§122。ne- dimensional waves and the equation of waves 3. Periodic waves (a)Space period Y(, t A snapshot at some instant to (b) Time period (x,D) An oscillation at any given position o I 812.2 one-dimensional waves and the equation of waves n one time period, some oscillatory state propagate one space period. T Phase speed:v=y=元v T 6
6 3. Periodic waves x ( , ) 0 Ψ x t λ o λ (a) Space period A snapshot at some instant t0 o t ( , ) 0 Ψ x t T T (b) Time period An oscillation at any given position x0 §12.2 one-dimensional waves and the equation of waves In one time period, some oscillatory state propagate one space period. Phase speed: λ ν λ = = ⋅ T v λ T §12.2 one-dimensional waves and the equation of waves
§122。ne- dimensional waves and the equation of waves (c Traveling wave y(x,) A=vT =νT 4. Sinusoidal (harmonic)waves Y(x, t)=Acos k(r-v) Y(x, t) 812.2 one-dimensional waves and the equation of waves 4. The physical meaning of the harmonic waves Model: One-dimensional harmonic waves If the oscillatory equation at point o is Yo=Acos(@t +o)--harmonic motion y=平(x,t)=?
7 x Ψ(x,t) v r o λ = vT v∆t (c) Traveling wave ν = λ = vT v =νλ T 1 4. Sinusoidal (harmonic) waves Ψ(x,t) = Acos[k(x − vt)] x Ψ(x,t) v r o λ §12.2 one-dimensional waves and the equation of waves 4. The physical meaning of the harmonic waves Model: One-dimensional harmonic waves x v r o If the oscillatory equation at point o is cos( ) Ψ0 = A ωt + φ Ψ = Ψ( x,t) = ? --harmonic motion §12.2 one-dimensional waves and the equation of waves
§122。ne- dimensional waves and the equation of waves Choose a ray of wave as x axis, construct a one-dimensional coordinate system. O is the origin, v is the velocity along the tx direction. Method 1 P(r) ifY0=Acos(ot+φ) The propagating time of the、△ state from point o to point P(c) The phase of point P(r)at instant t is the same as the point o at instant t-4t 812.2 one-dimensional waves and the equation of waves therefore Yn(x,)=0(0,t-△)=Acoo(t-)+外 (x,D)= Acosta(-1)+y(1) Method 2: L P The phase difference between two points of space interval of n is 2T. The phase at point P lag .2 relative to point o 8
8 Method 1 if cos( ) Ψ0 = A ωt + φ′ v O P(x) x The propagating time of the state from point o to point P(x) v x ∆t = Choose a ray of wave as x axis, construct a one-dimensional coordinate system. O is the origin, v is the velocity along the +x direction. The phase of point P(x) at instant t is the same as the point o at instant t-∆t. §12.2 one-dimensional waves and the equation of waves ( , ) = cos[ω( − t) +φ] v x Ψ x t A (1) ( , ) (0, ) 0 Ψ x t Ψ t t p = − ∆ = cos[ω( − ) +φ′] v x A t therefore u O P(x) x Method 2: The phase at point P lag relative to point o π λ ⋅ 2 x The phase difference between two points of space interval of λ is 2π. §12.2 one-dimensional waves and the equation of waves
§122。ne- dimensional waves and the equation of waves p(r,t=AcoS(at -2.2z) x or平(x,t)=Ac0s(…·2丌-o+φ)(2) Y(x, t)=Acos o(--t)+9](1) Due to见 Equations(1)and (2)is identical. 812.2 one-dimensional waves and the equation of waves y(x,t)= A cos a(-t)+小 A cos(2-at+y A coS 2( )+小 见T A cos( lx -at+o) Acos{k(x-v)+小 where元=wTa= 2兀k= T 2λ
9 Equations (1) and (2) is identical. ( , ) cos( 2π ) λ = ω + φ′ − ⋅ x Ψ x t A t p ( , ) cos( 2π ω φ ) λ = ⋅ − t + x or Ψ x t A (2) ω π λ 2 Due to = vT = v §12.2 one-dimensional waves and the equation of waves ( , ) = cos[ω( − t) +φ] (1) v x Ψ x t A λ π π λ ω 2 2 = = k = T where vT §12.2 one-dimensional waves and the equation of waves ( , ) = cos[ω( − t) + φ ] v x Ψ x t A cos( 2 ω φ ) λ = π − t + x A cos[ 2 ( ) φ ] λ = π − + T x t A = LL = Acos[ k( x − vt ) + φ ] = Acos( kx −ωt + φ )
812.2 one-dimensional waves and the equation L of waves 6. The comparison of graphs of oscillation and waveform T, A, o, the moving direction of point (b) The graphs of waveform u AFoAZ A,a, determine o from the graph of waveform at t=0, the moving direction of point 812.2 one-dimensional waves and the equation of waves Example: figure depicts the waveform of a traveling sinusoidal wave at instants t=0 S, find(a) the oscillatory equation of point o; (b) the wavefunction P(x, t); (c) the oscillatory equation of point P;(d) the moving directions of points a and b y(x,0) v=0.08m/s 0.04 b 10
10 §12.2 one-dimensional waves and the equation of waves 6. The comparison of graphs of oscillation and waveform T, A, φ, the moving direction of point (a) The graphs of oscillation A, λ, determine φ from the graph of waveform at t=0, the moving direction of point (b) The graphs of waveform • • a b x(m) 0.04 Ψ(x,0) o 0.2 P v = 0.08m/s Example: figure depicts the waveform of a traveling sinusoidal wave at instants t=0 s, find (a) the oscillatory equation of point o; (b) the wavefunction ;(c) the oscillatory equation of point P; (d) the moving directions of points a and b. Ψ(x,t) §12.2 one-dimensional waves and the equation of waves