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18,7 → 18,7 |
\end{abstract} |
\section{Úvod} |
\begin{itemize} |
\item Spočítejte vlstní frekvenci struny v praktiku a změřte její harmonické frekvence, z nich dopočítejte lineární hustotu struny. |
\item Spočítejte vlastní frekvenci struny v praktiku a změřte její harmonické frekvence, z nich dopočítejte lineární hustotu struny. |
\item Najděte základní a vyšší harmonické frekvence v Kundtově trubici. Ze známé délky trubice dopočítejte rychlost zvuku. |
\item Pro 10 ryzných frekvencí hledejte interferenční minima prodlužováním a zkracováním Quinckovy trubice. Vyneste do grafu závislost vlnové délky zvuku na rezonanční frekvenci. Z naměřených údajů dopočítejte rychlost zvuku. |
\item Najděte vlastní frekvence Helmzholtova dutinového rezonátoru. Vyneste závislost vlastní frekvence na objemu rezonátoru. |
27,7 → 27,8 |
\end{itemize} |
\section{Postup měření} |
Začali jsme hledáním harmonických frekvencí struny v praktiku. Podle teoretického výpočtu z hodnot lineární hustoty uvedené v \cite{zadani} |
\subsection{struna} |
Začali jsme hledáním harmonických frekvencí struny v praktiku. Podle teoretického výpočtu z hodnot lineární hustoty uvedené v \cite{akustika} nam vyšla rezonančni frekvence $f_0=24 [Hz]$. Skutečné naměřené hodnoty ale shrnuje tabulka \ref{rezonance_struna}. |
\begin{table}[htbp] |
\caption{Rezonanční frekvence 1,316m dlouhé struny} |
51,13 → 52,46 |
\label{rezonance_struna} |
\end{table} |
\begin{figure} |
\begin{center} |
\includegraphics[width=150mm]{struna.pdf} |
\end{center} |
\caption{Harmonické frekvence struny} |
\end{figure} |
Při regresi těchto hodnot fyzikální závislostí $f = \frac{1}{2} \cdot \sqrt{\frac{F}{\varrho}}$ , kde $F = 4,91 [N]$, jsme zjistili, že skutečná hodnota lineární hustoty struny výchází spíše o něco větší $0.00268895 \pm 5.369e-06 [kg/m]$. |
\subsection{Kundtova trubice} |
Další úkol byl principiálně podobný s tím rozdílem, že se jednalo o podélné vlnění v Kundtově trubici a naměřené výsledky uvádí tabulka \ref{rezonance_kund}. |
\begin{table}[htbp] |
\caption{Rezonanční frekvence 70cm Kundtovy trubice} |
\begin{center} |
\begin{tabular}{|c|c|c|} |
\hline |
Harmonická & Frekvence [Hz] & Rychlost zvuku [m/s] \\ \hline |
0 & 287,2 & 402,08 \\ \hline |
1 & 459,7 & 321,79 \\ \hline |
2 & 642 & 299,6 \\ \hline |
3 & 810 & 283,5 \\ \hline |
4 & 1059,8 & 296,74 \\ \hline |
\end{tabular} |
\end{center} |
\label{rezonance_kund} |
\end{table} |
Rychlost zvuku byla dopočtena použitím výrazu $ v_z=\frac{2 f L}{k}$. |
\subsection{Quinckova trubice} |
Následovalo měření vlnové délky v Quinckově trubici. |
\begin{table}[htbp] |
\caption{Hodnoty z měření Quinckovy trubice} |
\begin{center} |
\begin{tabular}{|c|c|c|c|c|} |
\hline |
Frekvence [Hz] & Minima & Vzdálenost [cm] & Vlnová délka [m] & Rychlost zvuku [m/s] \\ \hline |
5733 & 7 & 18,5 & 0,0529 & 303,03 \\ \hline |
5733 & 6 & 18,5 & 0,0612 & 303,03 \\ \hline |
5441,7 & 7 & 22 & 0,0629 & 342,05 \\ \hline |
5199 & 7 & 23,5 & 0,0671 & 349,08 \\ \hline |
5040,6 & 7 & 25 & 0,0714 & 360,04 \\ \hline |
72,147 → 106,45 |
\label{interference_Quinck} |
\end{table} |
%Graficke vyjadreni techto dat vypada takto: |
%\begin{figure}[h] \caption{Odrazy ultrazvuku od kovove desky} \label{obr1} |
% \begin{center} \includegraphics[width=5cm]{plot.ps} \end{center} |
%\end{figure} |
Dalším naším úkolem bylo změření rychlosti zvuku a pomocí této experimentálně zjištěné rychlosti se pak pokusit určit neznámou vzdálenost. Naše měření jsme prováděli odrazem. a jeho výsledky zobrazuje tabulka \ref{rychlost}. |
Výpočtem s využitím informací z \cite{sonar} jsme z naměřených hodnot určili rychlost zvuku na $v_{z}=(321,8\pm6,8)m/s$ |
\begin{table}[htbp] \caption{Měření rychlosti zvuku} |
\begin{center} |
\begin{tabular}{|c|c|} |
\hline Vzdalenost [cm] & cas[us] \\ \hline |
5 & 420 \\ \hline |
10 & 681 \\ \hline |
15 & 1010 \\ \hline |
20 & 1260 \\ \hline |
25 & 1620 \\ \hline |
30 & 1870 \\ \hline |
35 & 2160 \\ \hline |
40 & 2470 \\ \hline |
45 & 2750 \\ \hline |
50 & 3020 \\ \hline |
\end{tabular} |
\end{center} |
\label{rychlost} |
\end{table} |
