12.2 one-dimensional waves and the equation of waves yP(x,0)=yp(x,t) r-vt= constant phase d v=0→p= dt d x(x) Phase speed 812.2 one-dimensional waves and the equation of waves 2. The classical wave equation y(x,t)=平(x-v)=(u) a ay au a 0y202y ax au ax au at a 02y、au or au au a a-p ap 0=0 2=(,o、an_n0 at au at ay a v au au at u5 x y o x(x´) y y´ o o´ x • • vt x′ P P y (x,0) y (x,t) P = P x − vt = constant phase t x v v t x d d 0 d d − = ⇒ = Phase speed §12.2 one-dimensional waves and the equation of waves 2. The classical wave equation u x vt y x t Ψ x vt Ψ u = − ( , ) = ( − ) = ( ) 2 2 2 2 ( ) u Ψ x u u Ψ x u Ψ u Ψ x u u Ψ x Ψ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ 2 2 2 2 2 ( ) u Ψ v t u u Ψ v t u Ψ u Ψ v t u u Ψ t Ψ ∂ ∂ = ∂ ∂ ∂ ∂ − ∂ ∂ = ∂ ∂ ∂ ∂ = − ∂ ∂ ∂ ∂ = ∂ ∂ 2 2 2 2 2 x Ψ v t Ψ ∂ ∂ = ∂ ∂ 0 1 2 2 2 2 2 = ∂ ∂ − ∂ ∂ t Ψ x v Ψ or §12.2 one-dimensional waves and the equation of waves