Applets:Bessel Functions of the First Kind - Versionsgeschichte

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Textersetzung - „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29“ durch „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29“

Höfler am 26. Oktober 2023 um 10:05 Uhr

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Tasnad am 10. Juni 2020 um 08:04 Uhr

2020-06-10T08:04:17Z

Tasnad am 9. Juni 2020 um 20:46 Uhr

2020-06-09T20:46:53Z

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2018-08-27T16:31:51Z

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2018-08-27T16:28:52Z

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Guenter am 27. August 2018 um 15:43 Uhr

2018-08-27T15:43:23Z

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 ==About the Authors==
 This interactive applet was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the  [https://www.tum.de/ Technische Universität München].
-*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using  &bdquo;FlashMX&ndash;Actionscript&rdquo; (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
+*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using  &bdquo;FlashMX&ndash;Actionscript&rdquo; (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29|Günter Söder]]).
 *In 2018 this Applet was redesigned and updated to &bdquo;HTML5&rdquo; by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29|Tasnád Kernetzky]]).

@@ Zeile 181: / Zeile 181: @@
 This interactive applet was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the  [https://www.tum.de/ Technische Universität München].
 *The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using  &bdquo;FlashMX&ndash;Actionscript&rdquo; (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
-*In 2018 this Applet was redesigned and updated to &bdquo;HTML5&rdquo; by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).
+*In 2018 this Applet was redesigned and updated to &bdquo;HTML5&rdquo; by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29|Tasnád Kernetzky]]).
 ==Once again: Open Applet in new Tab==
 {{LntAppletLinkEn|besselFuns_en}}

← Nächstältere Version		Version vom 10. Juni 2020, 08:04 Uhr
Zeile 13:		Zeile 13:


−	[[Applets:~~Besselfunktionen_erster_Art_(neues_Applet)~~\|'''German description''']]	+	[[Applets:Besselfunktionen_erster_Art\|'''German description''']]

	==Theoretical Background==		==Theoretical Background==

@@ Zeile 1: / Zeile 1: @@
-{{LntAppletLinkEn|bessel}}
+{{LntAppletLinkEn|besselFuns_en}}
 ==Applet Description==
@@ Zeile 185: / Zeile 185: @@
 ==Once again: Open Applet in new Tab==
-{{LntAppletLinkEn|bessel}}
+{{LntAppletLinkEn|besselFuns_en}}

@@ Zeile 1: / Zeile 1: @@
-{{LntAppletLink|bessel}}
+{{LntAppletLinkEn|bessel}}
 ==Applet Description==
@@ Zeile 150: / Zeile 150: @@
 *Also in the frequency modulation, the Bessel lines now occur at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there are now more significantly more Bessel lines at the bottom due to the larger modulation index $η = 2.5$ than in the upper right  (for $η = 1.5$ valid) chart.}}
 ==Applet Manual==
@@ Zeile 185: / Zeile 183: @@
 *In 2018 this Applet was redesigned and updated to &bdquo;HTML5&rdquo; by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).
-==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==
+==Once again: Open Applet in new Tab==
-{{LntAppletLink|bessel}}
+{{LntAppletLinkEn|bessel}}

