. . . . . . . . . "\u4EA5\u59C6\u970D\u5179\u65B9\u7A0B\uFF08\u82F1\u8A9E\uFF1AHelmholtz equation\uFF09\u662F\u4E00\u500B\u63CF\u8FF0\u7535\u78C1\u6CE2\u7684\u692D\u5706\u504F\u5FAE\u5206\u65B9\u7A0B\uFF0C\u4EE5\u5FB7\u56FD\u7269\u7406\u5B66\u5BB6\u8D6B\u5C14\u66FC\u00B7\u51AF\u00B7\u4EA5\u59C6\u970D\u5179\u7684\u540D\u5B57\u547D\u540D\u3002\u5176\u57FA\u672C\u5F62\u5F0F\u5982\u4E0B\uFF1A \u5176\u4E2D \u22072 \u662F\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\uFF0Ck \u662F\u6CE2\u6578\uFF0CA \u662F\u632F\u5E45\u3002"@zh . "Die Helmholtz-Gleichung (nach Hermann von Helmholtz) ist eine partielle Differentialgleichung. Sie lautet: in einem Gebiet mit vorgegebenen Randbedingungen auf dem Rand . Dabei ist der Laplace-Operator, die L\u00F6sungsfunktion (Eigenfunktion) und der Eigenwert. Die Gleichung ist ein kontinuierliches Analogon zum diskreten Eigenwertproblem. In der Regel wird die Gleichung von unendlich vielen Eigenwerten und zugeh\u00F6rigen Eigenfunktionen gel\u00F6st. Im Spezialfall kartesischer Koordinaten mit dem Index und der Anzahl der (r\u00E4umlichen) Dimensionen besitzt der Laplace-Operator die Gestalt . Die Helmholtz-Gleichung ist eine homogene partielle Differentialgleichung (PDGL) zweiter Ordnung aus der Klasse der elliptischen PDGL. Sie ergibt sich auch z. B. aus der Wellengleichung nach Trennung der Variablen und Annahme harmonischer Zeitabh\u00E4ngigkeit. Im eindimensionalen Fall ist die Gleichung vom Typ einer gew\u00F6hnlichen Differentialgleichung. In Fall reduziert sich die Gleichung zur Laplace-Gleichung. Wird die rechte Seite der Gleichung durch eine Funktion ersetzt, so wird die resultierende Gleichung, eine Poisson-Gleichung, inhomogen."@de . "\u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632 \u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u062C\u0632\u0626\u064A\u0629 \u0645\u0646 \u0627\u0644\u062F\u0631\u062C\u0629 \u0627\u0644\u062B\u0627\u0646\u064A\u0629 \u0648\u0633\u0645\u064A\u062A \u0628\u0647\u0630\u0627 \u0627\u0644\u0627\u0633\u0645 \u062A\u064A\u0645\u0646\u0627 \u0628\u0627\u0644\u0639\u0627\u0644\u0645 \u0627\u0644\u0623\u0644\u0645\u0627\u0646\u064A \u0647\u0631\u0645\u0627\u0646 \u0641\u0648\u0646 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632. \u0648\u0644\u0647\u0627 \u062A\u0637\u0628\u064A\u0642\u0627\u062A \u0641\u064A\u0632\u064A\u0627\u0626\u064A\u0629 \u0639\u062F\u064A\u062F\u0629 \u0648\u0647\u064A \u0645\u0639\u0627\u062F\u0644\u0629 \u0645\u0623\u0644\u0648\u0641\u0629 \u0639\u0646\u062F \u0627\u0644\u0628\u062D\u062B \u0639\u0646 \u062D\u0644\u0648\u0644 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0627\u062A \u0627\u0644\u0645\u0648\u062C\u064A\u0629 \u0641\u064A \u0627\u0644\u0643\u0647\u0631\u0648\u0645\u063A\u0646\u0627\u0637\u064A\u0633\u064A\u0629 \u0648\u0643\u0630\u0644\u0643 \u0641\u064A \u062C\u0647\u062F \u064A\u0648\u0643\u0627\u0648\u0627. \u0648\u0639\u0646\u062F \u062A\u0637\u0628\u064A\u0642 \u062A\u0646\u062A\u062C \u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632 \u062F\u0627\u0626\u0645\u0627 \u062D\u0644\u0627 \u0648\u062D\u064A\u062F\u0627\u060C \u0648\u062C\u062F\u062A \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0639\u0646 \u0637\u0631\u064A\u0642 \u0641\u0635\u0644 \u0627\u0644\u0645\u062A\u063A\u064A\u0631\u0627\u062A \u0648\u064A\u0633\u062A\u0639\u0645\u0644 \u0641\u064A \u062D\u0644\u0647\u0627 \u0648\u0633\u064A\u0644\u0629 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: BEM)\u200F. \u0648\u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0639\u0644\u0649 \u0647\u0630\u0627 \u0627\u0644\u0646\u062D\u0648. \u062D\u064A\u062B \u0647\u0648 \u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633 (\u0644\u0627\u0628\u0644\u0627\u0633\u064A\u0627\u0646) \u0648 \u0631\u0642\u0645 \u0627\u0644\u0645\u0648\u062C\u0629 \u0648 \u0647\u064A \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0645\u0648\u062C\u064A\u0629. \u0648\u062A\u0639\u062F \u0645\u0639\u0627\u062F\u0644\u0629 \u0644\u0627\u0628\u0644\u0627\u0633 \u062D\u0627\u0644\u0629 \u062E\u0627\u0635\u0629 \u0645\u0646 \u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632. \u062D\u064A\u062B \u0623\u0646 \u0645\u0639\u0627\u062F\u0644\u0629 \u0644\u0627\u0628\u0644\u0627\u0633 \u0647\u064A \u0630\u0627\u062A\u0647\u0627 \u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632 \u0639\u0646\u062F\u0645\u0627 \u062A\u0643\u0648\u0646"@ar . . "16617"^^ . "\u30D8\u30EB\u30E0\u30DB\u30EB\u30C4\u65B9\u7A0B\u5F0F\uFF08\u30D8\u30EB\u30E0\u30DB\u30EB\u30C4\u307B\u3046\u3066\u3044\u3057\u304D\u3001\u82F1: Helmholtz equation\uFF09\u306F\u3001\u30D8\u30EB\u30DE\u30F3\u30FB\u30D5\u30A9\u30F3\u30FB\u30D8\u30EB\u30E0\u30DB\u30EB\u30C4\u306E\u540D\u306B\u3061\u306A\u3080\u65B9\u7A0B\u5F0F\u3067\u3001 \u3068\u3044\u3046\u6955\u5186\u578B\u306E\u504F\u5FAE\u5206\u65B9\u7A0B\u5F0F\u3067\u3042\u308B\u3002\u3053\u3053\u3067\u306F\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3\u3001k \u306F\u5B9A\u6570\u3001A = A (x, y, z) \u306F3\u6B21\u5143\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u7A7A\u9593 R3 \u3067\u5B9A\u7FA9\u3055\u308C\u305F\u672A\u77E5\u95A2\u6570\u3067\u3042\u308B\u3002k = 0 \u306F\u30E9\u30D7\u30E9\u30B9\u65B9\u7A0B\u5F0F\u3067\u3042\u308B\u3002"@ja . "\uC218\uD559\uC5D0\uC11C \uD5EC\uB984\uD640\uCE20 \uBC29\uC815\uC2DD(Helmholtz equation)\uC740 2\uCC28 \uD3B8\uBBF8\uBD84 \uBC29\uC815\uC2DD\uC758 \uD558\uB098\uB2E4. \uBB3C\uB9AC\uD559\uC5D0\uC11C \uC790\uC8FC \uB4F1\uC7A5\uD55C\uB2E4. \uB3C5\uC77C\uC758 \uBB3C\uB9AC\uD559\uC790 \uBC0F \uC0DD\uB9AC\uD559\uC790 \uD5E4\uB974\uB9CC \uD3F0 \uD5EC\uB984\uD640\uCE20\uC758 \uC774\uB984\uC744 \uB544\uB2E4."@ko . . "Helmholtz ekvation"@sv . . . . . "\u0420\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0413\u0435\u043B\u044C\u043C\u0433\u043E\u043B\u044C\u0446\u0430 - \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0437 \u0447\u0430\u0441\u0442\u0438\u043D\u043D\u0438\u043C\u0438 \u043F\u043E\u0445\u0456\u0434\u043D\u0438\u043C\u0438 \u0435\u043B\u0456\u043F\u0442\u0438\u0447\u043D\u043E\u0433\u043E \u0442\u0438\u043F\u0443, \u0449\u043E \u043C\u0430\u0454 \u0432\u0438\u0433\u043B\u044F\u0434: , \u0434\u0435 - \u043D\u0435\u0432\u0456\u0434\u043E\u043C\u0430 \u0444\u0443\u043D\u043A\u0446\u0456\u044F, - \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430, k - \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440."