In the context of a physics calculation, I have a wave equation with the following form, where $v$ is a constant and $f$ depends on $\vec{x}=(x,y,z)$ and on the time $t$:
$$\bigg(\dfrac{\partial^2}{\partial t^2}+v^2\nabla^2\bigg)f=0$$
By writing the Fourier expansion of $f$ and solving in Fourier space, I was able to reach this expression for the general solution, which agrees with the result given in the book:
$$f(\vec{x},t)=\int\dfrac{d^3 k}{(2\pi)^3}\big(A_k\sin(\omega_kt-\vec{k}\cdot\vec{x})+B_k\cos(\omega_kt-\vec{k}\cdot\vec{x})\big)$$
However, I know that the general solution for a wave equation in one dimension has the form:
$$f(x,t)=p(x-vt)+q(x+vt)$$
where $p$ and $q$ are sufficiently smooth arbitrary functions. So, my question is, can this result be generalized to the case of three dimensions, obtaining something like the following?
$$f(x,t)=p(\vec{x}-vt)+q(\vec{x}+vt)$$
and, if so, is this expression equivalent to the general solution that I found solving the equation in Fourier space?