2445:
233:
2732:
1266:
1048:
31:
2207:
100 Hz, exactly one octave above the alternating current frequency in Europe and most countries in Africa and Asia, 50 Hz. In most countries of the
Americasâwhere the AC frequency is 60 Hzâaltering A# on the fifth string, first fret from 116.54 Hz to 120 Hz produces a similar effect.
2206:
or, perhaps more easily, by adjusting the machine heads, to obtain the same, or a multiple, of the AC frequency to achieve the same effect. For example, in the case of a guitar, the 6th (lowest pitched) string pressed to the third fret gives a G at 97.999 Hz. A slight adjustment can alter it to
2187:. (If the refresh rate of the screen equals the frequency of the string or an integer multiple thereof, the string will appear still but deformed.) In daylight and other non-oscillating light sources, this effect does not occur and the string appears still but thicker, and lighter or blurred, due to
726:
1099:
817:
1416:
549:
1261:{\displaystyle {\frac {1}{\Delta x}}\left(\left.{\frac {\partial y}{\partial x}}\right|^{x+\Delta x}-\left.{\frac {\partial y}{\partial x}}\right|^{x}\right)={\frac {\mu }{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}
1868:
1043:{\displaystyle -{\frac {T_{2}\sin(\beta )}{T_{2}\cos(\beta )}}+{\frac {T_{1}\sin(\alpha )}{T_{1}\cos(\alpha )}}=-\tan(\beta )+\tan(\alpha )={\frac {\mu \Delta x}{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}
2141:
1742:
1053:
According to the small-angle approximation, the tangents of the angles at the ends of the string piece are equal to the slopes at the ends, with an additional minus sign due to the definition of
450:
518:
1320:
1529:
216:
1585:
is not a good approximation for the length of the string piece, the horizontal component of tension is not necessarily constant. The horizontal tensions are not well approximated by
2058:
1488:
2005:
382:, for which the net horizontal force is zero. Accordingly, using the small angle approximation, the horizontal tensions acting on both sides of the string segment are given by
1646:
1583:
1292:
264:
1071:
809:
336:
1454:
1091:
789:
356:
1669:
2081:
1915:
312:
176:
1943:
1891:
1793:
1765:
1693:
1603:
1552:
1312:
769:
749:
541:
380:
288:
156:
132:
523:
From Newton's second law for the vertical component, the mass (which is the product of its linear density and length) of this piece times its acceleration,
2799:
721:{\displaystyle \Sigma F_{y}=T_{1y}-T_{2y}=-T_{2}\sin(\beta )+T_{1}\sin(\alpha )=\Delta ma\approx \mu \Delta x{\frac {\partial ^{2}y}{\partial t^{2}}}.}
1805:
2949:
2768:
2089:
3142:
2654:
1702:
1562:
for more about this). However, this derivation is only valid for small amplitude vibrations; for those of large amplitude,
2381:
388:
456:
2353:
2924:
2226:
2202:
to the frequency of vibration of the string. In a dark room, this clearly shows the waveform. Otherwise, one can use
2260:
Molteno, T. C. A.; N. B. Tufillaro (September 2004). "An experimental investigation into the dynamics of a string".
1411:{\displaystyle {\frac {\partial ^{2}y}{\partial x^{2}}}={\frac {\mu }{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}
751:
and substituting the first and second equations obtains (we can choose either the first or the second equation for
17:
2175:, and the rate at which the string seems to vibrate is the difference between the frequency of the string and the
3132:
1496:
232:
183:
2761:
67:
2013:
1459:
2795:
2493:
2330:
1966:
2903:
2614:
2155:
on a vibrating string if the frequency is low enough and the vibrating string is held in front of a
2754:
2669:
2914:
2862:
2549:
2406:
2374:
2247:
47:
2183:, at a rate that is the difference between the frequency of the string and the frequency of the
2880:
2488:
2398:
1631:
1565:
1274:
246:
2887:
2858:
2629:
2527:
2431:
2231:
2188:
1768:
1056:
794:
321:
39:
1424:
1076:
774:
341:
2298:
2269:
1654:
2746:
2066:
1900:
297:
161:
8:
3106:
2929:
2816:
2649:
2559:
2184:
2172:
1894:
359:
2345:
2302:
2273:
86:. If the length or tension of the string is correctly adjusted, the sound produced is a
2892:
2735:
2581:
2571:
2532:
2367:
1928:
1876:
1796:
1778:
1750:
1678:
1588:
1537:
1297:
754:
734:
526:
365:
273:
141:
117:
2444:
2875:
2786:
2705:
2537:
2221:
91:
2994:
2919:
2715:
2634:
2598:
2566:
2517:
2306:
2277:
2242:
2199:
2180:
221:
1294:
approaches zero, the left hand side is the definition of the second derivative of
2944:
2870:
2710:
2664:
2505:
2463:
59:
3137:
2982:
2909:
2833:
2778:
2659:
2644:
2591:
1918:
1772:
315:
3126:
3091:
3029:
2934:
2838:
2624:
2586:
2554:
2509:
2471:
2421:
2416:
2349:
2203:
1559:
83:
35:
30:
3111:
2619:
2176:
87:
2007:, then we easily get an expression for the frequency of the nth harmonic:
1795:
is half of the wavelength of the fundamental harmonic. Hence one obtains
3086:
3044:
2999:
2964:
2848:
2828:
2811:
2639:
2451:
2195:
1649:
135:
3024:
3019:
3014:
2843:
2821:
2782:
2700:
2690:
2476:
2236:
2160:
2156:
1626:
158:) and inversely proportional to the square root of the linear density (
2281:
1963:
Moreover, if we take the nth harmonic as having a wavelength given by
3066:
3004:
2959:
2897:
2481:
2411:
2390:
2152:
1672:
1614:
1456:, and the coefficient of the second time derivative term is equal to
79:
55:
2310:
2289:
Tufillaro, N. B. (1989). "Nonlinear and chaotic string vibrations".