@uk . . . "La equaci\u00F3 de Helmholtz , anomenada aix\u00ED per Hermann von Helmholtz ve donada per: on \u00E9s el laplaci\u00E0, \u00E9s una constant (nombre d'ona), i un camp escalar, \u00E9s aquest cas, el camp magn\u00E8tic i el\u00E8ctric."@ca . . . . . "p/h046920"@en . . . . "\u0423\u0440\u0430\u0432\u043D\u0435\u0301\u043D\u0438\u0435 \u0413\u0435\u043B\u044C\u043C\u0433\u043E\u0301\u043B\u044C\u0446\u0430 \u2014 \u044D\u0442\u043E \u044D\u043B\u043B\u0438\u043F\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0432 \u0447\u0430\u0441\u0442\u043D\u044B\u0445 \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u044B\u0445: \u0433\u0434\u0435 \u2014 \u044D\u0442\u043E \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430, \u0430 \u043D\u0435\u0438\u0437\u0432\u0435\u0441\u0442\u043D\u0430\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u044F \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0430 \u0432 (\u043D\u0430 \u043F\u0440\u0430\u043A\u0442\u0438\u043A\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0413\u0435\u043B\u044C\u043C\u0433\u043E\u043B\u044C\u0446\u0430 \u043F\u0440\u0438\u043C\u0435\u043D\u044F\u0435\u0442\u0441\u044F \u0434\u043B\u044F )."@ru . . . . . . . . . . . . . "\u4EA5\u59C6\u970D\u5179\u65B9\u7A0B\uFF08\u82F1\u8A9E\uFF1AHelmholtz equation\uFF09\u662F\u4E00\u500B\u63CF\u8FF0\u7535\u78C1\u6CE2\u7684\u692D\u5706\u504F\u5FAE\u5206\u65B9\u7A0B\uFF0C\u4EE5\u5FB7\u56FD\u7269\u7406\u5B66\u5BB6\u8D6B\u5C14\u66FC\u00B7\u51AF\u00B7\u4EA5\u59C6\u970D\u5179\u7684\u540D\u5B57\u547D\u540D\u3002\u5176\u57FA\u672C\u5F62\u5F0F\u5982\u4E0B\uFF1A \u5176\u4E2D \u22072 \u662F\u62C9\u666E\u62C9\u65AF\u7B97\u5B50\uFF0Ck \u662F\u6CE2\u6578\uFF0CA \u662F\u632F\u5E45\u3002"@zh . "\u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632 \u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u062C\u0632\u0626\u064A\u0629 \u0645\u0646 \u0627\u0644\u062F\u0631\u062C\u0629 \u0627\u0644\u062B\u0627\u0646\u064A\u0629 \u0648\u0633\u0645\u064A\u062A \u0628\u0647\u0630\u0627 \u0627\u0644\u0627\u0633\u0645 \u062A\u064A\u0645\u0646\u0627 \u0628\u0627\u0644\u0639\u0627\u0644\u0645 \u0627\u0644\u0623\u0644\u0645\u0627\u0646\u064A \u0647\u0631\u0645\u0627\u0646 \u0641\u0648\u0646 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632. \u0648\u0644\u0647\u0627 \u062A\u0637\u0628\u064A\u0642\u0627\u062A \u0641\u064A\u0632\u064A\u0627\u0626\u064A\u0629 \u0639\u062F\u064A\u062F\u0629 \u0648\u0647\u064A \u0645\u0639\u0627\u062F\u0644\u0629 \u0645\u0623\u0644\u0648\u0641\u0629 \u0639\u0646\u062F \u0627\u0644\u0628\u062D\u062B \u0639\u0646 \u062D\u0644\u0648\u0644 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0627\u062A \u0627\u0644\u0645\u0648\u062C\u064A\u0629 \u0641\u064A \u0627\u0644\u0643\u0647\u0631\u0648\u0645\u063A\u0646\u0627\u0637\u064A\u0633\u064A\u0629 \u0648\u0643\u0630\u0644\u0643 \u0641\u064A \u062C\u0647\u062F \u064A\u0648\u0643\u0627\u0648\u0627. \u0648\u0639\u0646\u062F \u062A\u0637\u0628\u064A\u0642 \u062A\u0646\u062A\u062C \u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632 \u062F\u0627\u0626\u0645\u0627 \u062D\u0644\u0627 \u0648\u062D\u064A\u062F\u0627\u060C \u0648\u062C\u062F\u062A \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0639\u0646 \u0637\u0631\u064A\u0642 \u0641\u0635\u0644 \u0627\u0644\u0645\u062A\u063A\u064A\u0631\u0627\u062A \u0648\u064A\u0633\u062A\u0639\u0645\u0644 \u0641\u064A \u062D\u0644\u0647\u0627 \u0648\u0633\u064A\u0644\u0629 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: BEM)\u200F. \u0648\u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0639\u0644\u0649 \u0647\u0630\u0627 \u0627\u0644\u0646\u062D\u0648. \u062D\u064A\u062B \u0647\u0648 \u0645\u0624\u062B\u0631 \u0644\u0627\u0628\u0644\u0627\u0633 (\u0644\u0627\u0628\u0644\u0627\u0633\u064A\u0627\u0646) \u0648 \u0631\u0642\u0645 \u0627\u0644\u0645\u0648\u062C\u0629 \u0648 \u0647\u064A \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0645\u0648\u062C\u064A\u0629. \u0648\u062A\u0639\u062F \u0645\u0639\u0627\u062F\u0644\u0629 \u0644\u0627\u0628\u0644\u0627\u0633 \u062D\u0627\u0644\u0629 \u062E\u0627\u0635\u0629 \u0645\u0646 \u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632. \u062D\u064A\u062B \u0623\u0646 \u0645\u0639\u0627\u062F\u0644\u0629 \u0644\u0627\u0628\u0644\u0627\u0633 \u0647\u064A \u0630\u0627\u062A\u0647\u0627 \u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632 \u0639\u0646\u062F\u0645\u0627 \u062A\u0643\u0648\u0646"@ar . . "\u0423\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0413\u0435\u043B\u044C\u043C\u0433\u043E\u043B\u044C\u0446\u0430"@ru . . . . . . . "\u0420\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0413\u0435\u043B\u044C\u043C\u0433\u043E\u043B\u044C\u0446\u0430"@uk . "Ecuaci\u00F3n de Helmholtz"@es . "1117741633"^^ . . . "\u4EA5\u59C6\u970D\u5179\u65B9\u7A0B"@zh . . . . . "Equa\u00E7\u00E3o de Helmholtz"@pt . . "1156215"^^ . "L'\u00E9quation de Helmholtz (d'apr\u00E8s le physicien Hermann von Helmholtz) est une \u00E9quation aux d\u00E9riv\u00E9es partielles elliptique qui appara\u00EEt lorsque l'on cherche des solutions harmoniques de l'\u00E9quation de propagation des ondes de D'Alembert, appel\u00E9es \u00AB modes propres \u00BB, sur un domaine : Pour que le probl\u00E8me math\u00E9matique soit bien pos\u00E9, il faut sp\u00E9cifier une condition aux limites sur le bord du domaine, par exemple : \n* une condition de Dirichlet, \n* une condition de Neumann, \n* un m\u00E9lange des deux pr\u00E9c\u00E9dentes etc. Lorsque le domaine est compact, le spectre du Laplacien est discret, et les modes propres forment un ensemble d\u00E9nombrable infini : L'\u00E9quation de Helmholtz se g\u00E9n\u00E9ralise en g\u00E9om\u00E9trie non euclidienne en rempla\u00E7ant le Laplacien par l'op\u00E9rateur de Laplace-Beltrami sur une vari\u00E9t\u00E9 riemannienne."@fr . . . . . . . "In analisi matematica, l'equazione agli autovalori del laplaciano si chiama equazione di Helmholtz. Si tratta di un'equazione differenziale alle derivate parziali ellittica del secondo ordine a cui si pu\u00F2 ricondurre in alcuni casi per esempio l'equazione delle onde: in questo caso permette di ricavare rapidamente la relazione di dispersione. Altri casi notevoli in cui l'equazione agli autovalori del laplaciano \u00E8 uno strumento utile sono l'equazione della diffusione e le equazioni ellittiche del secondo ordine. Anche la teoria della trave elastica, e in particolare i problemi di carico di punta secondo Eulero sono riconducibili a casi pratici dell'equazione di Helmholtz."