Wave number can be easily obtained in one dimensional situation. $$k=\omega/c$$ In the book, Fundamentals of Acoustics, the method of separation of variables is applied to solve two dimensional wave equation. It follows that wave number is also decomposed. $$k_x^{2}+k_z^{2}=k^{2}$$ $k_x$ and $k_z$ are determined by the boundary conditions.
But in one dimensional wave equation, it seems that we don't need additional conditions to get wave number. Given a specific frequency $f$ and the velocity $c$ of wave is determined by medium. Then we can get $k$.
I wonder how to determine the value of the decomposed wave number if there are no boundary conditions, for example, an infinite area. Any help would be greatly appreciated.
I got the explanation in the book [1].
$k_x$ is called the trace wavenumber in the $x$ direction. $\lambda_x=2\pi/k_x$ is called the trace wavelength in the $x$ direction. $c_x=\omega/k_x$ is called the phase speed of the trace wave along the $x$ axis.
Since $k$ is a constant, the trace wavenumbers are not independent of one another. Importantly, there is no restriction on the value of $k_x$ and $k_z$; they can extend over all real numbers from $-\infty$ to $+\infty$.
If the wave propagates in the direction given by $\theta$, the angle between the direction of propagation and the Z-axis, the trace wavenumbers follow that: $$k_x =k sin\theta$$ $$k_z =k cos\theta$$
For more imformation, you can read the book.
[1] Earl G. Williams, Fourier Acoustics, Academic Press, 1999, https://doi.org/10.1016/B978-012753960-7/50009-7