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¤W¦¡¥k¤âÃä¬Û·í©ó¬O¬O¤G¦¸·L¤À¡A¦Ü©ón°£¤§ªº (Δx)2 ±N·|¨Ó¦Û Δk = k/Δx ¤Î Δm = μ Δx ¦U¤@Ó¡C¦p¦¹¡A¦¡¤l¤¤¦U¦³¹ï®É¶¡¤Î¹ïªÅ¶¡·L¤À¨â¦¸¡A³o§Y¬Oªi°Ê¤èµ{¦¡ªº¯S¼x¡C
¥H¤Wªº±À¾É¤£ºÞ¬OÁaªi©Î¬O¾îªi³£¾A¥Î¡Aì¦]¡G¼u©Ê¾Ç¤w¾Ç¹L°ÅÀ³¤O»P°ÅÀ³ÅÜ¡A¦]¦¹¦³¤@¼Ëªº½u©Ê¤½¦¡¥i¥H¨Ï¥Î¡C
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ªiªº¼Æ¾Ç´yz
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³Ì±`¥Î°ª«nªº¬O¥¿©¶ªi (sinusolidal waves)
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©P´Á T ¡GÅý¤T¨¤¨ç¼Æ¨«§¹¤@Ó¨ç¼ÆÈÅܤơ]¤§«á«ÂС^©Ò»Ýnªº®É¶¡¡A¨£¤W¹Ï
A sin ( ω(t=0) + φ0 ) = A sin ( 2π + φ0 )
2π= ωT
¨¤ÀW²v ω¡Gω= 2π/ T
ÀW²v f ¡Gf = 1 / T
ªiªø λ¡G¬Û¾F¨âӳ̤jȶ¡ªº¶ZÂ÷¡A¨£¤W¹Ï
ªi³t v ¡Gv = λν
¥Ñ«e¤wª¾¨«¤@Óªiªøªº®É¶¡¬O¤@©P´Á¡A¬Gªi³t¬°ªiªø°£¥H©P´Á¡A§Yªø¼¥HÀW²v
ªi§Î¡Bªi¼Æ¡B¬Û¦ì
«e±§Q¥Î®¶Àúªº¹Ï¹³¦U¦Û«Ø¥ß¤F®É¶¡»PªÅ¶¡«ÂЩʪº±¡§Î¡A§Y·í®É¼È¥ý·Q¹³®¶¤l¦ì©ó x=0 ³B¡Cn·Q¥H¼Æ¾Ç¨ç¼Æ§Î¦¡§â§¹¾ã¦Ó¥i¦bªÅ¶¡¤¤¶Ç¼½ªºªiªí¥Ü¥X¨Ó¡A¦Ò¼{ªi«enªá t = x / v ªº®É¶¡¨Ó²¾°Ê x ¨º»ò¦hªº¶ZÂ÷¡C¥u¶·n§â t §ï¼g¬° t - x / v ¡A¤@¼Ë©ñ¨ì쥻®É¶¡¨ç¼Æ t ©Ò¦bªº¦a¤è¡C
y(t) = A sin ( ω ( t - x / v ) + θ0 )
«e¤w¦³ ω = 2πf
y(t) = A sin ( 2πf ( t - x / v ) + θ0)
y(t) = A sin ( 2πf t - 2πfx / v + θ0)
y(t) = A sin ( 2πf t - 2πf x /(λf) + θ0)
y(t) = A sin ( 2πf t - 2πx /λ + θ0)
y(t) = A sin ( (2π/T) t - (2π/λ) x + θ0)
¨ä¤¤ 2π/T = ω
§Ṳ́ñ·Ó¨Ó©w 2π/λ= κ
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y(t) = A sin (ω t - κx + θ0)
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y(t) = A sin ( κx - ωt + φ0 )
³o»P¤W¦¡¬Oµ¥®Äªºµ²ªG¡A¦]®³¬° sin(-x) = - sin(x) = sin(π + x)
φ0 = π - φ0
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µù¡G¦b«e±§Ṳ́w¸g©w¥X¤F¨¤ÀW²v»P©P´ÁªºÃö«Y¡A¨ä·N¸q¬O³æ¦ì®É¶¡ªø«×ùØ·|¦³´XÓªi¡A¦p¤U
ω= 2π/ T
¦b¤T¨¤¨ç¼Æªi A sin ( κx - ωt + φ0 ) ªº©w¸q°ì κx - ωt + φ0 ¥Î»P¨¤ÀW²v¡]®É¶¡¡^¬Û¦Pªº¤è¦¡¨Ó¬Ý«ÝªÅ¶¡ªº«Y¼Æ¡A§ÚÌ´N¦³ªi¼Æ¡]¤S¥s°µªi¦V¶q¡^¡A
κ = 2π/ λ
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y(t) = A sin ( κx - ωt + φ0 )
¨ä¤¤¡Aªiªº¬Û¦ì φ = κx - ωt + φ0
"ªi«e" ¦ì©ó¬Y©T©w "¬Û¦ì" ¨¤«×
φ = C = κx - ωt + φ0
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x = (ω/κ) t + (φ0+ C ) /κ
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dx/dt = ω/κ
³o¥s°µ "¬Û" ³t«×¡A¤]´N¬O "ªi«e" ªº³t«×¡A§Y¤U±«nªºÃö«Y¦¡¡G
v = ω/ κ