1863:{\displaystyle f={\frac {v}{2L}}={1 \over 2L}{\sqrt {T \over \mu }}}
3081:
3034:
2977:
2866:
2576:
2544:
2164:
1959:
the lighter the string, the higher the frequency of the fundamental
1956:
the higher the tension, the higher the frequency of the fundamental
1953:
the shorter the string, the higher the frequency of the fundamental
43:
3101:
3076:
3059:
3054:
2987:
2972:
2522:
1558:
of propagation of the wave in the string (see the article on the
3071:
3049:
3039:
3009:
2194:
A similar but more controllable effect can be obtained using a
1946:
267:
95:
2136:{\displaystyle f_{n}={\frac {n}{2L}}{\sqrt {\frac {T}{\mu }}}}
3096:
2695:
1622:
1618:
1555:
771:, so we conveniently choose each one with the matching angle
103:
99:
75:
2359:
2853:
2776:
2685:
2216:
1922:
1171:
1125:
291:
63:
2063:
And for a string under a tension T with linear density
2259:
2171:
of an analog oscilloscope). This effect is called the
1737:{\displaystyle v={\frac {\lambda }{\tau }}=\lambda f.}
362:
on either side can both be approximated by a constant
2092:
2069:
2016:
1969:
1931:
1903:
1879:
1808:
1781:
1753:
1705:
1681:
1657:
1634:
1591:
1568:
1540:
1499:
1462:
1427:
1323:
1300:
1277:
1102:
1079:
1059:
820:
797:
777:
757:
737:
552:
529:
459:
391:
368:
344:
324:
300:
276:
249:
186:
164:
144:
120:
2239:(harmonic resonance based on equal string divisions)
2198:. This device allows matching the frequency of the
445:{\displaystyle T_{1x}=T_{1}\cos(\alpha )\approx T.}
114:
The velocity of propagation of a wave in a string (
2135:
2075:
2052:
1999:
1937:
1909:
1885:
1862:
1787:
1759:
1736:
1687:
1663:
1640:
1597:
1577:
1546:
1523:
1482:
1448:
1410:
1306:
1286:
1260:
1085:
1065:
1042:
803:
783:
763:
743:
720:
535:
513:{\displaystyle T_{2x}=T_{2}\cos(\beta )\approx T.}
512:
444:
374:
350:
330:
306:
282:
258:
210:
170:
150:
126:
3124:
1949:of the vibrating part of the string. Therefore:
543:, will be equal to the net force on the piece:
2146:
1621:produced by the string can be calculated. The
2762:
2375:
358:are small, then the horizontal components of
1613:Once the speed of propagation is known, the
1093:. Using this fact and rearranging provides
1771:is the one produced by the vibration whose
2769:
2755:
2382:
2368:
2179:of the screen. The same can happen with a
2288:
1625:of propagation of a wave is equal to the
1524:{\displaystyle v={\sqrt {T \over \mu }},}
211:{\displaystyle v={\sqrt {T \over \mu }}.}
1608:
29:
138:of the force of tension of the string (
14:
3125:
2053:{\displaystyle f_{n}={\frac {nv}{2L}}}
2750:
2363:
90:. Vibrating strings are the basis of
220:This relationship was discovered by
1775:are the two ends of the string, so
