I'm a little stuck with understanding the properties of 2D waves.

I have the wave

$e^{2\pi i(jx+ky-\omega_{j,k}t)}$=$\exp\left(2\pi i\left(\left[\begin{array}{l} j \\ k \end{array}\right]\cdot\left[\begin{array}{l}x \\ y \end{array}\right]-\omega_{j,k} t\right)\right)$.

The wavenumber in the $x$-direction is $j$ and the wavenumber in the $y$-direction is $k$. As wavenumber is "frequency in the space sense", this gives a wavelength of $\frac{1}{j}$ and $\frac{1}{k}$ in the $x$ and $y$ directions respectively. $\omega_{j,k}$ has a $j$ and $k$ dependency and is the frequency of the wave, which gives a period of $\frac{1}{\omega_{j,k}}$.

The speed in the $x$-direction is, speed=distance/time =wavelength/period=$\frac{\omega_{j,k}}{j}$.

Similarly in the $y$ direction, the speed is $\frac{\omega_{j,k}}{k}$.

In the 2D linear advection equation $u_t-c_1u_x-c_2u_y=0$, $\omega_{j,k}=c_1j+c_2k$. This gives,

$\exp\left(2\pi i\left[\begin{array}{l} j \\ k \end{array}\right]\cdot\left[\begin{array}{l}x-c_1t \\ y-c_2t \end{array}\right]\right)$ as my wave. This indicates that the speed in the $x$-direction is $c_1$ and in the $y$-direction $c_2$. However, this is not what is calculated by $\frac{\omega_{j,k}}{j}$ and $\frac{\omega_{j,k}}{k}$ respectively. I need to understand this so that I can calculate the wave speed in the two directions when $\omega_{j,k}$is not a linear function of $j$ and $k$. Should I be using the dot product somewhere?

I'm also a little unsure why the speed in the direction of travel can't be calculated using pythagoras' theorem. I've seen literature giving this speed as $\frac{\omega_{j,k}}{\sqrt{j^2+k^2}}$, rather than $\omega_{j,k}\sqrt{\left(\frac{1}{j}\right)^2+\left(\frac{1}{k}\right)^2}$.

Thanks in advance.


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