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SSC Applied Mathematics QUESTION #4159
Question 1
The D'Alembert solution of the wave equation $u_{tt} = c^2 u_{xx}$ with $u(x,0) = f(x)$ and $u_t(x,0) = g(x)$ is:
  • $u(x,t) = \frac{f(x+ct) + f(x-ct)}{2} + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)ds$✔️
  • $u(x,t) = f(x+ct) + f(x-ct)$
  • $u(x,t) = \frac{1}{2}[f(x+ct) - f(x-ct)]$
  • $u(x,t) = \frac{1}{c}\int_{x-ct}^{x+ct} g(s)ds$
Correct Answer Explanation
D'Alembert's formula: $u(x,t) = \frac{f(x+ct) + f(x-ct)}{2} + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)ds$. First term from initial displacement, second from initial velocity.