Temporally quasiperiodic data, propagating in the laboratory frame, can be rendered periodic by Galilean transformation

For a broad class of distributions of temperature, concentration, or another quantity propagating rectilinearly, we show that temporally quasiperiodic behavior in the laboratory frame can be rendered periodic by Galilean transformation. The approach is illustrated analytically and numerically using as an example a closed-form model distribution generated from a one-dimensional partial differential equation, and a detailed process is developed to determine frame speed from more general quasiperiodic, one-dimensional, temporally- and spatially-discretized data. The approach is extended to two- and three-dimensional rectilinear propagation, and its application to nonrectilinear propagation, along with implications for interpreting noise-corrupted data, are also discussed.

Figure 1

(a) Time series and (b) power spectrum (arbitrary units) for quasiperiodic response $u(\eta =0.4,\tau )$ with an intrinsic frequency $\omega =\pi /\phantom{\pi 5}5$ (incommensurable with the extrinsic frequency 1), $\mu =10$, and $u_0\left(\eta \right)=1$. In the power spectrum (from a time series over $0\le \tau \le 1000$ sampled uniformly with time increment ${10}^{-3}$), power levels, in arbitrary units, of 1 and $2.79\times {10}^{-4}$ at $f=0$ and 6, respectively, are omitted for graphical clarity.

Figure 2

Variation of $u(\eta ,\tau )$ with $\mu =10$ and $u_0\left(\eta \right)=1$ for two values of the intrinsic frequency: (a) a commensurable case with $\omega =2/\phantom{23}3$ producing periodic response, and (b) an incommensurable case with $\omega =\pi /\phantom{\pi 5}5$ producing quasiperiodic response. The $\alpha =0.2$ characteristic is shown as a continuous white curve. $u(\eta ,\tau )$ has been scaled by $8.007=max\left[u\right(\eta ,\tau ;\omega =\pi /\phantom{\pi 5}5\left)\right] >max\left[u\right(\eta ,\tau ;\omega =2/\phantom{23}3\left)\right]$.

Figure 3

Spatiotemporal variation of $U({{\eta }^{\prime }}_{\mathrm{PGT}},{\tau }^{\prime })$, with intrinsic frequency $\omega =\pi /\phantom{\pi 5}5$ (incommensurable with the extrinsic frequency 1), $\mu =10$, and $u_0\left(\eta \right)=1$ [case shown in Fig. 2], with magnitude scaled by $8.007=max\left[U({{\eta }^{\prime }}_{\mathrm{PGT}},{\tau }^{\prime })\right]$. In this PGT frame, the $\alpha =0.2$ characteristic is shown as a continuous solid white curve and is time periodic.

Figure 4

Power spectrum (arbitrary units) of moving-frame time series $U({{\eta }^{\prime }}_{\mathrm{PGT}}=0.35,{\tau }^{\prime })$ obtained from the quasiperiodic laboratory-frame distribution with incommensurable frequencies 1 and $\omega =\pi /\phantom{\pi 5}5, \mu =10$, and $u_0\left(\eta \right)=1$, shown on (a) linear and (b) semilog bases. In (a), power levels of 1 and $9.95\times {10}^{-4}$ at $f/\phantom{f\omega }\omega =0$ and 10, respectively, are omitted for graphical clarity. In (b), asymptotic decay with frequency is exponential.

Figure 5

Four moving-frame time series constructed from laboratory-frame data. Each consists of discrete points along a line whose slope is the reciprocal of frame velocity ($v=0.5$ here). Data are sampled with a time increment of $\mathrm{\Delta }{\tau }_{\mathrm{mf}}=\mathrm{\Delta }{\eta }_{\mathrm{mf}}/\phantom{\mathrm{\Delta }{\eta }_{\mathrm{mf}}v}v=0.1$. Intersections of the moving-frame time series with the $\tau =0$ axis are at ${\eta }_1=0, {\eta }_2=0.05, {\eta }_3=0.175$, and ${\eta }_4=0.25$), with no requirement that ${\eta }_i-{\eta }_{i-1}$ be uniform. For $v>0$, moving-frame time series with larger ${\eta }_i$ have fewer points than those with smaller ${\eta }_i$.

Figure 6

Moving frames that can be chosen beginning at a particular spatiotemporal point for a dataset with low spatial and temporal resolution. Frames indicated are nontrivial in the sense that each passes through at least one point for $\tau >{\tau }_1$ and has nonzero velocity.

Figure 7

Extracted frequencies, their sum, and the absolute value of their difference ($•$ : $f_{\mathrm{E},\mathrm{A}}$ and $f_{\mathrm{E},\mathrm{B}}$; $\nabla$ : $f_{\mathrm{E},\mathrm{A}}+f_{\mathrm{E},\mathrm{B}}$; $\mathrm{\Delta }$ : $|f_{\mathrm{E},\mathrm{A}}-f_{\mathrm{E},\mathrm{B}}|$) for 16 frame velocities. The “primary frequency lines,” i.e., basis frequencies vs frame velocity, are shown as solid lines (equations given in text), and two additional frequency lines are shown as a dotted line and as a dashed line (“paired equations” given in text). The two parts of each frequency line have slopes of equal magnitude. The absence of a marker on the $f_1\left(v\right)$ primary frequency line at $v=0.08$ is due to our choice to plot only the four frequencies above. At $v=0.08$, the basis frequency pair $f_{\mathrm{E},\mathrm{A}}+3f_{\mathrm{E},\mathrm{B}}$ and $f_{\mathrm{E},\mathrm{B}}$, with $f_{\mathrm{E},\mathrm{A}}=0.02832$ and $f_{\mathrm{E},\mathrm{B}}=0.20000$, lies on the primary frequency lines $f_1\left(v\right)$ and $f_2\left(v\right)$, respectively.

Figure 8

Variation of $U({\eta }^{\prime },{\tau }^{\prime })$, with the intrinsic frequency $\omega =\pi /\phantom{\pi 5}5$ (incommensurable with the extrinsic frequency 1), $\mu =10$, and $u_0\left(\eta \right)=1$, in the moving frames with (a) $v=(1-0.62832)/\phantom{(1-0.62832)10}10$ and (c) $v=(1+0.62832)/\phantom{(1+0.62832)10}10$, with magnitude scaled by $8.007=max\left[U({\eta }^{\prime },{\tau }^{\prime })\right]$. These frame velocities were determined as intersections of the frequency-line pair $f_1\left(v\right)=0.62832$ and $f_2\left(v\right)=\left|1-10v\right|$, i.e., by solving (13) with $r=m/\phantom{mn}n=1$. As expected, the distribution is time periodic in both frames, as shown by the power spectra of the $U({\eta }^{\prime }=0,{\tau }^{\prime })$ times series for (b) $v=(1-0.62832)/\phantom{(1-0.62832)10}10$ and (d) $v=(1+0.62832)/\phantom{(1+0.62832)10}10$. In (b) and (d), the peak at $f=0$ is omitted for graphical clarity. The symbol $F$ denotes the value 0.62832 in each subfigure.