1. **Problem Statement:** Solve the wave equation $$\frac{\partial^2 u}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2}$$ with $$c^2 = 1$$ and initial displacement $$u(x,0) = f(x)$$ given by a piecewise linear function: $$f(x) = \begin{cases} \frac{x}{10}, & 0 \leq x \leq 10 \\ 2 - \frac{x}{10}, & 10 < x \leq 20 \\ 0, & \text{otherwise} \end{cases}$$ 2. **Formula and Method:** The general solution to the 1D wave equation with speed $$c=1$$ is given by d'Alembert's formula: $$u(x,t) = \frac{1}{2} \left[ f(x - t) + f(x + t) \right] + \frac{1}{2} \int_{x - t}^{x + t} g(s) \, ds$$ where $$g(x) = u_t(x,0)$$ is the initial velocity. Since no initial velocity is given, assume $$g(x) = 0$$. 3. **Applying the formula:** Since $$g(x) = 0$$, the solution simplifies to: $$u(x,t) = \frac{1}{2} \left[ f(x - t) + f(x + t) \right]$$ 4. **Evaluate $$f(x)$$ for shifted arguments:** Recall the piecewise definition of $$f(x)$$: - For $$0 \leq x \leq 10$$, $$f(x) = \frac{x}{10}$$ - For $$10 < x \leq 20$$, $$f(x) = 2 - \frac{x}{10}$$ - Elsewhere, $$f(x) = 0$$ 5. **Final solution:** $$u(x,t) = \frac{1}{2} \left[ f(x - t) + f(x + t) \right]$$ where $$f$$ is evaluated piecewise as above. This represents two waves traveling left and right without change in shape, each scaled by $$\frac{1}{2}$$ and summed. **Summary:** The wave motion $$u(x,t)$$ is the average of the initial shape shifted left and right by $$t$$: $$$u(x,t) = \frac{1}{2} f(x - t) + \frac{1}{2} f(x + t)$$$ This completes the solution for the given initial displacement and zero initial velocity.