Uniqueness of coefficients in plane wave representation of wave equation's solution 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.
 A: 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
$$ B (\boldsymbol{k}) = b (\boldsymbol{k}, t) \mathrm{e}^{\mathrm{i} c \lvert \boldsymbol{k} \rvert t} = [\mathcal{F} p(\cdot,t)] (\boldsymbol{k}) \mathrm{e}^{\mathrm{i} c \lvert \boldsymbol{k} \rvert t} = \frac{\mathrm{e}^{\mathrm{i} c \lvert \boldsymbol{k} \rvert t}}{(2 \pi)^3} \int \limits_{\mathbb{R}^3} p (\boldsymbol{r},t) \, \mathrm{e}^{-\mathrm{i} \boldsymbol{k} \cdot \boldsymbol{r}} \, \mathrm{d} \boldsymbol{r} \, .$$