@it . . . . . . . . "\u0423\u0440\u0430\u0432\u043D\u0435\u0301\u043D\u0438\u0435 \u0413\u0435\u043B\u044C\u043C\u0433\u043E\u0301\u043B\u044C\u0446\u0430 \u2014 \u044D\u0442\u043E \u044D\u043B\u043B\u0438\u043F\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0432 \u0447\u0430\u0441\u0442\u043D\u044B\u0445 \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u0434\u043D\u044B\u0445: \u0433\u0434\u0435 \u2014 \u044D\u0442\u043E \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430, \u0430 \u043D\u0435\u0438\u0437\u0432\u0435\u0441\u0442\u043D\u0430\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u044F \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0430 \u0432 (\u043D\u0430 \u043F\u0440\u0430\u043A\u0442\u0438\u043A\u0435 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435 \u0413\u0435\u043B\u044C\u043C\u0433\u043E\u043B\u044C\u0446\u0430 \u043F\u0440\u0438\u043C\u0435\u043D\u044F\u0435\u0442\u0441\u044F \u0434\u043B\u044F )."@ru . . "Helmholtz equation"@en . . . . . . . . . "\u0645\u0639\u0627\u062F\u0644\u0629 \u0647\u0644\u0645\u0647\u0648\u0644\u062A\u0632"@ar . "In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation where \u22072 is the Laplace operator (or \"Laplacian\"), k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences."@en . . . . . . . . . . . . "Helmholtz ekvation \u00E4r en partiell differentialekvation som lyder d\u00E4r \u22072 \u00E4r laplaceoperatorn, k \u00E4r ett v\u00E5gtal och A \u00E4r en amplitud. Ekvationen \u00E4r tidsinvariant, och anv\u00E4nds bland annat f\u00F6r att l\u00F6sa v\u00E5gekvationen. Denna artikel om matematisk analys saknar v\u00E4sentlig information. Du kan hj\u00E4lpa till genom att l\u00E4gga till den."@sv . . . . "Die Helmholtz-Gleichung (nach Hermann von Helmholtz) ist eine partielle Differentialgleichung. Sie lautet: in einem Gebiet mit vorgegebenen Randbedingungen auf dem Rand . Dabei ist der Laplace-Operator, die L\u00F6sungsfunktion (Eigenfunktion) und der Eigenwert. Die Gleichung ist ein kontinuierliches Analogon zum diskreten Eigenwertproblem. In der Regel wird die Gleichung von unendlich vielen Eigenwerten und zugeh\u00F6rigen Eigenfunktionen gel\u00F6st. Im Spezialfall kartesischer Koordinaten mit dem Index und der Anzahl der (r\u00E4umlichen) Dimensionen besitzt der Laplace-Operator die Gestalt ."@de . . "L'\u00E9quation de Helmholtz (d'apr\u00E8s le physicien Hermann von Helmholtz) est une \u00E9quation aux d\u00E9riv\u00E9es partielles elliptique qui appara\u00EEt lorsque l'on cherche des solutions harmoniques de l'\u00E9quation de propagation des ondes de D'Alembert, appel\u00E9es \u00AB modes propres \u00BB, sur un domaine : Pour que le probl\u00E8me math\u00E9matique soit bien pos\u00E9, il faut sp\u00E9cifier une condition aux limites sur le bord du domaine, par exemple : \n* une condition de Dirichlet, \n* une condition de Neumann, \n* un m\u00E9lange des deux pr\u00E9c\u00E9dentes etc."