Traveling Waves: Energy Transport

Transcription

1 Traveling Waves: Energ Transport wave is a traveling disturbance that transports energ but not matter. Intensit: I P power rea Intensit I power per unit area (measured in Watts/m 2 ) Intensit is proportional to the square of the amplitude! v Variation with Distance: If sound is emitted isotropicall (i.e. equal intensit in all directions) from a point source with power P source and if the mechanical energ of the wave is conserved then I P 4πr source 2 (intensit from isotropic point source) Source P S r Sphere with radius r Speed of Propagation: The speed of an mechanical wave depends on both the inertial propert of the medium (stores kinetic energ) and the elastic propert (stores potential energ). v elastic inertial (wave speed) v F T µ Transverse wave on a string: F T string tension µ M/L linear mass densit Universit of Florida PHY 2053 Page 1

2 Constructing Traveling Waves f(x) at time t0 f(x-v v x 0 Constructing Traveling Waves: To construct a wave with shape f(x) at time t 0 traveling to the right with speed v simpl make the replacement: Traveling Harmonic Waves: Harmonic waves have the form sin(kx + ) at time t 0, where k is the "wave number, k 2π/λ, λ is the "wave length". and is the "amplitude". To construct a harmonic wave traveling to the right with speed v, replace x b x-vt as follows: sin( k( x v + ) sin( kx ωt + ) ( x, ( x, sin( kx ωt + ) sin( kx + ωt + ) where ω kv. The phase angle determines at x t 0, (xt0) sin. If (xt0) 0 then 0. PHY 2053 Page 2 Universit of Florida x vt λ sin(kx) kx (radians) Speed of propagation! Harmonic wave traveling to the right Harmonic wave traveling to the left v ω k

3 Waves: Mathematical Description -axis ( x, sin( sin( kx ω wave traveling to the right (x, Φ kx-ωt Φ Vector with length undergoing uniform circular motion with phase Φ kx ωt. The projection onto the -axis gives sin(kx ω. If t 0 then -axis ( x, t 0) sin( kx) λ (x) Φ kx x One circular revolution corresponds to Φ 2π kλ, and hence k 2π/λ ( wave number ). Universit of Florida PHY 2053 Page 3

4 Waves: Mathematical Description -axis ( x, sin( sin( kx ω wave traveling to the right (x, Φ kx-ωt Φ Vector with length undergoing uniform circular motion with phase Φ kx ωt. The projection onto the -axis gives sin(kx ω. If x 0 then -axis T ( x 0, sin( ω ( Φ -ωt t One circular revolution corresponds to Φ 2π ωt and hence T 2π/ω ( period ). Universit of Florida PHY 2053 Page 4

5 Waves: Mathematical Description In General -axis ( x, sin( sin( kx ω t + ) wave traveling to the right (x, Φ kx-ωt+ Φ Overall phase Φ kx ωt +. (x, -axis Φ kx+ωt+ ( x, sin( sin( kx +ω t + ) wave traveling to the left Φ Period (in s) T 2π ω Universit of Florida Frequenc (in Hz) f 1 T ngular Frequenc (in rad/s) ω 2πf Wave Number (in rad/m) k 2π λ Wave Speed (in m/s) ω v λf k PHY 2053 Page 5

6 Waves: Mathematical Description ( x, sin( sin( kx ω t + ) wave traveling to the right v node x point on string n th node Waves Propagation: node is a point on the wave where (x, vanishes: Φ kx ωt + nπ n 0,1,2,L kx1 ωt1 + nπ k( x2 x2) ω ( t2 t1) dx ω v node kx2 ωt2 + nπ k x ω t dt k (wave speed) Transverse Speed & cceleration: For transverse waves the points on the string move up and down while the wave moves to the right. transverse speed of a point on the string d u ω cos( kx ωt + ) dt u ω max Universit of Florida transverse acceleration of a point on the string du 2 a ω sin( kx ωt + ) dt a ω 2 max SHM PHY 2053 Page 6 a 2 ω

7 Waves: Example Problems transverse wave on a taught string has amplitude, wavelength λ and speed v. point on the string onl moves in the transverse direction. If its maximum transverse speed is u max, what is the ratio u max /v? nswer: 2π/λ The function (x, cos(kx - ω describes a wave on a taut string with the x-axis parallel to the string. The wavelength is λ 3.14 cm and the amplitude is 0.1 cm. If the maximum transverse speed of an point on the string is 10 m/s, what is the speed of propagation of the travelling wave in the x-direction? nswer: 50 m/s umax ω umax ω ω ωλ v k 2π v ω k u ω 2π max k v ω / k λ λumax (3.14cm)(10m / s) _ v 50m / s 2π 2π (0.1cm) Universit of Florida PHY 2053 Page 7

8 Waves: Example Problems sinusoidal wave moving along a string is shown twice in the figure. Crest travels in the positive direction along the x-axis and moves a distance d 12 cm in 3 ms. If the tick marks along the x-axis are 10 cm apart, what is the frequenc of the traveling wave? nswer: 100 Hz v d 12cm 40m s t 3ms / v 40m / s f 100Hz λ 0.4m λ 4 x 4(10 cm) 0. 4m Universit of Florida PHY 2053 Page 8