812.2 one-dimensional waves and the equation of waves Solution: (a) A=0.04(m)2=0.4(m)/(, 0 v=00Sm/s O=2=0.4z(rads-) afm) The initial phase angle of point 0: po=n 2 y(0,t)=0.04c0s(0.4m+ (b) the wavefunction y(x,t)=0.04c0s04x(t- 0.08 812.2 one-dimensional waves and the equation of waves (c) the oscillatory equation of point P 0.4、兀 甲(xp,1)=0.04c00mx =004c0s0.4m+ (d)the moving directions of points a and b (x,1) 0.04 0.2
11 (a) 2 0.4 (rad s ) 0.04( ) 0.4(m) −1 = = ⋅ = = π λ ω π λ v A m • a •b x(m) 0.04 Ψ(x,t) o 0.2 P v=0.08m/s Solution: ) 2 (0, ) 0.04 cos( 0.4 π Ψ t = πt + 2 0 π The initial phase angle of point o: ϕ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = − + 2 ) 0.08 ( , ) 0.04cos 0.4 ( π π x Ψ x t t (b) the wavefunction §12.2 one-dimensional waves and the equation of waves Ψ a b x 0.04 Ψ(x,t) o 0.2 p • • ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = − + 2 0.04cos 0.4 2 ) 0.08 0.4 ( , ) 0.04cos 0.4 ( π π π π t Ψ x t t P (c) the oscillatory equation of point P (d) the moving directions of points a and b. §12.2 one-dimensional waves and the equation of waves
812.3 Energy transport via mechanical waves The velocity of wave on a string The wave is viewed from a reference frame moving with the wave so it is seen to be at rest. The disturbance is motionless but the string is observed to move to the left at a constant speed △m(v2/r)=2Tsin(61/2)≈T6 △m=MM,b=M/r (v2/r)=T v=√T/ C s 12. 3 Energy transport via mechanical waves Generalize to the other waves The general relationship for the speed of mechanical waves in material media magnitude of a force factor v OC mass factor Sound wave in solid: vsolid module P Sound wave in lianid Www/B12
12 §12.3 Energy transport via mechanical waves 1. The velocity of wave on a string µ µ µ θ θ θ / ( / ) , / ( / ) 2 sin( / 2) 2 2 v T r l l v r T m l l r m v r T T = ∆ ∴ ∆ = ∆ = ∆ = ∆ Q∆ = ≈ The wave is viewed from a reference frame moving with the wave so it is seen to be at rest. The disturbance is motionless but the string is observed to move to the left at a constant speed v. x y T r T′ r θ θ / 2 ∆l r C θ / 2 §12.3 Energy transport via mechanical waves The general relationship for the speed of mechanical waves in material media: 1/ 2 mass factor magnitude of a force factor ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ v ∝ 1/ 2 solid ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ρ E v 1/ 2 liquid ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ρ B v Sound wave in solid: Sound wave in liquid: young´s modulus bulk modulus Generalize to the other waves: