I have a series of wave-readings which show wave amplitudes pr. time unit for different events. So on the $x $-axis we have seconds, and on the $y$-axis, wave height.
If I want to convert this to a time-independent plot, and go therefore from:
$$\frac{A}{t}\longrightarrow \lambda/\kappa,$$
where $A$ is the amplitude, $t$ is time (in sec), $\lambda$ is the wavelength, and $\kappa$ is the wavenumber, how do I do that?
Any suggestions appreciated.
Your graph, amplitude as a function of time, does not include any factor of distance. It is only an oscillation at the one point being measured for amplitude. Wave number and wavelength are distance-based relationships. In a standard sine wave, wave number is $\kappa=1/\lambda$. Wave number and wavelength do not relate to amplitude. Unless you know the wave speed, they do not relate to time.
If the variation of amplitude with respect to time shows the summation of various frequencies, then you could use a Fourier transformation to find amplitude as a function of frequency. For this to work, however, the wave would need a repeating cycle. The period of that repeating cycle would give you the fundamental frequency on which to build your Fourier analysis.
The plot of the function is generated from a series of $(x,y)$-coordinates, $\{(x_1,y_1), (x_2,y_2),\cdots(x_n,y_n)\}$, where $x$=time (t), $y$=Amplitude (A). In order to convert this plot into a stattionary plot, where $x$ is the position, we have to convert the set of $x$ coordinates from time to position.
We use the conversion operator:
$$Q:\frac{A}{t}\longrightarrow \frac{A}{x}$$
which is an operation on the set of coordinates of the plot:
\begin{equation} Q: \{(x_1,y_1), (x_2,y_2),\cdots(x_n,y_n)\}\longrightarrow \{v,1\}\cdot\{(x_1,y_1), (x_2,y_2),\cdots,(x_n,y_n)\} \end{equation}
Where $v$ is the average velocity of the wavetrain. This yields a new set of coordinates:
$$\{(v\cdot x_1, y_1), (v\cdot x_2, y_2),\cdots,(v\cdot x_n, y_n)\}$$
In this way, we have converted the time-axis (x-axis) into a position-axis, and when we perform an interpolation we obtain a stationary plot of the waves (waveprofile), where $\lambda$ is readable on the $x$-axis. This plot is in effect a "stretched" out version of the time-dependent plot, stretched on the x-axis.
