Practice Final Form of D'Alembert’s Solution (with initial conditions) - 14.5 | 14. D’Alembert’s Solution of Wave Equation | Mathematics - iii (Differential Calculus) - Vol 2
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Final Form of D'Alembert’s Solution (with initial conditions)

14.5 - Final Form of D'Alembert’s Solution (with initial conditions)

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Learning

Practice Questions

Test your understanding with targeted questions

Question 1 Easy

What is D'Alembert's solution?

💡 Hint: Think about how waves behave with given initial states.

Question 2 Easy

What do the terms \( \phi(x) \) and \( \psi(x) \) represent?

💡 Hint: What do we predict when we start with a wave?

4 more questions available

Interactive Quizzes

Quick quizzes to reinforce your learning

Question 1

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)

💡 Hint: Think about how waves can part ways.

Question 2

True or False: D'Alembert's solution applies only in the scenario of non-linear wave forms.

True
False

💡 Hint: Recall the types of equations we discussed.

1 more question available

Challenge Problems

Push your limits with advanced challenges

Challenge 1 Hard

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.

💡 Hint: Recognize how the exponential function behaves when traveling across the wave.

Challenge 2 Hard

Analyze how increasing the wave speed \( c \) affects the propagation of \( u(x,0) = ext{sin}(x) \) and \( \partial u / \partial t(x,0) = 0 \).

💡 Hint: Think about a race—how does speed make a difference?

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