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2 $BFC@-6J@~(B

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$$u_t+a u_x=0 \hspace{1zw}(a \mbox{ $B$ODj?t(B})$$ (7)

$B$r2r$$$F!"$=$N2r$,(B

$$u(t,x)=f(x-at)$$ (8)

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  $$u(0,x)=f(x) \hspace{1zw}(\mbox{$B$9$Y$F$N<B?t(B $x$\ $B$KBP$7$F(B})$$ (9)

  
  

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$$\frac{dx}{dt}=a$$

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$$v(t)=u(t,at+x_0)$$

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$$\frac{d }{d t}v(t) = \frac{d }{d t} u(t,at+x_0)$$

$$= \frac{\partial u}{\partial t}(t,at+x_0)\frac{d t}{d t}+\frac{\partial u}{\partial x}(t,at+x_0)\frac{d (at+x_0)}{d t}$$

$$= u_t(t,at+x_0)+a u_x(t,at+x_0)$$ (10)

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$$\frac{d v(t)}{d t}=0$$

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$$u(t,at+x_0)=u(0,x_0)=f(x_0)$$

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$$u(t,x)=u(0,x-at)=f(x-at)$$

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