**Proof plan.** The idea is to decompose the forced problem into infinitesimal homogeneous problems. At each time $s$, the source $f(\cdot, s)$ acts as an initial velocity impulse. We solve the homogeneous [wave equation](/page/Wave%20Equation) with zero displacement and initial velocity $f(\cdot, s)$, starting at time $s$, and integrate these contributions over $s \in [0, t]$. Verification reduces to differentiating under the [integral](/page/Integral) sign and using the fact that each constituent solves the homogeneous equation. **Step 1: Define the family of corrector problems.** For each $s \in [0, t]$, let $w(\cdot, \cdot; s): \mathbb{R}^n \times [s, \infty) \to \mathbb{R}$ be the unique solution of \begin{align*} \begin{cases} \partial_t^2 w - \Delta w = 0 & \text{in } \mathbb{R}^n \times (s, \infty), \\ w(\cdot, s; s) = 0, \\ \partial_t w(\cdot, s; s) = f(\cdot, s). \end{cases} \end{align*} This is a standard homogeneous Cauchy problem with zero displacement and velocity $f(\cdot, s)$, which has a unique $C^2$ solution by the representation formulas (d'Alembert for $n = 1$, Kirchhoff/Poisson for $n = 2, 3$, and their extensions for general $n$). **Step 2: Form the superposition.** Define \begin{align*} u(x, t) := \int_0^t w(x, t; s) \, d\mathcal{L}^1(s). \end{align*} We verify $u$ solves the stated problem. **Step 3: Check the initial conditions.** At $t = 0$ the integral has empty range: $u(x, 0) = \int_0^0 w(x, 0; s) \, d\mathcal{L}^1(s) = 0$. For the time derivative, by the Leibniz rule: \begin{align*} \partial_t u(x, t) = w(x, t; t) + \int_0^t \partial_t w(x, t; s) \, d\mathcal{L}^1(s). \end{align*} At $t = 0$: $\partial_t u(x, 0) = w(x, 0; 0) + 0 = 0$, since $w(\cdot, s; s) = 0$ by definition. **Step 4: Verify the equation.** Differentiating once more: \begin{align*} \partial_t^2 u(x, t) = \partial_t w(x, t; t) + \int_0^t \partial_t^2 w(x, t; s) \, d\mathcal{L}^1(s) = f(x, t) + \int_0^t \Delta w(x, t; s) \, d\mathcal{L}^1(s), \end{align*} where we used $\partial_t w(x, t; t)\big|_{s = t} = f(x, t)$ (from the initial velocity condition on $w$) and $\partial_t^2 w = \Delta w$ (since $w$ solves the homogeneous wave equation in the variable $t$ for each fixed $s$). Meanwhile: \begin{align*} \Delta u(x, t) = \int_0^t \Delta w(x, t; s) \, d\mathcal{L}^1(s). \end{align*} Subtracting: $\partial_t^2 u - \Delta u = f(x, t)$. **Step 5: Regularity.** The regularity $u \in C^2$ follows from the smoothness of $f$ and standard estimates on the homogeneous propagator, together with the fact that [differentiation](/page/Derivative) under the integral sign is justified by the uniform bounds on $w$ and its derivatives (guaranteed by the compact support and smoothness of $f$).