Známou rychlost jsme následně využili k dopočtení neznámé vzdálenosti od překážky za pomoci časového posunu změřeného echa. Jak ukazuje tabulka \ref{vzdalenost}. |
\begin{table}[htbp] |
\caption{Měření vzdálenosti odrazem} |
\begin{figure} |
\begin{center} |
\begin{tabular}{|c|c|c|} |
\hline cas[us] & \multicolumn{ 2}{|c|}{skutecna / zmerena vzdalenost [cm]}\\ \hline |
1720 & 27 & 27,67 \\ \hline |
1360 & 21 & 21,88 \\ \hline |
2000 & 32 & 32,18 \\ \hline |
2230 & 36 & 35,88 \\ \hline |
2410 & 39 & 38,78 \\ \hline |
2640 & 43 & 42,48 \\ \hline |
\end{tabular} |
\includegraphics[width=150mm]{quinck.pdf} |
\caption{Závislost vlnové délky na frekvenci v Quinckově trubici} |
\end{center} |
\label{vzdalenost} |
\end{table} |
U všech těchto měření bylo vhodné odečíst 50us spoždění měřící aparatury (hlavně zesilovače). |
\end{figure} |
Dalším úkolem bylo proměření Dopplerova posuvu, zde šlo již o náročnější měření s pohybujícím se vozíkem na kolejové dráze. Naměřené výsledky shrnuje tabulka \ref{doppler}. |
Proložením $ l = \frac{v_z}{f} $ byla získána hodnota rychlosti zvuku $v_z=(349.379 \pm 2 [m/s])$. |
\subsection{Helmholtzův rezonátor} |
\begin{table}[htbp] |
\caption{Měření Dopplerova posuvu} |
\caption{Rezonance Helmholtzova rezonátoru (laboratorní baňky) v závislosti na objemu vlité vody} |
\begin{center} |
\begin{tabular}{|c|l|} \hline |
$f_0=40,42[kHz]$ & $v=0,61[m/s]$ \\ \hline |
\multicolumn{ 2}{|c|}{40,48} \\ \hline |
\multicolumn{ 2}{|c|}{40,49} \\ \hline |
\multicolumn{ 2}{|c|}{40,48} \\ \hline |
\multicolumn{ 2}{|c|}{40,48} \\ \hline |
\multicolumn{ 2}{|c|}{40,49} \\ \hline |
\multicolumn{ 2}{|c|}{40,48} \\ \hline |
\begin{tabular}{|c|c|} |
\hline |
Objem vody & Rezonance [Hz] \\ \hline |
0 & 178 \\ \hline |
200 & 197 \\ \hline |
300 & 208 \\ \hline |
400 & 223 \\ \hline |
600 & 262 \\ \hline |
800 & 336 \\ \hline |
\end{tabular} |
\begin{tabular}{|c|l|} \hline |
$f_0=40,39[kHz]$ & $v=0,46[m/s]$ \\ \hline |
\multicolumn{ 2}{|c|}{40,45} \\ \hline |
\multicolumn{ 2}{|c|}{40,44} \\ \hline |
\multicolumn{ 2}{|c|}{40,45} \\ \hline |
\multicolumn{ 2}{|c|}{40,44} \\ \hline |
\multicolumn{ 2}{|c|}{40,44} \\ \hline |
\multicolumn{ 2}{|c|}{40,44} \\ \hline |
\end{tabular} |
\begin{tabular}{|c|l|} \hline |
$f_0=40,48[kHz]$ & $v=0,4[m/s]$ \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\multicolumn{ 2}{|c|}{40,52} \\ \hline |
\multicolumn{ 2}{|c|}{40,52} \\ \hline |
\multicolumn{ 2}{|c|}{40,52} \\ \hline |
\end{tabular} |
\begin{tabular}{|c|l|} \hline |
$f_0=40,47[kHz]$ & $v=0,5 [m/s]$ \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\multicolumn{ 2}{|c|}{40,53} \\ \hline |
\end{tabular} |
\end{center} |
\label{doppler} |
\label{rezonance_helmholtz} |
\end{table} |
Nakonec následoval nejproblematičtější úkol a to měření difrakce. Zde bylo prakticky vyloučeno dodržet podmínky ze zadání ulohy \cite{sonar}, které specifikují vzdálenost mikrofonu od mřížky v rozsahu 3-4m. Z důvodu omezeného prostoru v laboratoři jsme tak měříli poize ve vzdálenosti 1,75m |
\begin{table}[htbp] |
\caption{Měření difrakce na mřížce m=10mm} |
\begin{figure} |
\begin{center} |
\begin{tabular}{|c|c|} |
\hline |
\multicolumn{ 2}{|c|}{Počet štěrbin N=1} \\ \hline |
offset[mm] & Intenzita[1] \\ \hline |
268 & 0,3 \\ \hline |
273 & 0,48 \\ \hline |
277 & 0,64 \\ \hline |
281 & 0,5 \\ \hline |
285 & 0,29 \\ \hline |
293 & 0,48 \\ \hline |
300 & 1,17 \\ \hline |
306 & 0,68 \\ \hline |
322 & 0,19 \\ \hline |
333 & 0,68 \\ \hline |
\end{tabular} |
\begin{tabular}{|c|c|} |
\hline |
\multicolumn{ 2}{|c|}{Počet štěrbin N=2} \\ \hline |
offset[mm] & Intenzita[1] \\ \hline |
292 & 0,79 \\ \hline |
326 & 0,95 \\ \hline |
271 & 0,94 \\ \hline |
301 & 0,45 \\ \hline |
264 & 0,43 \\ \hline |
33,2 & 0,47 \\ \hline |
\end{tabular} |
\begin{tabular}{|c|c|} |
\hline |
\multicolumn{ 2}{|c|}{Počet štěrbin N=3} \\ \hline |
offset[mm] & Intenzita[1] \\ \hline |
300 & 3,15 \\ \hline |
292 & 2,66 \\ \hline |
282 & 3,8 \\ \hline |
311 & 2,82 \\ \hline |
324 & 3,35 \\ \hline |
305 & 3,06 \\ \hline |
295 & 3,01 \\ \hline |
\end{tabular} |
\includegraphics[width=150mm]{helmholtz.pdf} |
\end{center} |
\label{difrakce} |
\end{table} |
\caption{Závislost rezonanční frekvence Helmholtzova oscilátoru na objemu vlité vody} |