@fr . . "Helmholtz-Gleichung"@de . . "A equa\u00E7\u00E3o de Helmholtz \u00E9 um tipo de equa\u00E7\u00E3o diferencial parcial que \u00E9 expressa da seguinte forma: onde \u22072 \u00E9 o Laplaciano, k \u00E9 o n\u00FAmero de onda, e A \u00E9 a amplitude. A equa\u00E7\u00E3o, que recebeu o nome de Hermann von Helmholtz, surge em v\u00E1rios dom\u00EDnios da f\u00EDsica e engenharia, tipicamente para descrever fen\u00F3menos f\u00EDsicos que s\u00E3o dependentes do tempo. Ela corresponde a um caso geral da Equa\u00E7\u00E3o de Laplace."@pt . . . . "\u30D8\u30EB\u30E0\u30DB\u30EB\u30C4\u65B9\u7A0B\u5F0F\uFF08\u30D8\u30EB\u30E0\u30DB\u30EB\u30C4\u307B\u3046\u3066\u3044\u3057\u304D\u3001\u82F1: Helmholtz equation\uFF09\u306F\u3001\u30D8\u30EB\u30DE\u30F3\u30FB\u30D5\u30A9\u30F3\u30FB\u30D8\u30EB\u30E0\u30DB\u30EB\u30C4\u306E\u540D\u306B\u3061\u306A\u3080\u65B9\u7A0B\u5F0F\u3067\u3001 \u3068\u3044\u3046\u6955\u5186\u578B\u306E\u504F\u5FAE\u5206\u65B9\u7A0B\u5F0F\u3067\u3042\u308B\u3002\u3053\u3053\u3067\u306F\u30E9\u30D7\u30E9\u30B7\u30A2\u30F3\u3001k \u306F\u5B9A\u6570\u3001A = A (x, y, z) \u306F3\u6B21\u5143\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u7A7A\u9593 R3 \u3067\u5B9A\u7FA9\u3055\u308C\u305F\u672A\u77E5\u95A2\u6570\u3067\u3042\u308B\u3002k = 0 \u306F\u30E9\u30D7\u30E9\u30B9\u65B9\u7A0B\u5F0F\u3067\u3042\u308B\u3002"@ja . . "Equaci\u00F3 de Helmholtz"@ca . . . . . . . . "La ecuaci\u00F3n de Helmholtz, nombrada as\u00ED por Hermann von Helmholtz, viene dada por: La ecuaci\u00F3n aparece en varios contextos de la f\u00EDsica donde se interpreta como el n\u00FAmero de onda. As\u00ED tambi\u00E9n, dicha ecuaci\u00F3n es com\u00FAnmente encontrada en problemas de electromagnetismo, en la teor\u00EDa del potencial de Yukawa y, como caso particular, en la ecuaci\u00F3n de Klein-Gordon para un campo estacionario."@es . . . . . . . . . . "Equazione di Helmholtz"@it . "Helmholtz equation"@en . . . . . "\u00C9quation de Helmholtz"@fr . . . "Helmholtz ekvation \u00E4r en partiell differentialekvation som lyder d\u00E4r \u22072 \u00E4r laplaceoperatorn, k \u00E4r ett v\u00E5gtal och A \u00E4r en amplitud. Ekvationen \u00E4r tidsinvariant, och anv\u00E4nds bland annat f\u00F6r att l\u00F6sa v\u00E5gekvationen. Denna artikel om matematisk analys saknar v\u00E4sentlig information. Du kan hj\u00E4lpa till genom att l\u00E4gga till den."@sv . . . "In analisi matematica, l'equazione agli autovalori del laplaciano si chiama equazione di Helmholtz. Si tratta di un'equazione differenziale alle derivate parziali ellittica del secondo ordine a cui si pu\u00F2 ricondurre in alcuni casi per esempio l'equazione delle onde: in questo caso permette di ricavare rapidamente la relazione di dispersione. Altri casi notevoli in cui l'equazione agli autovalori del laplaciano \u00E8 uno strumento utile sono l'equazione della diffusione e le equazioni ellittiche del secondo ordine. Anche la teoria della trave elastica, e in particolare i problemi di carico di punta secondo Eulero sono riconducibili a casi pratici dell'equazione