236:Illustration for a vibrating string
24:
2354:The Wolfram Demonstrations Project
1569:
1483:{\displaystyle {\frac {1}{v^{2}}}}
1389:
1375:
1342:
1328:
1278:
1239:
1225:
1184:
1176:
1158:
1138:
1130:
1109:
1021:
1007:
991:
699:
685:
675:
660:
553:
250:
231:
25:
3154:
2338:
2000:{\displaystyle \lambda _{n}=2L/n}
2731:
2730:
2443:
2331:The wave equation and wave speed
1747:If the length of the string is
3143:String instrument construction
2324:
1443:
1431:
1421:This is the wave equation for
979:
973:
961:
955:
937:
931:
910:
904:
876:
870:
849:
843:
654:
648:
626:
620:
498:
492:
430:
424:
13:
1:
2389:
2253:
2227:Vibrations of a circular drum
227:
731:Dividing this expression by
7:
2291:American Journal of Physics
2262:American Journal of Physics
2210:
2147:Observing string vibrations
10:
3159:
2352:and Mark Robertson-Tessi,
2958:
2904:Music On A Long Thin Wire
2807:
2793:
2726:
2678:
2607:
2504:
2462:
2438:
2397:
134:) is proportional to the
2800:HornbostelâSachs numbers
1641:{\displaystyle \lambda }
1578:{\displaystyle \Delta x}
1287:{\displaystyle \Delta x}
259:{\displaystyle \Delta x}
2407:Architectural acoustics
2248:Reflection phase change
1671:, or multiplied by the
1066:{\displaystyle \alpha }
804:{\displaystyle \alpha }
331:{\displaystyle \alpha }
109:
2881:Long-string instrument
2494:FletcherâMunson curves
2489:Equal-loudness contour
2399:Acoustical engineering
2137:
2077:
2054:
2001:
1939:
1925:per unit length), and
1911:
1887:
1864:
1789:
1761:
1738:
1689:
1665:
1642:
1599:
1579:
1548:
1525:
1484:
1450:
1449:{\displaystyle y(x,t)}
1412:
1308:
1288:
1262:
1087:
1086:{\displaystyle \beta }
1067:
1044:
805:
785:
784:{\displaystyle \beta }
765:
745:
722:
537:
514:
446:
376:
352:
351:{\displaystyle \beta }
332:
308:
284:
270:of a piece of string,
260:
237:
212:
172:
152:
128:
51:
3133:Mechanical vibrations
2630:Hermann von Helmholtz
2528:Fundamental frequency
2432:Sympathetic resonance
2189:persistence of vision
2138:
2078:
2055:
2002:
1940:
1912:
1888:
1865:
1790:
1762:
1739:
1690:
1666:
1664:{\displaystyle \tau }
1643:
1609:Frequency of the wave
1600:
1580:
1549:
1526:
1485:
1451:
1413:
1309:
1289:
1263:
1088:
1068:
1045:
806:
786:
766:
746:
723:
538:
515:
447:
377:
353:
333:
309:
285:
261:
235:
213:
173:
153:
129:
33:
2346:The Vibrating String
2090:
2076:{\displaystyle \mu }
2067:
2014:
1967:
1929:
1910:{\displaystyle \mu }
1901:
1877:
1806:
1779:
1769:fundamental harmonic
1751:
1703:
1679:
1655:
1632:
1589:
1566:
1538:
1497:
1460:
1425:
1321:
1298:
1275:
1100:
1077:
1057:
818:
795:
775:
755:
735:
550:
527:
457:
389:
366:
342:
322:
307:{\displaystyle \mu }
298:
274:
247:
224:in the late 1500s.