\end{figure} |
\section{Diskuse} |
Díky našim měřícím podmínkám bych výsleky měření hodnotil spíše, jako velice informativní, neboť například zvláště při měření difrakce se v datech uplaťnovala jakákoli změna měřeného prostředí. (procházející kolegové, přesun přívodních vodičů, i samotný přesun měřícího mikrofonu). Při ověřivání zákonu odrazu byla zase problematická neznalost vyzařívacích charakteristik reproduktoru. Navíc díky absenci jakéhokoli mechanického vedení docházelo k vyosení snímače z jeho původní pozice. Tento jev by sice bylo možné částečně eliminovat hledáním maxima signálu vždy pod zvoleným reflexním úhlem ale tato metoda by asi značně přesáhla měřící čas, který i tak byl velice napjatý. |
Díky našim měřícím podmínkám bych výsleky měření hodnotil spíše, jako velice informativní, neboť například zvláště při měření difrakce se v datech uplaťnovala jakákoli změna měřeného prostředí. (procházející kolegové, přesun přívodních vodičů, i samotný přesun měřícího mikrofonu). Při ověřivání zákonu odrazu byla zase problematická neznalost vyzařívacích charakteristik reproduktoru. Navíc díky absenci jakéhokoli mechanického vedení docházelo k vyosení snímače z jeho původní pozice. Tento jev by sice bylo možné částečně eliminovat hledáním maxima signálu vždy pod zvoleným reflexním úhlem ale tato metoda by asi značně přesáhla měřící čas, který i tak byl velice napjatý. |
\section*{Závěr} |
Měřením jsme ověřili platnost zákona odrazu z geometrické optiky i pro zvukové vlny. Dále jsme zjistili, že rychlosti zvuku v našich laboratorních podmínkách se nijak zásadně neliší od tabulkových hodnot a též Dopplerův efekt je reálnou vlastností vlnění. |
Měřením byly v podstatě potvrzeny tabulkové ho dnoty rychlostí zvuku a ověřeny rezonanční vlastnosti vlnění. |
\begin{thebibliography}{99} |
\bibitem{akustika}{\it Zadání úlohy 9 - Základní experimenty akustiky}. \href{http://fyzika.fjfi.cvut.cz/Praktika/Akustika/akustikaPRA.pdf}{http://fyzika.fjfi.cvut.cz/Praktika/Akustika/akustikaPRA.pdf}. |
\bibitem{sctripta_vlneni} |
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set terminal postscript |
set output "helmholtz.ps" |
set xlabel "[ml]" |
set ylabel "[Hz]" |
set key off |
plot "helmholtz.txt" using 1:2:3 with errorbars |
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/hpt2 hpt 2 mul def |
/Lshow {currentpoint stroke M 0 vshift R |
Blacktext {gsave 0 setgray show grestore} {show} ifelse} def |
/Rshow {currentpoint stroke M dup stringwidth pop neg vshift R |
Blacktext {gsave 0 setgray show grestore} {show} ifelse} def |
/Cshow {currentpoint stroke M dup stringwidth pop -2 div vshift R |
Blacktext {gsave 0 setgray show grestore} {show} ifelse} def |
/UP {dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def |
/hpt2 hpt 2 mul def /vpt2 vpt 2 mul def} def |
/DL {Color {setrgbcolor Solid {pop []} if 0 setdash} |
{pop pop pop 0 setgray Solid {pop []} if 0 setdash} ifelse} def |
/BL {stroke userlinewidth 2 mul setlinewidth |
Rounded {1 setlinejoin 1 setlinecap} if} def |
/AL {stroke userlinewidth 2 div setlinewidth |
Rounded {1 setlinejoin 1 setlinecap} if} def |
/UL {dup gnulinewidth mul /userlinewidth exch def |
dup 1 lt {pop 1} if 10 mul /udl exch def} def |
/PL {stroke userlinewidth setlinewidth |
Rounded {1 setlinejoin 1 setlinecap} if} def |
% Default Line colors |
/LCw {1 1 1} def |
/LCb {0 0 0} def |
/LCa {0 0 0} def |
/LC0 {1 0 0} def |
/LC1 {0 1 0} def |
/LC2 {0 0 1} def |
/LC3 {1 0 1} def |
/LC4 {0 1 1} def |
/LC5 {1 1 0} def |
/LC6 {0 0 0} def |
/LC7 {1 0.3 0} def |
/LC8 {0.5 0.5 0.5} def |
% Default Line Types |
/LTw {PL [] 1 setgray} def |
/LTb {BL [] LCb DL} def |
/LTa {AL [1 udl mul 2 udl mul] 0 setdash LCa setrgbcolor} def |
/LT0 {PL [] LC0 DL} def |
/LT1 {PL [4 dl1 2 dl2] LC1 DL} def |
/LT2 {PL [2 dl1 3 dl2] LC2 DL} def |
/LT3 {PL [1 dl1 1.5 dl2] LC3 DL} def |
/LT4 {PL [6 dl1 2 dl2 1 dl1 2 dl2] LC4 DL} def |
/LT5 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC5 DL} def |
/LT6 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC6 DL} def |
/LT7 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC7 DL} def |
/LT8 {PL [2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 4 dl2] LC8 DL} def |