di Helmholtz. Molte funzioni speciali sono ottenute cercando soluzioni dell'equazione di Helmholtz con il metodo di separazione delle variabili in coordinate curvilinee. Alcuni esempi sono le armoniche cilindriche, le funzioni paraboliche del cilindro e le armoniche sferiche. dimostr\u00F2 nel 1934 che esistono solamente undici sistemi di coordinate curvilinee che permettono di trovare soluzioni dell'equazione di Helmholtz con il metodo di separazione delle variabili."@it . . "A equa\u00E7\u00E3o de Helmholtz \u00E9 um tipo de equa\u00E7\u00E3o diferencial parcial que \u00E9 expressa da seguinte forma: onde \u22072 \u00E9 o Laplaciano, k \u00E9 o n\u00FAmero de onda, e A \u00E9 a amplitude. A equa\u00E7\u00E3o, que recebeu o nome de Hermann von Helmholtz, surge em v\u00E1rios dom\u00EDnios da f\u00EDsica e engenharia, tipicamente para descrever fen\u00F3menos f\u00EDsicos que s\u00E3o dependentes do tempo. Ela corresponde a um caso geral da Equa\u00E7\u00E3o de Laplace."@pt . . . . . "\u0420\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0413\u0435\u043B\u044C\u043C\u0433\u043E\u043B\u044C\u0446\u0430 - \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0437 \u0447\u0430\u0441\u0442\u0438\u043D\u043D\u0438\u043C\u0438 \u043F\u043E\u0445\u0456\u0434\u043D\u0438\u043C\u0438 \u0435\u043B\u0456\u043F\u0442\u0438\u0447\u043D\u043E\u0433\u043E \u0442\u0438\u043F\u0443, \u0449\u043E \u043C\u0430\u0454 \u0432\u0438\u0433\u043B\u044F\u0434: , \u0434\u0435 - \u043D\u0435\u0432\u0456\u0434\u043E\u043C\u0430 \u0444\u0443\u043D\u043A\u0446\u0456\u044F, - \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u041B\u0430\u043F\u043B\u0430\u0441\u0430, k - \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440."@uk . "La equaci\u00F3 de Helmholtz , anomenada aix\u00ED per Hermann von Helmholtz ve donada per: on \u00E9s el laplaci\u00E0, \u00E9s una constant (nombre d'ona), i un camp escalar, \u00E9s aquest cas, el camp magn\u00E8tic i el\u00E8ctric."@ca . "In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation where \u22072 is the Laplace operator (or \"Laplacian\"), k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences."@en . "\u30D8\u30EB\u30E0\u30DB\u30EB\u30C4\u65B9\u7A0B\u5F0F"@ja . . . . . "La ecuaci\u00F3n de Helmholtz, nombrada as\u00ED por Hermann von Helmholtz, viene dada por: La ecuaci\u00F3n aparece en varios contextos de la f\u00EDsica donde se interpreta como el n\u00FAmero de onda. As\u00ED tambi\u00E9n, dicha ecuaci\u00F3n es com\u00FAnmente encontrada en problemas de electromagnetismo, en la teor\u00EDa del potencial de Yukawa y, como caso particular, en la ecuaci\u00F3n de Klein-Gordon para un campo estacionario."@es . "\uD5EC\uB984\uD640\uCE20 \uBC29\uC815\uC2DD"@ko . . . . . "\uC218\uD559\uC5D0\uC11C \uD5EC\uB984\uD640\uCE20 \uBC29\uC815\uC2DD(Helmholtz equation)\uC740 2\uCC28 \uD3B8\uBBF8\uBD84 \uBC29\uC815\uC2DD\uC758 \uD558\uB098\uB2E4. \uBB3C\uB9AC\uD559\uC5D0\uC11C \uC790\uC8FC \uB4F1\uC7A5\uD55C\uB2E4. \uB3C5\uC77C\uC758 \uBB3C\uB9AC\uD559\uC790 \uBC0F \uC0DD\uB9AC\uD559\uC790 \uD5E4\uB974\uB9CC \uD3F0 \uD5EC\uB984\uD640\uCE20\uC758 \uC774\uB984\uC744 \uB544\uB2E4."@ko .