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ªi°Ê¤èµ{¦¡ªº¯S¼x¡A¬O¹ï®É¶¡·L¤À¨â¦¸¤]¹ïªÅ¶¡·L¤À¨â¦¸¡A¦p¤U¡G
∂2 y(x,t) /∂t2 - v2 ∂2 y(x,t) /∂x2 = 0
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ªi°Ê¤èµ{¦¡¤¤¡Aªi³t¦ÛµM¥X²{¡A¬G¥¦¬O¤ñ¸û°ò¥»ªº¶q¡C¤@Ó¤¶½è¦³¤@©wªºªi³t¡]¦pªÅ®ð¡B¤ô¤Î¿û¡^¡A¦Ó¤£¦P·f°tªºÀW²v»Pªiªø³£¦³¥i¯à¡A¥un«O«ù¼¿n¬Oªiªø§Y¥i¡C
³o»P¤@Ó³æ¤@®¶¤l¡A¥u¦³¤@Ó¦ÛµMÀW²v¡A¨Ã¤£¬Û¦P¡C¨Æ¹ê¤W¡A¨âӦۥѫתº½¢¦X®¶¤l¹ï¡A¦³¨âÓ¦ÛµMÀW²v¡An Ӧۥѫתº½¢¦X®¶¤l¸s¡A¦@¦³ n Ó¦ÛµMÀW²v¡C¤@Ó³sÄò¤¶½è¥iµø¬°¦³µL¼ÆÓ®¶¤l½¢¦X¦b¤@°_¡A¦]¦¹ÀW²v¦³µL¦hºØ¡]¥t¥~¡A¥H´¹Å鬰¨Ò¡Aªiªø¦³³Ì¤pÈ¡A¦]¦¹ÀW²v¦³³Ì¤jÈ¡^¡C
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¥H¥¿©¶ªi y(t) = A sin ( κx - ωt + φ0 ) ¥N¤JÅçÃÒ¡A·L¤À sin ±o cos¡A¦A·L¤À cos ±o -sin ¡A¥i¬Ý¥X½T©w½T¬°¨ä¸Ñ¡A¥un¨ä¤¤ªº°Ñ¼Æº¡¨¬
ω2 - v2 κ2 = 0
¦Ó«e±¤w¸g¥Ñªi«eªº²¾°Ê±À±o v = ω/ κ¡A¦]¦¹¤W¦¡¦Û°Ê¦¨¥ß¡A¬G¥¿©¶ªi½T¹ê¬Oªi°Ê¤èµ{¦¡ªº¸Ñ¡C
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¤@Óȱoª`·Nªº¨Æ±¡¬O¡A¥¿©¶ªi¤w¸g¬O³Ì¤@¯ë©Êªº±¡ªp¤F¶Ü¡H¤£¬Oªº¡A¥ô¦ó¥i·L¤À¨ç¼Æ¨ä©w¸q°ì¨ã¦³ κx - ωt ªº§Î¦¡¡A³£¬Oªi°Ê¤èµ{¦¡ªº¸Ñ¡C¥t¥~¡A²ßºD¤W κ©M ω§Ú̳£¥u¥Î¥¿È¡A¦]¦¹ Y(κx - ωt ) »P Y( κx + ωt ) ¡A«eªÌ¥Nªí¦V¥¿ x ¤è¦V¶Ç¼½¡B«áªÌ¥Nªí¦Vt x ¤è¦V¶Ç¼½¡A³£¬O¤@¯ë¸Ñ¡C
±q·L¤ÀªºÃìÂê²v ¥i¥H¬Ý¥X¡A
df(g) / dx = (df/dg) (dg/dx)
df(g) / dt = (df/dg) (dg/dt)
§Ṳ́£ºÞ¬O¦b°µ¹ï®É¶¡®y¼Ð·L¤Àªº³¡¤À¥ç©Î¬O°µ¹ïªÅ¶¡®y¼Ð·L¤Àªº³¡¤À¡A
g(x,t) = κx - ωt ¤@©w·|¾ÉP ω2 - v2 κ2 = 0 ªºº¡¨¬
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¦pªG§Ú̧⩶ªº®¶Àú¬Ý§@¬O½¢¦X¦b¤@°_ªº®¶¤l¡A¦^´_¤O»P¦ùªø¶q¦¨¥¿¤ñ¡A§Y÷±i¤O¡]ª`·N¤£¬O©P´Á T¡^T = k Δx ¡A¨ä¤¤ k = √() «h§Ú̪¾¹D¨äªi³t¬O
k Δx2 = m v2
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v = √(k/m) Δx =
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¥H y(x,t) = A exp {- (x - vt)2 } ¬°¨Ò¡A¥¦¥s°µ °ª´µ¨ç¼Æ ¡A¨ä©w¸q°ì¤]¨ã¦³ x - v t ªº§Î¦¡¡A¦]¦¹¤]¬Oªi°Ê¤èµ{¦¡ªº¸Ñ¡C
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¨âÓ®¶´T¬Û¦P¡B¦æ¶i³t«×¬Û¤Ïªº¥¿©¶ªi¡A¨äÅ|¥[ªº©M
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sin α ± sin β = sin α cos β± cos αsin β
¥i±o
y(t) = A sin ( κx - ωt ) + A sin ( κx + ωt ) = 2A sin (κx ) cos (ωt )
§Y¾nªi¡C
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λn = 2L / n , n ∈N
v = √(T/μ)
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fn = v / λn = n √[ T / ( 4 L2μ) ]
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