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The equation for a travelling wave is usually taken in the form of,

$$y = A\mathrm{e}^{i(kx - \omega t)}$$

When a standing wave is formed by the interference of two counter-propagating waves, then it implies that the equation of the reflected wave is in the form of,

$$y = A\mathrm{e}^{i(-kx - \omega t)}$$

Now, given a two-dimensional point-wise distribution of a pressure field, we can use the FFT to extract the spatial frequencies and the relative amplitudes corresponding to each spatial frequency (assuming that the temporal dependence is harmonic and can be neglected). My question is, in order to reconstruct the pressure field distribution using a finite number of "standing" wave components, is it okay to find out the spatial frequencies along the first quadrant of the $(k_x,k_y)$ plot and write the corresponding standing wave component as,

$$z = A_i \cos(k_xx + k_yy - \omega t),$$ where $A_i$ is the amplitude corresponding to the $(k_x,k_y)$ pair.

Or does this introduce an error since the amplitude of the "complementary" wavenumber pair $(-k_x,-k_y)$ is not necessarily the same as the original pair ?

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I'm just going to focus on the spatial Fourier transform / spatial dependence in this answer, so the answer can be written in terms of properties of the Fourier transform; calculating the time dependence is a different story. I assume that you're working with the wave equation on a domain where you can do separation of variables in terms of Cartesian coordinates so that you can infer the time dependence from the spatial Fourier transform of the initial conditions.

For a real-valued function of one variable $f(x)$, the Fourier transform $F(k)$ satisfies $F(k)=F(-k)^*$. Geometrically, this says that if you know the Fourier transform for positive $k$, you can reflect the function about the origin and complex conjugate the values to reconstruct the negative values of the Fourier transform. So you only need to know the Fourier transform for positive frequencies to computer the inverse Fourier transform.

The generalization to two variables is that when $f(x,y)$ is real, then $F(k_1, k_2) = -F(-k_1, -k_2)^*$. Geometrically, this says that if you know $F(k_1, k_2)$ in some region, then you can reflect that region through the origin and complex conjugate the values to compute the missing values of the Fourier transform in the reflected region. Therefore, in general, you need more information about $F(k_1, k_2)$ than the region $k_1, k_2>0$ to compute the inverse Fourier transform. You also need to know the Fourier transform in a region where $k_1$ and $k_2$ have opposite signs. In a $k_1-k_2$ plane, this would amount to saying you either need to know $F(k_1, k_2)$ in the entire upper half plane, or the entire right half plane -- or at least some half plane that includes the origin. Knowing the Fourier transform in a half plane that includes the origin is enough information to reconstruct the rest of the Fourier transform by the reflection property $F(k_1, k_2) = -F(-k_1, -k_2)^*$.

In your notation, the problem with only knowing $k_1$ and $k_2$ in the upper right quadrant is that it would only allow plane waves $\sim \cos(k_1 x + k_2 y)$ where wavefronts satisfied $k_1 x + k_2 y = {\rm const}$ for $k_1, k_2>0$, or in other words, the wavefronts would only be allowed to have a negative slope in the $k_1-k_2$ plane. In order to have positive sloping wavefronts, you have to allow one of $k_1$ or $k_2$ to be negative.

A side note: in your notation, you should allow for a phase offset in addition to an amplitude, like $\sim A_i \cos(k_x x + k_y y + \varphi_i)$.

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  • $\begingroup$ Thank you so much for the clarification. That definitely helps. And thanks for the quick tip as well. I completely forgot about the phase spectrum that is associated with the Fourier Transform. $\endgroup$ Mar 17 at 17:56

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