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SSC Applied Mathematics QUESTION #1328
Question 1
In the D'Alembert solution, the function \(\dfrac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds\) represents the contribution from:
  • The initial displacement \(f(x)\)
  • The initial velocity \(g(x)\)✔️
  • The boundary conditions
  • The wave speed \(c\)
Correct Answer Explanation
In the D'Alembert solution \(u(x,t)=\dfrac{f(x+ct)+f(x-ct)}{2}+\dfrac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds\), the first term comes from initial displacement \(u(x,0)=f(x)\) and the second term arises from the initial velocity condition \(u_t(x,0)=g(x)\).