/Pnt {stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore} def |
/Dia {stroke [] 0 setdash 2 copy vpt add M |
hpt neg vpt neg V hpt vpt neg V |
hpt vpt V hpt neg vpt V closepath stroke |
Pnt} def |
/Pls {stroke [] 0 setdash vpt sub M 0 vpt2 V |
currentpoint stroke M |
hpt neg vpt neg R hpt2 0 V stroke |
} def |
/Box {stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M |
0 vpt2 neg V hpt2 0 V 0 vpt2 V |
hpt2 neg 0 V closepath stroke |
Pnt} def |
/Crs {stroke [] 0 setdash exch hpt sub exch vpt add M |
hpt2 vpt2 neg V currentpoint stroke M |
hpt2 neg 0 R hpt2 vpt2 V stroke} def |
/TriU {stroke [] 0 setdash 2 copy vpt 1.12 mul add M |
hpt neg vpt -1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt 1.62 mul V closepath stroke |
Pnt} def |
/Star {2 copy Pls Crs} def |
/BoxF {stroke [] 0 setdash exch hpt sub exch vpt add M |
0 vpt2 neg V hpt2 0 V 0 vpt2 V |
hpt2 neg 0 V closepath fill} def |
/TriUF {stroke [] 0 setdash vpt 1.12 mul add M |
hpt neg vpt -1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt 1.62 mul V closepath fill} def |
/TriD {stroke [] 0 setdash 2 copy vpt 1.12 mul sub M |
hpt neg vpt 1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt -1.62 mul V closepath stroke |
Pnt} def |
/TriDF {stroke [] 0 setdash vpt 1.12 mul sub M |
hpt neg vpt 1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt -1.62 mul V closepath fill} def |
/DiaF {stroke [] 0 setdash vpt add M |
hpt neg vpt neg V hpt vpt neg V |
hpt vpt V hpt neg vpt V closepath fill} def |
/Pent {stroke [] 0 setdash 2 copy gsave |
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat |
closepath stroke grestore Pnt} def |
/PentF {stroke [] 0 setdash gsave |
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat |
closepath fill grestore} def |
/Circle {stroke [] 0 setdash 2 copy |
hpt 0 360 arc stroke Pnt} def |
/CircleF {stroke [] 0 setdash hpt 0 360 arc fill} def |
/C0 {BL [] 0 setdash 2 copy moveto vpt 90 450 arc} bind def |
/C1 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 90 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C2 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 90 180 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C3 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 180 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C4 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 180 270 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C5 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 90 arc |
2 copy moveto |
2 copy vpt 180 270 arc closepath fill |
vpt 0 360 arc} bind def |
/C6 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 90 270 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C7 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 270 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C8 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 270 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C9 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 270 450 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill |
2 copy moveto |
2 copy vpt 90 180 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C11 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 180 arc closepath fill |
2 copy moveto |
2 copy vpt 270 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C12 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 180 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C13 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 90 arc closepath fill |
2 copy moveto |
2 copy vpt 180 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C14 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 90 360 arc closepath fill |
vpt 0 360 arc} bind def |
/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto |
neg 0 rlineto closepath} bind def |
/Square {dup Rec} bind def |
/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def |
/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def |
/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def |
/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def |
/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def |
/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def |