@@ Zeile 57: / Zeile 57: @@
 {{GraueBox|TEXT=
 $\text{Example (B):}$&nbsp; &nbsp;For the spectrum of the analytic signal, the following applies to ''Phase Modulation of a sinusoidal signal'':
-[[Datei:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
+[[Datei:Mod_T_3_1_S4_version2.png|right|frame|Phase Modulation: &nbsp; Spectrum of the Analytic Signal]]
 :$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
 Denote this
-*$f_{\rm T}$ the carrier frequency,
+*$f_{\rm T}$ the carrier frequency &nbsp; (German: &nbsp; $\rm T$rägerfrequenz),
-*$f_{\rm N}$ the message frequency,
+*$f_{\rm N}$ the frequency of the source signal &nbsp; (German: &nbsp; $\rm N$achrichtenfrequenz),
-* $A_{\rm T}$ the carrier amplitude.
+* $A_{\rm T}$ the carrier amplitude &nbsp; (German: &nbsp; $\rm T$rägeramplitude).
@@ Zeile 68: / Zeile 68: @@
 Based on the graphic, the following statements are possible:
-*$S_+(f)$ consists here of infinitely many discrete lines at a distance of $f_{\rm N}$.
+*$S_+(f)$ consists here of an infinite number of discrete lines at a distance of $f_{\rm N}$.
-*It is thus in principle infinitely extended.
+*Thus the sectrum is in principle infinitely extended.
 *The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
 *The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetrical about $f_{\rm T}$ for even $n$.
 *In the case of odd $n$, a change of sign corresponding to $\text{Property (B)}$ must be taken into account.
-*The phase modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
+*The Phase Modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
 <br><br>
@@ Zeile 79: / Zeile 79: @@
 The applications of the Bessel functions in nature and engineering are diverse and play an important role in physics, for example:
-* Investigation of natural vibrations of cylindrical resonators,
+* Electromagnetic waves in a cylindrical waveguide,
-* Solution of the radial Schrödinger-equation,
+* Solutions to the radial Schrödinger equation,
-* Sound pressure amplitudes of low-viscosity rotational flows,
+* Pressure amplitudes of inviscid rotational flows,
-* Heat conduction in cylindrical bodies,
+* Heat conduction in a cylindrical object,
-* Scattering problem of a grid,
+* Diffusion problems on a lattice,
-* Dynamics of oscillating bodies,
+* Dynamics of floating bodies,
 * Angular resolution.
 The Bessel functions are counted among the special functions because of their many applications in mathematical physics.
 In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.
 {{GraueBox|TEXT=

@@ Zeile 30: / Zeile 30: @@
 The functions  ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German: &nbsp; ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.
-''Annotation:'' &nbsp; There are a number of modifications of the Bessel functions, including the second kind Bessel functions named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$&ndash;functions. However, in this applet, only the first kind Bessel functions are &nbsp; &rArr; &nbsp; ${\rm J}_n (x)$ is considered.
+''Annotation:'' &nbsp; There are a number of modifications of the Bessel functions, including the Bessel functions of the second kind named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$ functions. However, in this applet, only the first kind Bessel functions ${\rm J}_n (x)$ are considered.
 <br><br>
@@ Zeile 56: / Zeile 56: @@
 {{GraueBox|TEXT=
-$\text{Example (B):}$&nbsp; &nbsp;For the spectrum of the analytic signal, the following applies to [[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals|phase modulation of a sinusoidal signal]]:
+$\text{Example (B):}$&nbsp; &nbsp;For the spectrum of the analytic signal, the following applies to ''Phase Modulation of a sinusoidal signal'':
 [[Datei:Mod_T_3_1_S4_version2.png|right|frame|Spectrum of the analytic signal in phase modulation]]
 :$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$

@@ Zeile 23: / Zeile 23: @@
 \cdot {\rm J}_n (x)= 0. $$
-This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://de.wikipedia.org/wiki/Friedrich_Wilhelm_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:
+This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://en.wikipedia.org/wiki/Friedrich_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:
 :$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha

@@ Zeile 6: / Zeile 6: @@
 :$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2
-\hspace{0.02cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$
+\hspace{0.05cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$
 *The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.
 *The left output provides the function values ${\rm J}_0 (x = x_1)$, ... , ${\rm J}_9 (x = x_1)$ for a slider-settable value $x_1$ in the range $0 \le x_1 \le 15$ with increment $0.5$.
-*The right output provides the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value) Increment like on the left).
+*The right output provides the function values ${\rm J}_0 (x = x_2)$, ... , ${\rm J}_9 (x = x_2)$ for a slider-settable value $x_2$ (same range and value increment like on the left).