1. **State the problem:** Solve the wave equation for the function $u(x,t) = \sin(x - at)$ where $a$ is a constant. 2. **Recall the wave equation:** The standard one-dimensional wave equation is $$\frac{\partial^2 u}{\partial t^2} = a^2 \frac{\partial^2 u}{\partial x^2}$$ where $a$ is the wave speed. 3. **Calculate the first partial derivatives:** - With respect to $x$: $$\frac{\partial u}{\partial x} = \cos(x - at)$$ - With respect to $t$: $$\frac{\partial u}{\partial t} = -a \cos(x - at)$$ 4. **Calculate the second partial derivatives:** - With respect to $x$: $$\frac{\partial^2 u}{\partial x^2} = -\sin(x - at)$$ - With respect to $t$: $$\frac{\partial^2 u}{\partial t^2} = -a^2 \sin(x - at)$$ 5. **Substitute into the wave equation:** $$-a^2 \sin(x - at) = a^2 (-\sin(x - at))$$ which simplifies to $$-a^2 \sin(x - at) = -a^2 \sin(x - at)$$ 6. **Conclusion:** The function $u(x,t) = \sin(x - at)$ satisfies the wave equation. This shows that $\sin(x - at)$ represents a wave traveling in the positive $x$-direction with speed $a$.