184:
171:{\displaystyle \mu }
162:
142:
118:
2650:Werner Meyer-Eppler
2560:Missing fundamental
2303:1989AmJPh..57..408T
2274:2004AmJPh..72.1157M
2217:Fretted instruments
2185:alternating current
2173:stroboscopic effect
2888:Melde's experiment
2533:Frequency spectrum
2232:Melde's experiment
2133:
2073:
2050:
1997:
1935:
1907:
1883:
1860:
1785:
1757:
1734:
1685:
1661:
1638:
1595:
1575:
1544:
1521:
1480:
1446:
1408:
1304:
1284:
1271:In the limit that
1258:
1083:
1063:
1040:
801:
781:
761:
741:
718:
533:
510:
442:
372:
348:
328:
304:
280:
256:
238:
208:
168:
148:
124:
92:string instruments
52:
3120:
3119:
2876:Longitudinal wave
2744:
2743:
2706:Musical acoustics
2538:harmonic spectrum
2282:10.1119/1.1764557
2222:Musical acoustics
2159:such as one of a
2131:
2130:
2119:
2048:
1938:{\displaystyle L}
1886:{\displaystyle T}
1858:
1857:
1846:
1828:
1788:{\displaystyle L}
1760:{\displaystyle L}
1720:
1688:{\displaystyle f}
1598:{\displaystyle T}
1547:{\displaystyle v}
1516:
1515:
1478:
1403:
1369:
1356:
1307:{\displaystyle y}
1253:
1219:
1191:
1145:
1116:
1035:
1001:
941:
880:
764:{\displaystyle T}
744:{\displaystyle T}
713:
536:{\displaystyle a}
375:{\displaystyle T}
283:{\displaystyle m}
203:
202:
178:) of the string:
151:{\displaystyle T}
127:{\displaystyle v}
38:in a string. The
16:(Redirected from
3150:
2940:String vibration
2771:
2764:
2757:
2748:
2747:
2734:
2733:
2635:Carleen Hutchins
2567:Combination tone
2454:
2447:
2427:String vibration
2384:
2377:
2370:
2361:
2360:
2333:
2328:
2314:
2285:
2268:(9): 1157â1169.
2243:String resonance
2200:xenon flash lamp
2181:fluorescent lamp
2151:One can see the
2142:
2140:
2139:
2134:
2132:
2123:
2122:
2120:
2118:
2107:
2102:
2101:
2082:
2080:
2079:
2074:
2059:
2057:
2056:
2051:
2049:
2047:
2039:
2031:
2026:
2025:
2006:
2004:
2003:
1998:
1993:
1979:
1978:
1944:
1942:
1941:
1936:
1916:
1914:
1913:
1908:
1892:
1890:
1889:
1884:
1869:
1867:
1866:
1861:
1859:
1850:
1849:
1847:
1845:
1834:
1829:
1827:
1816:
1794:
1792:
1791:
1786:
1766:
1764:
1763:
1758:
1743:
1741:
1740:
1735:
1721:
1713:
1696:
1694:
1692:
1691:
1686:
1670:
1668:
1667:
1662:
1647:
1645:
1644:
1639:
1604:
1602:
1601:
1596:
1584:
1582:
1581:
1576:
1553:
1551:
1550:
1545:
1530:
1528:
1527:
1522:
1517:
1508:
1507:
1489:
1487:
1486:
1481:
1479:
1477:
1476:
1464:
1455:
1453:
1452:
1447:
1417:
1415:
1414:
1409:
1404:
1402:
1401:
1400:
1387:
1383:
1382:
1372:
1370:
1362:
1357:
1355:
1354:
1353:
1340:
1336:
1335:
1325:
1313:
1311:
1310:
1305:
1293:
1291:
1290:
1285:
1267:
1265:
1264:
1259:
1254:
1252:
1251:
1250:
1237:
1233:
1232:
1222:
1220:
1212:
1207:
1203:
1202:
1201:
1196:
1192:
1190:
1182:
1174:
1165:
1164:
1150:
1146:
1144:
1136:
1128:
1117:
1115:
1104:
1092:
1090:
1089:
1084:
1072:
1070:
1069:
1064:
1049:
1047:
1046:
1041:
1036:
1034:
1033:
1032:
1019:
1015:
1014:
1004:
1002:
997:
986:
942:
940:
924:
923:
913:
897:
896:
886:
881:
879:
863:
862:
852:
836:
835:
825:
810:
808:
807:
802:
790:
788:
787:
782:
770:
768:
767:
762:
750:
748:
747:
742:
727:
725:
724:
719:
714:
712:
711:
710:
697:
693:
692:
682:
641:
640:
613:
612:
597:
596:
581:
580:
565:
564:
542:
540:
539:
534:
519:
517:
516:
511:
485:
484:
472:
471:
451:
449:
448:
443:
417:
416:
404:
403:
381:
379:
378:
373:
357:
355:
354:
349:
337:
335:
334:
329:
313:
311:
310:
305:
289:
287:
286:
281:
265:
263:
262:
257:
222:Vincenzo Galilei
217:
215:
214:
209:
204:
195:
194:
177:
175:
174:
169:
157:
155:
154:
149:
133:
131:
130:
125:
82:, i.e. constant
72:vibrating string
42:and the first 5
21:
18:Vibrating string
3158:
3157:
3153:
3152:
3151:
3149:
3148:
3147:
3123:
3122:
3121:
3116:
3025:Japanese fiddle
2963:
2954:
2945:Transverse wave
2893:Mersenne's laws
2871:String harmonic
2803:
2789:
2775:
2745:
2740:
2722:
2674:
2665:D. Van Holliday
2603:
2572:Mersenne's laws
2506:Audio frequency
2500:
2464:Psychoacoustics
2458:
2457:
2450:
2436:
2393:
2388:
2341:
2336:
2329:
2325:
2311:10.1119/1.16011
2256:
2213:
2149:
2121:
2111:
2106:
2097:
2093:
2091:
2088:
2087:
2068:
2065:
2064:
2040:
2032:
2030:
2021:
2017:
2015:
2012:
2011:
1989:
1974:
1970:
1968:
1965:
1964:
1930:
1927:
1926:
1902:
1899:
1898:
1878:
1875:
1874:
1848:
1838:
1833:
1820:
1815:
1807:
1804:
1803:
1797:Mersenne's laws
1780:
1777:
1776:
1752:
1749:
1748:
1712:
1704:
1701:
1700:
1680:
1677:
1676:
1675:
1656:
1653:
1652:
1648:divided by the
1633:
1630:
1629:
1611:
1590:
1587:
1586:
1567:
1564:
1563:
1539:
1536:
1535:
1506:
1498:
1495:
1494:
1472:
1468:
1463:
1461:
1458:
1457:
1426:
1423:
1422:
1396:
1392:
1388:
1378:
1374:
1373:
1371:
1361:
1349:
1345:
1341:
1331:
1327:
1326:
1324:
1322:
1319:
1318:
1299:
1296:
1295:
1276:
1273:
1272:
1246:
1242:
1238:
1228:
1224:
1223:
1221:
1211:
1197:
1183:
1175:
1173:
1170:
1169:
1151:
1137:
1129:
1127:
1124:
1123:
1122:
1118:
1108:
1103:
1101:
1098:
1097:
1078:
1075:
1074:
1058:
1055:
1054:
1028:
1024:
1020:
1010:
1006:
1005:
1003:
987:
985:
919:
915:
914:
892:
888:
887:
885:
858:
854:
853:
831:
827:
826:
824:
819:
816:
815:
796:
793:
792:
776:
773:
772:
756:
753:
752:
736:
733:
732:
706:
702:
698:
688:
684:
683:
681:
636:
632:
608:
604:
589:
585:
573:
569:
560:
556:
551:
548:
547:
528:
525:
524:
480:
476:
464:
460:
458:
455:
454:
412:
408:
396:
392:
390:
387:
386:
367:
364:
363:
343:
340:
339:
323:
320:
319:
299:
296:
295:
275:
272:
271:
248:
245:
244:
230:
193:
185:
182:
181:
163:
160:
159:
143:
140:
139:
119:
116:
115:
112:
48:harmonic series
28:
23:
22:
15:
12:
11:
5:
3156:
3146:
3145:
3140:
3135:
3118:
3117:
3115:
3114:
3109:
3104:
3099:
3094:
3089:
3084:
3079:
3074:
3069:
3064:
3063:
3062:
3057:
3052:
3047:
3042:
3037:
3027:
3022:
3017:
3012:
3007:
3002:
2997:
2992:
2991:
2990:
2983:Bladder fiddle
2980:
2975:
2969:
2967:
2956:
2955:
2953:
2952:
2947:
2942:
2937:
2932:
2927:
2922:
2917:
2912:
2907:
2900:
2895:
2890:
2885:
2884:
2883:
2873:
2856:
2851:
2846:
2841:
2836:
2831:
2826:
2825:
2824:
2814:
2808:
2805:
2804:
2794:
2791:
2790:
2774:
2773:
2766:
2759:
2751:
2742:
2741:
2739:
2738:
2727:
2724:
2723:
2721:
2720:
2719:
2718:
2713:
2703:
2698:
2693:
2688:
2682:
2680:
2679:Related topics
2676:
2675:
2673:
2672:
2667:
2662:
2660:Joseph Sauveur
2657:
2652:
2647:
2645:Marin Mersenne
2642:
2637:
2632:
2627:
2622:
2617:
2611:
2609:
2605:
2604:
2602:
2601:
2596:
2595:
2594:
2584:
2579:
2574:
2569:
2564:
2563:
2562:
2557:
2552:
2542:
2541:
2540:
2530:
2525:
2520:
2514:
2512:
2502:
2501:
2499:
2498:
2497:
2496:
2486:
2485:
2484:
2479:
2468:
2466:
2460:
2459:
2456:
2455:
2448:
2440:
2439:
2437:
2435:
2434:
2429:
2424:
2419:
2414:
2409:
2403:
2401:
2395:
2394:
2387:
2386:
2379:
2372:
2364:
2358:
2357:
2340:
2339:External links
2337:
2335:
2334:
2322:
2321:
2320:
2316:
2315:
2286:
2255:
2252:
2251:
2250:
2245:
2240:
2234:
2229:
2224:
2219:
2212:
2209:
2148:
2145:
2144:
2143:
2129:
2126:
2117:
2114:
2110:
2105:
2100:
2096:
2072:
2061:
2060:
2046:
2043:
2038:
2035:
2029:
2024:
2020:
1996:
1992:
1988:
1985:
1982:
1977:
1973:
1961:
1960:
1957:
1954:
1934:
1921:(that is, the
1919:linear density
1906:
1897:(in Newtons),
1882:
1871:
1870:
1856:
1853:
1844:
1841:
1837:
1832:
1826:
1823:
1819:
1814:
1811:
1784:
1756:
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1348:
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1206:
1200:
1195:
1189:
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1168:
1163:
1160:
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1154:
1149:
1143:
1140:
1135:
1132:
1126:
1121:
1114:
1111:
1107:
1082:
1062:
1051:
1050:
1039:
1031:
1027:
1023:
1018:
1013:
1009:
1000:
996:
993:
990:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
939:
936:
933:
930:
927:
922:
918:
912:
909:
906:
903:
900:
895:
891:
884:
878:
875:
872:
869:
866:
861:
857:
851:
848:
845:
842:
839:
834:
830:
823:
800:
780:
760:
740:
729:
728:
717:
709:
705:
701:
696:
691:
687:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
639:
635:
631:
628:
625:
622:
619:
616:
611:
607:
603:
600:
595:
592:
588:
584:
579:
576:
572:
568:
563:
559:
555:
532:
521:
520:
509:
506:
503:
500:
497:
494:
491:
488:
483:
479:
475:
470:
467:
463:
452:
441:
438:
435:
432:
429:
426:
423:
420:
415:
411:
407:
402:
399:
395:
371:
347:
327:
316:linear density
303:
279:
255:
252:
229:
226:
207:
201:
198:
192:
189:
167:
147:
123:
111:
108:
78:with constant
36:standing waves
26:
9:
6:
4:
3:
2:
3155:
3144:
3141:
3139:
3136:
3134:
3131:
3130:
3128:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3092:Tromba marina
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3033:
3032:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2996:
2993:
2989:
2986:
2985:
2984:
2981:
2979:
2976:
2974:
2971:
2970:
2968:
2966:
2961:
2957:
2951:
2948:
2946:
2943:
2941:
2938:
2936:
2935:Standing wave
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2905:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2882:
2879:
2878:
2877:
2874:
2872:
2868:
2864:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2823:
2820:
2819:
2818:
2815:
2813:
2810:
2809:
2806:
2801:
2797:
2792:
2788:
2784:
2780:
2772:
2767:
2765:
2760:
2758:
2753:
2752:
2749:
2737:
2729:
2728:
2725:
2717:
2714:
2712:
2709:
2708:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2687:
2684:
2683:
2681:
2677:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2655:Lord Rayleigh
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2625:Ernst Chladni
2623:
2621:
2618:
2616:
2613:
2612:
2610:
2606:
2600:
2597:
2593:
2590:
2589:
2588:
2587:Standing wave
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2561:
2558:
2556:
2555:Inharmonicity
2553:
2551:
2548:
2547:
2546:
2543:
2539:
2536:
2535:
2534:
2531:
2529:
2526:
2524:
2521:
2519:
2516:
2515:
2513:
2511:
2507:
2503:
2495:
2492:
2491:
2490:
2487:
2483:
2480:
2478:
2475:
2474:
2473:
2470:
2469:
2467:
2465:
2461:
2453:
2449:
2446:
2442:
2441:
2433:
2430:
2428:
2425:
2423:
2422:Soundproofing
2420:
2418:
2417:Reverberation
2415:
2413:
2410:
2408:
2405:
2404:
2402:
2400:
2396:
2392:
2385:
2380:
2378:
2373:
2371:
2366:
2365:
2362:
2355:
2351:
2350:Alain Goriely
2347:
2343:
2342:
2332:
2327:
2323:
2318:
2317:
2312:
2308:
2304:
2300:
2296:
2292:
2287:
2283:
2279:
2275:
2271:
2267:
2263:
2258:
2257:
2249:
2246:
2244:
2241:
2238:
2235:
2233:
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2214:
2208:
2205:
2201:
2197:
2192:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2127:
2124:
2115:
2112:
2108:
2103:
2098:
2094:
2086:
2085:
2084:
2070:
2044:
2041:
2036:
2033:
2027:
2022:
2018:
2010:
2009:
2008:
1994:
1990:
1986:
1983:
1980:
1975:
1971:
1958:
1955:
1952:
1951:
1950:
1948:
1932:
1924:
1920:
1904:
1896:
1880:
1854:
1851:
1842:
1839:
1835:
1830:
1824:
1821:
1817:
1812:
1809:
1802:
1801:
1800:
1798:
1782:
1774:
1770:
1754:
1731:
1728:
1725:
1722:
1717:
1714:
1709:
1706:
1699:
1698:
1697:
1682:
1674:
1658:
1651:
1635:
1628:
1624:
1620:
1616:
1606:
1592:
1572:
1561:
1560:wave equation
1557:
1541:
1518:
1512:
1509:
1503:
1500:
1493:
1492:
1491:
1473:
1469:
1465:
1440:
1437:
1434:
1428:
1405:
1397:
1393:
1384:
1379:
1366:
1363:
1358:
1350:
1346:
1337:
1332:
1317:
1316:
1315:
1301:
1281:
1255:
1247:
1243:
1234:
1229:
1216:
1213:
1208:
1204:
1198:
1193:
1187:
1179:
1166:
1161:
1155:
1152:
1147:
1141:
1133:
1119:
1112:
1105:
1096:
1095:
1094:
1080:
1060:
1037:
1029:
1025:
1016:
1011:
998:
994:
988:
982:
976:
970:
967:
964:
958:
952:
949:
946:
943:
934:
928:
925:
920:
916:
907:
901:
898:
893:
889:
882:
873:
867:
864:
859:
855:
846:
840:
837:
832:
828:
821:
814:
813:
812:
798:
778:
758:
738:
715:
707:
703:
694:
689:
678:
672:
669:
666:
663:
657:
651:
645:
642:
637:
633:
629:
623:
617:
614:
609:
605:
601:
598:
593:
590:
586:
582:
577:
574:
570:
566:
561:
557:
546:
545:
544:
530:
507:
504:
501:
495:
489:
486:
481:
477:
473:
468:
465:
461:
453:
439:
436:
433:
427:
421:
418:
413:
409:
405:
400:
397:
393:
385:
384:
383:
369:
361:
345:
325:
318:. If angles
317:
301:
293:
277:
269:
253:
241:
234:
225:
223:
218:
205:
199:
196:
190:
187:
179:
165:
145:
137:
121:
107:
105:
101:
97:
93:
89:
85:
81:
77:
74:to produce a
73:
69:
65:
61:
57:
49:
45:
41:
37:
32:
19:
3112:Washtub bass
2965:musical bows
2939:
2925:Scale length
2902:
2822:Third bridge
2670:Thomas Young
2620:Jens Blauert
2608:Acousticians
2426:
2326:
2294:
2290:
2265:
2261:
2193:
2177:refresh rate
2168:
2150:
2062:
1962:
1872:
1746:
1612:
1533:
1420:
1270:
1052:
730:
522:
242:
239:
219:
180:
113:
88:musical tone
71:
53:
3087:Psalmodicon
3000:Diddley bow
2859:Fundamental
2849:Fingerboard
2829:Chordophone
2787:instruments
2640:Franz Melde
2615:John Backus
2599:Subharmonic
2452:Spectrogram
2196:stroboscope
136:square root
40:fundamental
34:Vibration,
3127:Categories
3020:Ichigenkin
3015:Ground bow
2960:Monochords
2950:Tuning peg
2930:Soundboard
2844:Enharmonic
2701:Ultrasound
2691:Infrasound
2477:Bark scale
2297:(5): 408.
