| /dokumenty/PRA1/9-Akustika/akustika.pdf |
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| /dokumenty/PRA1/9-Akustika/akustika.tex |
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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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| 0 -4501 R |
| ( 6000) Cshow |
| 1.000 UL |
| LTb |
| 1.000 UL |
| LTb |
| 966 4872 N |
| 966 448 L |
| 6080 0 V |
| 0 4424 V |
| -6080 0 V |
| Z stroke |
| LCb setrgbcolor |
| 308 2660 M |
| currentpoint gsave translate 90 rotate 0 0 M |
| (wavelenght [m]) Cshow |
| grestore |
| LTb |
| LCb setrgbcolor |
| 4006 98 M |
| (frequency [Hz]) Cshow |
| LTb |
| 1.000 UP |
| 1.000 UL |
| LTb |
| 1.000 UL |
| LT0 |
| /Helvetica findfont 140 scalefont setfont |
| 1371 4188 M |
| 52 -55 V |
| 52 -53 V |
| 52 -53 V |
| 52 -53 V |
| 52 -51 V |
| 51 -51 V |
| 52 -49 V |
| 52 -50 V |
| 52 -48 V |
| 52 -48 V |
| 52 -47 V |
| 52 -46 V |
| 51 -45 V |
| 52 -45 V |
| 52 -45 V |
| 52 -43 V |
| 52 -43 V |
| 52 -43 V |
| 52 -42 V |
| 51 -41 V |
| 52 -41 V |
| 52 -40 V |
| 52 -40 V |
| 52 -39 V |
| 52 -38 V |
| 52 -38 V |
| 51 -38 V |
| 52 -37 V |
| 52 -37 V |
| 52 -36 V |
| 52 -35 V |
| 52 -36 V |
| 52 -34 V |
| 51 -35 V |
| 52 -34 V |
| 52 -33 V |
| 52 -33 V |
| 52 -33 V |
| 52 -32 V |
| 51 -32 V |
| 52 -31 V |
| 52 -31 V |
| 52 -31 V |
| 52 -31 V |
| 52 -30 V |
| 52 -29 V |
| 51 -30 V |
| 52 -29 V |
| 52 -28 V |
| 52 -28 V |
| 52 -28 V |
| 52 -28 V |
| 52 -28 V |
| 51 -27 V |
| 52 -26 V |
| 52 -27 V |
| 52 -26 V |
| 52 -26 V |
| 52 -26 V |
| 52 -25 V |
| 51 -25 V |
| 52 -25 V |
| 52 -24 V |
| 52 -25 V |
| 52 -24 V |
| 52 -23 V |
| 52 -24 V |
| 51 -23 V |
| 52 -23 V |
| 52 -23 V |
| 52 -22 V |
| 52 -23 V |
| 52 -22 V |
| 52 -22 V |
| 51 -21 V |
| 52 -22 V |
| 52 -21 V |
| 52 -21 V |
| 52 -21 V |
| 52 -21 V |
| 52 -20 V |
| 51 -20 V |
| 52 -20 V |
| 52 -20 V |
| 52 -20 V |
| 52 -19 V |
| 52 -20 V |
| 51 -19 V |
| 52 -19 V |
| 52 -18 V |
| 52 -19 V |
| 52 -18 V |
| 52 -19 V |
| 52 -18 V |
| 51 -17 V |
| 52 -18 V |
| 52 -18 V |
| 52 -17 V |
| 52 -17 V |
| 1.000 UP |
| stroke |
| LT1 |
| /Helvetica findfont 140 scalefont setfont |
| 6505 871 M |
| 0 632 V |
| 6474 871 M |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 5915 947 M |
| 0 632 V |
| 5884 947 M |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 5423 1213 M |
| 0 632 V |
| -31 -632 R |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 5102 1484 M |
| 0 632 V |
| -31 -632 R |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 4837 1396 M |
| 0 632 V |
| -31 -632 R |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 4499 1605 M |
| 0 632 V |
| -31 -632 R |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 4168 1819 M |
| 0 632 V |
| -31 -632 R |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 3398 2281 M |
| 0 632 V |
| -31 -632 R |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 2790 2660 M |
| 0 632 V |
| -31 -632 R |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 1371 3798 M |
| 0 632 V |
| -31 -632 R |
| 62 0 V |
| -62 632 R |
| 62 0 V |
| 6505 1187 Pls |
| 5915 1263 Pls |
| 5423 1529 Pls |
| 5102 1800 Pls |
| 4837 1712 Pls |
| 4499 1921 Pls |
| 4168 2135 Pls |
| 3398 2597 Pls |
| 2790 2976 Pls |
| 1371 4114 Pls |
| 1.000 UL |
| LTb |
| 966 4872 N |
| 966 448 L |
| 6080 0 V |
| 0 4424 V |
| -6080 0 V |
| Z stroke |
| 1.000 UP |
| 1.000 UL |
| LTb |
| stroke |
| grestore |
| end |
| showpage |
| %%Trailer |
| %%DocumentFonts: Helvetica |
| %%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 |
| 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 |
| /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 |
| 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 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 |
| 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 |
| /Helvetica findfont 140 scalefont setfont |
| 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 |