3D wave equation $$ \Delta p(\boldsymbol{r},t) - \frac{1}{c^{2}} \frac{\partial^{2} p}{\partial t^{2}}(\boldsymbol{r},t) = 0 \tag{1} $$ By applying the spatial Fourier transform to this PDE, we obtain the following ODE $$ c^{2} |\boldsymbol{k}|^{2} \hat{p}(\boldsymbol{k},t) - \frac{\partial^{2} \hat{p}}{\partial t^{2}}(\boldsymbol{k},t) = 0 \tag{2} $$ Since $(2)$ is ODE of harmonic oscillator, the solution can be written as $$ \hat{p}(\boldsymbol{k},t) = A(\boldsymbol{k}) e^{ic|\boldsymbol{k}|t} + B(\boldsymbol{k}) e^{-ic|\boldsymbol{k}|t} \tag{3} $$ Since $A=0$ by physical interpretation, apply the inverse Fourier transform to $(3)$. $$ p(\boldsymbol{r},t) = \int_{\mathbb{R}^{3}} B(\boldsymbol{k}) e^{i(\boldsymbol{k}\cdot \boldsymbol{r} - c|\boldsymbol{k}| t)} d\boldsymbol{k} \tag{4} $$ This is plane wave representation of 3D wave equation's solution.
Now if sound pressure $p(\boldsymbol{r},t)$ is known, can the coefficients $\{B(\boldsymbol{k})\}$ be uniquely determined?
In other words, I want to know invertibility of following integral operator $\mathcal{G}$ and its inverse operator if $\mathcal{G}$ is invertible. $$ \mathcal{G}\hat{f} (\boldsymbol{r},t) = \int_{\mathbb{R}^{3}} \hat{f}(\boldsymbol{k}) e^{i(\boldsymbol{k}\cdot \boldsymbol{r} - c|\boldsymbol{k}| t)} d\boldsymbol{k} \tag{5} $$ In the first place, it is difficult for me to calculate multiple integrals with polar coordinate component and I'm not familiar with integral operator theory.
Thank you for your advice.
You can write $$ p(\boldsymbol{r},t) = \int \limits_{\mathbb{R}^3} B(\boldsymbol{k}) \,\mathrm{e}^{\mathrm{i} (\boldsymbol{k} \cdot \boldsymbol{r} - c \lvert \boldsymbol{k} \rvert t)} \, \mathrm{d} \boldsymbol{k} = \int \limits_{\mathbb{R}^3} B(\boldsymbol{k}) \mathrm{e}^{-\mathrm{i} c \lvert \boldsymbol{k} \rvert t} \,\mathrm{e}^{\mathrm{i} \boldsymbol{k} \cdot \boldsymbol{r}} \, \mathrm{d} \boldsymbol{k} = [\mathcal{F}^{-1} b (\cdot, t)] (\boldsymbol{r}) \, , $$ where $b (\boldsymbol{k}, t) = B(\boldsymbol{k}) \mathrm{e}^{-\mathrm{i} c \lvert \boldsymbol{k} \rvert t}$. Since the Fourier transform $\mathcal{F}$ is invertible (on suitable function spaces), $B$ is uniquely determined by $$ B (\boldsymbol{k}) = b (\boldsymbol{k}, t) \mathrm{e}^{\mathrm{i} c \lvert \boldsymbol{k} \rvert t} = [\mathcal{F} p(\cdot,t)] (\boldsymbol{k}) \mathrm{e}^{\mathrm{i} c \lvert \boldsymbol{k} \rvert t} = \frac{\mathrm{e}^{\mathrm{i} c \lvert \boldsymbol{k} \rvert t}}{(2 \pi)^3} \int \limits_{\mathbb{R}^3} p (\boldsymbol{r},t) \, \mathrm{e}^{-\mathrm{i} \boldsymbol{k} \cdot \boldsymbol{r}} \, \mathrm{d} \boldsymbol{r} \, .$$