/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill |
exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def |
/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def |
/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill |
2 copy vpt Square fill Bsquare} bind def |
/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def |
/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def |
/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill |
Bsquare} bind def |
/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill |
Bsquare} bind def |
/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def |
/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill |
2 copy vpt Square fill Bsquare} bind def |
/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill |
2 copy exch vpt sub exch vpt Square fill Bsquare} bind def |
/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def |
/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def |
/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def |
/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def |
/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def |
/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def |
/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def |
/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def |
/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def |
/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def |
/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def |
/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def |
/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def |
/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def |
/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def |
/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def |
/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def |
/DiaE {stroke [] 0 setdash vpt add M |
hpt neg vpt neg V hpt vpt neg V |
hpt vpt V hpt neg vpt V closepath stroke} def |
/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M |
0 vpt2 neg V hpt2 0 V 0 vpt2 V |
hpt2 neg 0 V closepath stroke} def |
/TriUE {stroke [] 0 setdash vpt 1.12 mul add M |
hpt neg vpt -1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt 1.62 mul V closepath stroke} def |
/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M |
hpt neg vpt 1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt -1.62 mul V closepath stroke} def |
/PentE {stroke [] 0 setdash gsave |
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat |
closepath stroke grestore} def |
/CircE {stroke [] 0 setdash |
hpt 0 360 arc stroke} def |
/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def |
/DiaW {stroke [] 0 setdash vpt add M |
hpt neg vpt neg V hpt vpt neg V |
hpt vpt V hpt neg vpt V Opaque stroke} def |
/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M |
0 vpt2 neg V hpt2 0 V 0 vpt2 V |
hpt2 neg 0 V Opaque stroke} def |
/TriUW {stroke [] 0 setdash vpt 1.12 mul add M |
hpt neg vpt -1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt 1.62 mul V Opaque stroke} def |
/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M |
hpt neg vpt 1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt -1.62 mul V Opaque stroke} def |
/PentW {stroke [] 0 setdash gsave |
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat |
Opaque stroke grestore} def |
/CircW {stroke [] 0 setdash |
hpt 0 360 arc Opaque stroke} def |
/BoxFill {gsave Rec 1 setgray fill grestore} def |
/Density { |
/Fillden exch def |
currentrgbcolor |
/ColB exch def /ColG exch def /ColR exch def |
/ColR ColR Fillden mul Fillden sub 1 add def |
/ColG ColG Fillden mul Fillden sub 1 add def |
/ColB ColB Fillden mul Fillden sub 1 add def |
ColR ColG ColB setrgbcolor} def |
/BoxColFill {gsave Rec PolyFill} def |
/PolyFill {gsave Density fill grestore grestore} def |
/h {rlineto rlineto rlineto gsave closepath fill grestore} bind def |
% |
% PostScript Level 1 Pattern Fill routine for rectangles |
% Usage: x y w h s a XX PatternFill |
% x,y = lower left corner of box to be filled |
% w,h = width and height of box |
% a = angle in degrees between lines and x-axis |
% XX = 0/1 for no/yes cross-hatch |