2254:References
2237:3rd bridge
2161:television
2157:CRT screen
1627:wavelength
228:Derivation
3067:Langeleik
3005:Duxianqin
2898:Monochord
2867:Overtones
2863:Harmonics
2582:Resonance
2482:Mel scale
2412:Monochord
2391:Acoustics
2153:waveforms
2128:μ
2071:μ
1972:λ
1905:μ
1855:μ
1726:λ
1718:τ
1715:λ
1673:frequency
1659:τ
1636:λ
1615:frequency
1570:Δ
1513:μ
1390:∂
1376:∂
1364:μ
1343:∂
1329:∂
1279:Δ
1240:∂
1226:∂
1214:μ
1185:∂
1177:∂
1167:−
1159:Δ
1139:∂
1131:∂
1110:Δ
1081:β
1061:α
1022:∂
1008:∂
992:Δ
989:μ
977:α
971:
959:β
953:
947:−
935:α
929:
908:α
902:
874:β
868:
847:β
841:
822:−
799:α
779:β
700:∂
686:∂
676:Δ
673:μ
670:≈
661:Δ
652:α
646:
624:β
618:
602:−
583:−
554:Σ
502:≈
496:β
490:
434:≈
428:α
422:
346:β
326:α
302:μ
251:Δ
200:μ
166:μ
80:frequency
70:causes a
68:Resonance
56:vibration
44:overtones
3082:Onavillu
3035:Genggong
3030:Jaw harp
2978:Berimbau
2920:Re-entry
2777:Musical
2736:Category
2577:Overtone
2545:Harmonic
2319:Specific
2211:See also
2165:computer
240:Source:
94:such as
3102:Umuduri
3077:Masenqo
3060:Mukkuri
3055:Morsing
2995:ÄĂ n bầu
2988:Boom-ba
2973:Ahardin
2779:strings
2523:Formant
2299:Bibcode
2270:Bibcode
2204:bending
2083:, then
1945:is the
1917:is the
1895:tension
1893:is the
1617:of the
1554:is the
1490:; thus
360:tension
266:be the
96:guitars
46:in the
3107:Unitar
3072:Lesiba
3050:Kubing
3045:Khomuz
3040:Gogona
3010:Ektara
2834:Course
2817:Bridge
2785:, and
2716:Violin
2550:Series
1947:length
1873:where
1767:, the
1650:period
1534:Where
294:, and
268:length
104:pianos
102:, and
100:cellos
60:string
27:A wave
3138:Sound
3097:Tumbi
2839:Drone
2783:wires
2711:Piano
2696:Sound
2510:pitch
2472:Pitch
2348:" by
2163:or a
1773:nodes
1623:speed
1619:sound
1556:speed
84:pitch
76:sound
62:is a
58:in a
2910:Node
2854:Fret
2796:List
2686:Echo
2592:Node
2518:Beat
2508:and
1923:mass
1073:and
791:and
338:and
314:its
292:mass
290:its
243:Let
110:Wave
64:wave
2962:and
2915:Nut
2812:Bow
2307:doi
2278:doi
2169:not
968:tan
950:tan
926:cos
899:sin
865:cos
838:sin
643:sin
615:sin
487:cos
419:cos
3129::
2781:,
2305:.
2295:57
2293:.
2276:.
2266:72
2264:.
2191:.
1799::
1605:.
1314::
811:)
106:.
98:,
66:.
54:A
2869:/
2865:/
2861:/
2802:)
2798:(
2770:e
2763:t
2756:v
2383:e
2376:t
2369:v
2356:.
2344:"
2313:.
2309::
2301::
2284:.
2280::
2272::
2167:(
2125:T
2116:L
2113:2
2109:n
2104:=
2099:n
2095:f
2045:L
2042:2
2037:v
2034:n
2028:=
2023:n
2019:f
1995:n
1991:/
1987:L
1984:2
1981:=
1976:n
1933:L
1881:T
1852:T
1843:L
1840:2
1836:1
1831:=
1825:L
1822:2
1818:v
1813:=
1810:f
1783:L
1755:L
1732:.
1729:f
1723:=
1710:=
1707:v
1695::
1683:f
1593:T
1573:x
1542:v
1519:,
1510:T
1504:=
1501:v
1474:2
1470:v
1466:1
1444:)
1441:t
1438:,
1435:x
1432:(
1429:y
1406:.
1398:2
1394:t
1385:y
1380:2
1367:T
1359:=
1351:2
1347:x
1338:y
1333:2
1302:y
1282:x
1256:.
1248:2
1244:t
1235:y
1230:2
1217:T
1209:=
1205:)
1199:x
1194:|
1188:x
1180:y
1162:x
1156:+
1153:x
1148:|
1142:x
1134:y
1120:(
1113:x
1106:1
1038:.
1030:2
1026:t
1017:y
1012:2
999:T
995:x
983:=
980:)
974:(
965:+
962:)
956:(
944:=
938:)
932:(
921:1
917:T
911:)
905:(
894:1
890:T
883:+
877:)
871:(
860:2
856:T
850:)
844:(
833:2
829:T
759:T
739:T
716:.
708:2
704:t
695:y
690:2
679:x
667:a
664:m
658:=
655:)
649:(
638:1
634:T
630:+
627:)
621:(
610:2
606:T
599:=
594:y
591:2
587:T
578:y
575:1
571:T
567:=
562:y
558:F
531:a
508:.
505:T
499:)
493:(
482:2
478:T
474:=
469:x
466:2
462:T
440:.
437:T
431:)
425:(
414:1
410:T
406:=
401:x
398:1
394:T
370:T
278:m
254:x
206:.
197:T
191:=
188:v
146:T
122:v
50:.
20:)
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