→ GHJK777 :p幣給你囉 06/01 22:05
※ 引述《GHJK777 (GHJK777)》之銘言：
: Show that z = f(x + at) + g(x - at)
: is a solution of the wave equation
: 2 2
: δ z δz
: -------- = a^2 * -------
: δ t^2 δx^2
: 300p有獎徵答!!
2 2
d [ f(x + at) + g(x - at) ] 2 d [ f(x + at) + g(x - at) ]
─────────────── = a ────────────────
dt^2 dx^2
d 在這裡都代表偏微分運算
假設 u = x + at
v = x - at
df df du
── = ── ──
dt du dt
= f' ×a
2
d f d(a f') du
─── = ──── ──
dt^2 du dt
2
= a f''
同樣的~~
2
d g 2
─── = a g''
dt^2
2 2
d 2 d f(x+at)
故 ── [ f + g ] = a (f'' + g'') (注意唷~ f'' = ───── , g也一樣)
dt^2 d(x + at)^2
df df du
── = ── ── = f'
dx du dx
2
d f
─── = f''
dx^2
2
2 d 2
故可知道 a ── [f + g] = a (f'' + g'')
dx^2
所以證明了 z = f(x+at) + g(x-at) 為 wave equation 的解
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