% |
/PatternFill {gsave /PFa [ 9 2 roll ] def |
PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate |
PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec |
gsave 1 setgray fill grestore clip |
currentlinewidth 0.5 mul setlinewidth |
/PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def |
0 0 M PFa 5 get rotate PFs -2 div dup translate |
0 1 PFs PFa 4 get div 1 add floor cvi |
{PFa 4 get mul 0 M 0 PFs V} for |
0 PFa 6 get ne { |
0 1 PFs PFa 4 get div 1 add floor cvi |
{PFa 4 get mul 0 2 1 roll M PFs 0 V} for |
} if |
stroke grestore} def |
% |
/languagelevel where |
{pop languagelevel} {1} ifelse |
2 lt |
{/InterpretLevel1 true def} |
{/InterpretLevel1 Level1 def} |
ifelse |
% |
% PostScript level 2 pattern fill definitions |
% |
/Level2PatternFill { |
/Tile8x8 {/PaintType 2 /PatternType 1 /TilingType 1 /BBox [0 0 8 8] /XStep 8 /YStep 8} |
bind def |
/KeepColor {currentrgbcolor [/Pattern /DeviceRGB] setcolorspace} bind def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke} |
>> matrix makepattern |
/Pat1 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke |
0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke} |
>> matrix makepattern |
/Pat2 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L |
8 8 L 8 0 L 0 0 L fill} |
>> matrix makepattern |
/Pat3 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L |
0 12 M 12 0 L stroke} |
>> matrix makepattern |
/Pat4 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L |
0 -4 M 12 8 L stroke} |
>> matrix makepattern |
/Pat5 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L |
0 12 M 8 -4 L 4 12 M 10 0 L stroke} |
>> matrix makepattern |
/Pat6 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L |
0 -4 M 8 12 L 4 -4 M 10 8 L stroke} |
>> matrix makepattern |
/Pat7 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L |
12 0 M -4 8 L 12 4 M 0 10 L stroke} |
>> matrix makepattern |
/Pat8 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L |
-4 0 M 12 8 L -4 4 M 8 10 L stroke} |
>> matrix makepattern |
/Pat9 exch def |
/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def |
/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def |
/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def |
/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def |
/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def |
/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def |
/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def |
} def |
% |
% |
%End of PostScript Level 2 code |
% |
/PatternBgnd { |
TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse |
} def |
% |
% Substitute for Level 2 pattern fill codes with |
% grayscale if Level 2 support is not selected. |
% |
/Level1PatternFill { |
/Pattern1 {0.250 Density} bind def |
/Pattern2 {0.500 Density} bind def |
/Pattern3 {0.750 Density} bind def |
/Pattern4 {0.125 Density} bind def |
/Pattern5 {0.375 Density} bind def |
/Pattern6 {0.625 Density} bind def |
/Pattern7 {0.875 Density} bind def |
} def |
% |
% Now test for support of Level 2 code |
% |
Level1 {Level1PatternFill} {Level2PatternFill} ifelse |
% |
/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont |
dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall |
currentdict end definefont pop |
end |
%%EndProlog |
%%Page: 1 1 |
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( 3500) Cshow |
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(wavelenght [m]) Cshow |
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LTb |
LCb setrgbcolor |
4006 98 M |
(frequency [Hz]) Cshow |
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%%Pages: 1 |
/dokumenty/PRA1/9-Akustika/struna.gp |
---|
1,5 → 1,5 |
#set terminal postscript |
#set output "struna.ps" |
set terminal postscript |
set output "struna.ps" |
set key off |
set xlabel "# harmonics" |
set ylabel "frequency" |
6,6 → 6,6 |
F=4.91 |
f(x)=(x+1)*1/2*sqrt(F/Ro) |
fit f(x) "struna.txt" via Ro |
plot f(x) title "approximed", "struna.txt" using 1:2:3 with errorbars title "measured" |
pause 10 |
plot f(x), "struna.txt" using 1:2:3 with errorbars |
/dokumenty/PRA1/9-Akustika/struna.pdf |
---|
Cannot display: file marked as a binary type. |
