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CSS Applied Mathematics QUESTION #1307
Question 1
The D'Alembert solution of the wave equation \(u_{xx}=\dfrac{1}{c^2}u_{tt}\) with \(u(x,0)=f(x)\) and \(u_t(x,0)=g(x)\) is:
  • \(u(x,t)=\dfrac{f(x+ct)+f(x-ct)}{2}+\dfrac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds\)✔️
  • \(u(x,t)=f(x+ct)+f(x-ct)\)
  • \(u(x,t)=\dfrac{f(x+ct)-f(x-ct)}{2}+\dfrac{1}{c}\int_0^x g(s)\,ds\)
  • \(u(x,t)=f(x)g(t)\)
Correct Answer Explanation
D'Alembert's formula gives the general solution of the 1D wave equation: \(u(x,t)=\dfrac{f(x+ct)+f(x-ct)}{2}+\dfrac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds\). It represents two travelling waves moving in opposite directions.