the boundary conditions to derive the specific solution.

4 more questions in this section. Enroll free to practise them all.