svn:mime-type = application/octet-stream |
Property changes: |
Added: svn:mime-type |
+application/octet-stream |
\ No newline at end of property |
/dokumenty/PRA1/9-Akustika/struna.ps |
---|
1,7 → 1,7 |
%!PS-Adobe-2.0 |
%%Title: struna.ps |
%%Creator: gnuplot 4.2 patchlevel 4 |
%%CreationDate: Fri Oct 16 14:29:58 2009 |
%%CreationDate: Mon Oct 19 00:06:38 2009 |
%%DocumentFonts: (atend) |
%%BoundingBox: 50 50 554 770 |
%%Orientation: Landscape |
47,10 → 47,10 |
/Title (struna.ps) |
/Subject (gnuplot plot) |
/Creator (gnuplot 4.2 patchlevel 4 ) |
/Author (Jakub Kákona,,,) |
/Author (Jakub Kakona,,,) |
% /Producer (gnuplot) |
% /Keywords () |
/CreationDate (Fri Oct 16 14:29:58 2009) |
/CreationDate (Mon Oct 19 00:06:38 2009) |
/DOCINFO pdfmark |
end |
} ifelse |
436,15 → 436,15 |
(Helvetica) findfont 140 scalefont setfont |
1.000 UL |
LTb |
882 728 M |
882 448 M |
63 0 V |
6101 0 R |
-63 0 V |
798 728 M |
798 448 M |
( 0) Rshow |
1.000 UL |
LTb |
882 1557 M |
882 1333 M |
63 0 V |
6101 0 R |
-63 0 V |
452,7 → 452,7 |
( 50) Rshow |
1.000 UL |
LTb |
882 2386 M |
882 2218 M |
63 0 V |
6101 0 R |
-63 0 V |
460,7 → 460,7 |
( 100) Rshow |
1.000 UL |
LTb |
882 3214 M |
882 3102 M |
63 0 V |
6101 0 R |
-63 0 V |
468,7 → 468,7 |
( 150) Rshow |
1.000 UL |
LTb |
882 4043 M |
882 3987 M |
63 0 V |
6101 0 R |
-63 0 V |
484,51 → 484,51 |
( 250) Rshow |
1.000 UL |
LTb |
882 728 M |
882 448 M |
0 63 V |
0 4081 R |
0 4361 R |
0 -63 V |
882 588 M |
882 308 M |
( 0) Cshow |
1.000 UL |
LTb |
2115 728 M |
2115 448 M |
0 63 V |
0 4081 R |
0 4361 R |
0 -63 V |
0 -4221 R |
0 -4501 R |
( 2) Cshow |
1.000 UL |
LTb |
3348 728 M |
3348 448 M |
0 63 V |
0 4081 R |
0 4361 R |
0 -63 V |
0 -4221 R |
0 -4501 R |
( 4) Cshow |
1.000 UL |
LTb |
4580 728 M |
4580 448 M |
0 63 V |
0 4081 R |
0 4361 R |
0 -63 V |
0 -4221 R |
0 -4501 R |
( 6) Cshow |
1.000 UL |
LTb |
5813 728 M |
5813 448 M |
0 63 V |
0 4081 R |
0 4361 R |
0 -63 V |
0 -4221 R |
0 -4501 R |
( 8) Cshow |
1.000 UL |
LTb |
7046 728 M |
7046 448 M |
0 63 V |
0 4081 R |
0 4361 R |
0 -63 V |
0 -4221 R |
0 -4501 R |
( 10) Cshow |
1.000 UL |
LTb |
535,19 → 535,19 |
1.000 UL |
LTb |
882 4872 N |
882 728 L |
882 448 L |
6164 0 V |
0 4144 V |
0 4424 V |
-6164 0 V |
Z stroke |
LCb setrgbcolor |
308 2800 M |
308 2660 M |
currentpoint gsave translate 90 rotate 0 0 M |
(frequency) Cshow |
grestore |
LTb |
LCb setrgbcolor |
3964 378 M |
3964 98 M |
(# harmonics) Cshow |
LTb |
1.000 UP |
555,137 → 555,194 |
LTb |
1.000 UL |
LT0 |
LTb |
3397 133 M |
(approximed) Rshow |
LT0 |
3481 133 M |
399 0 V |
882 1081 M |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 35 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
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63 36 V |
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63 36 V |
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63 36 V |
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62 36 V |
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63 35 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 35 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
/Helvetica findfont 140 scalefont setfont |
882 826 M |
62 38 V |
63 38 V |
62 39 V |
62 38 V |
62 38 V |
63 38 V |
62 38 V |
62 39 V |
62 38 V |
63 38 V |
62 38 V |
62 38 V |
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63 38 V |
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62 38 V |
63 39 V |
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63 38 V |
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62 38 V |
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63 38 V |
62 38 V |
62 38 V |
62 38 V |
63 39 V |
62 38 V |
62 38 V |
62 38 V |
63 38 V |
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62 38 V |
62 38 V |
63 38 V |
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63 38 V |
62 38 V |
62 38 V |
62 38 V |
63 38 V |
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62 38 V |
62 38 V |
63 38 V |
62 38 V |
62 39 V |
62 38 V |
63 38 V |
62 38 V |
62 38 V |
62 39 V |
63 38 V |
62 38 V |
62 38 V |
62 38 V |
63 39 V |
62 38 V |
62 38 V |
63 38 V |
62 38 V |
62 38 V |
62 39 V |
63 38 V |
62 38 V |
62 38 V |
62 38 V |
63 39 V |
62 38 V |
62 38 V |
62 38 V |
63 38 V |
62 39 V |
62 38 V |
62 38 V |
63 38 V |
62 38 V |
62 39 V |
63 38 V |
62 38 V |
62 38 V |
62 38 V |
63 39 V |
62 38 V |
62 38 V |
62 38 V |
63 38 V |
62 38 V |
62 39 V |
62 38 V |
63 38 V |
62 38 V |
62 38 V |
62 39 V |
63 38 V |
62 38 V |
1.000 UP |
stroke |
LT1 |
LTb |
4804 133 M |
(measured) Rshow |
LT1 |
882 1060 Pls |
1498 1424 Pls |
2115 1789 Pls |
2731 2154 Pls |
3348 2485 Pls |
3964 2833 Pls |
4580 3214 Pls |
5197 3546 Pls |
5813 3911 Pls |
6430 4259 Pls |
7046 4623 Pls |
5087 133 Pls |
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882 763 M |
0 106 V |
851 763 M |
62 0 V |
851 869 M |
62 0 V |
585 280 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
586 280 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
585 278 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
586 253 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
585 271 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
585 294 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
586 255 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
585 281 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
586 257 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
585 285 R |
0 106 V |
-31 -106 R |
62 0 V |
-62 106 R |
62 0 V |
882 816 Pls |
1498 1202 Pls |
2115 1588 Pls |
2731 1972 Pls |
3348 2331 Pls |
3964 2708 Pls |
4580 3108 Pls |
5197 3469 Pls |
5813 3856 Pls |
6430 4219 Pls |
7046 4610 Pls |
1.000 UL |
LTb |
882 4872 N |
882 728 L |
882 448 L |
6164 0 V |
0 4144 V |
0 4424 V |
-6164 0 V |
Z stroke |
1.000 UP |
695,277 → 752,6 |
grestore |
end |
showpage |
%%Page: 2 2 |
gnudict begin |
gsave |
50 50 translate |
0.100 0.100 scale |
90 rotate |
0 -5040 translate |
0 setgray |
newpath |
(Helvetica) findfont 140 scalefont setfont |
1.000 UL |
LTb |
882 728 M |
63 0 V |
6101 0 R |
-63 0 V |
798 728 M |
( 0) Rshow |
1.000 UL |
LTb |
882 1557 M |
63 0 V |
6101 0 R |
-63 0 V |
-6185 0 R |
( 50) Rshow |
1.000 UL |
LTb |
882 2386 M |
63 0 V |
6101 0 R |
-63 0 V |
-6185 0 R |
( 100) Rshow |
1.000 UL |
LTb |
882 3214 M |
63 0 V |
6101 0 R |
-63 0 V |
-6185 0 R |
( 150) Rshow |
1.000 UL |
LTb |
882 4043 M |
63 0 V |
6101 0 R |
-63 0 V |
-6185 0 R |
( 200) Rshow |
1.000 UL |
LTb |
882 4872 M |
63 0 V |
6101 0 R |
-63 0 V |
-6185 0 R |
( 250) Rshow |
1.000 UL |
LTb |
882 728 M |
0 63 V |
0 4081 R |
0 -63 V |
882 588 M |
( 0) Cshow |
1.000 UL |
LTb |
2115 728 M |
0 63 V |
0 4081 R |
0 -63 V |
0 -4221 R |
( 2) Cshow |
1.000 UL |
LTb |
3348 728 M |
0 63 V |
0 4081 R |
0 -63 V |
0 -4221 R |
( 4) Cshow |
1.000 UL |
LTb |
4580 728 M |
0 63 V |
0 4081 R |
0 -63 V |
0 -4221 R |
( 6) Cshow |
1.000 UL |
LTb |
5813 728 M |
0 63 V |
0 4081 R |
0 -63 V |
0 -4221 R |
( 8) Cshow |
1.000 UL |
LTb |
7046 728 M |
0 63 V |
0 4081 R |
0 -63 V |
0 -4221 R |
( 10) Cshow |
1.000 UL |
LTb |
1.000 UL |
LTb |
882 4872 N |
882 728 L |
6164 0 V |
0 4144 V |
-6164 0 V |
Z stroke |
LCb setrgbcolor |
308 2800 M |
currentpoint gsave translate 90 rotate 0 0 M |
(frequency) Cshow |
grestore |
LTb |
LCb setrgbcolor |
3964 378 M |
(# harmonics) Cshow |
LTb |
1.000 UP |
1.000 UL |
LTb |
1.000 UL |
LT0 |
LTb |
3397 133 M |
(approximed) Rshow |
LT0 |
3481 133 M |
399 0 V |
882 1081 M |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 35 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 35 V |
62 36 V |
62 36 V |
62 36 V |
63 35 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 35 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 35 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
62 36 V |
62 36 V |
63 35 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 36 V |
62 35 V |
63 36 V |
62 36 V |
62 35 V |
62 36 V |
63 36 V |
62 35 V |
1.000 UP |
stroke |
LT1 |
LTb |
4804 133 M |
(measured) Rshow |
LT1 |
882 1060 Pls |
1498 1424 Pls |
2115 1789 Pls |
2731 2154 Pls |
3348 2485 Pls |
3964 2833 Pls |
4580 3214 Pls |
5197 3546 Pls |
5813 3911 Pls |
6430 4259 Pls |
7046 4623 Pls |
5087 133 Pls |
1.000 UL |
LTb |
882 4872 N |
882 728 L |
6164 0 V |
0 4144 V |
-6164 0 V |
Z stroke |
1.000 UP |
1.000 UL |
LTb |
stroke |
grestore |
end |
showpage |
%%Trailer |
%%DocumentFonts: Helvetica |
%%Pages: 2 |
%%Pages: 1 |