Phys0 Lectues 30, 3 Wave Motion Key points: Types of Waves: Tansvese and Longitudinal Mathematical Repesentation of a Taveling Wave The Pinciple of Supeposition Standing Waves; Resonance Ref: -7,8,9,0,,6,,3,6. Page
Chaacteistics of Wave Motion When a vibation popagates, we have a wave.
Chaacteistics of Wave Motion Amplitude, A Wavelength, λ Fequency, f and peiod, T f T Wave velocity,
Types of Waves: Tansvese and Longitudinal The motion of paticles in a wave can be eithe pependicula to the wave diection (tansvese) o paallel to it (longitudinal).
Types of Waves: Tansvese and Longitudinal Sound waves ae longitudinal waves:
Types of Waves: Tansvese and Longitudinal The velocity of a tansvese wave on a cod is given by: Whee F T is the tension in the cod and μ is the linea density (mass pe unit length, μ=m/l). The velocity inceases when the tension inceases, and deceases when the mass inceases.
Types of Waves: Tansvese and Longitudinal The velocity of a longitudinal wave depends on the elastic estoing foce of the medium and on the mass density. o Fo a solid od. Fo fluids E Young s Modulus; B Bulk modulus.
Example: Echolocation. Echolocation is a fom of sensoy peception used by animals such as bats, toothed whales, and dolphins. The animal emits a pulse of sound (a longitudinal wave) which, afte eflection fom objects, etuns and is detected by the animal. Echolocation waves can have fequencies of about 00,000 Hz. (a) Estimate the wavelength of a sea animal s echolocation wave. (b) If an obstacle is 00 m fom the animal, how long afte the animal emits a wave is its eflection detected? (a) v f v B. 00 05 9. 40 3 m / s v f. 40 5 0 3. 40 m (b) t d v 00. 40 3 0.4s
Enegy Tanspoted by Waves By looking at the enegy of a paticle of matte in the medium of a wave, we find: Then, assuming the entie medium has the same density, the powe (enegy though S pe unit time) is P Slf A / t Sv f A The intensity (powe pe unit aea) is f k 4 f m k m The intensity is popotional to the squae of the fequency and to the squae of the amplitude.
If a wave is able to spead out thee-dimensionally fom its souce, and the medium is unifom, the wave is spheical. Just fom geometical consideations, as long as the powe output is constant, we see: Enegy Tanspoted by Waves 0 : and at powe Total P P P 0 4 4 : at Intensity P P I 0 4 4 : at Intensity P P I 0 4 : at Intensity P I
Example: Eathquake intensity. The intensity of an eathquake P wave taveling though the Eath and detected 00 km fom the souce is.0 x 0 6 W/m. What is the intensity of that wave if detected 400 km fom the souce? I P P, I 4 4 0 0 I I I 5. 00 6 4. 00 6. 5 W / m 5 4. 00 I 0
Mathematical Repesentation of a Taveling Wave Suppose the shape of a wave is given by: Spacial dependence D x Asink x Time dependence D t Asin t T
Mathematical Repesentation of a Taveling Wave Afte a time t, the wave cest has taveled a distance vt, so we wite: O: with, (wavenumbe) kv o v k In geneal, o Asink x t D x,t Acosk x t D x,t
Example: A taveling wave. The left-hand end of a long hoizontal stetched cod oscillates tansvesely in SHM with fequency f = 50 Hz and amplitude.6 cm. The cod is unde a tension of 40 N and has a linea density μ = 0. kg/m. At t = 0, the end of the cod has an upwad displacement of.6 cm and is falling. Detemine (a) the wavelength of waves poduced and (b) the equation fo the taveling wave. (a) FT 40 v 34. 6m / s 0. v f 34. 6 50 0. 37m v f (b) D x,t Asink x t A 0. 06m, k, 45m, f 570 s at t 0: 0. 06 0. 06sin, 0. 66ad, t 0.06sin45x 570t 0.66 D x (falling)
The Pinciple of Supeposition Supeposition: The displacement at any point is the vecto sum of the displacements of all waves passing though that point at that instant. Fouie s theoem: Any complex peiodic wave can be witten as the sum of sinusoidal waves of diffeent amplitudes, fequencies, and phases.
Intefeence These gaphs show the sum of two waves. In (a) they add constuctively; in (b) they add destuctively; and in (c) they add patially destuctively.
Standing Waves; Resonance Standing waves occu when both ends of a sting ae fixed. In that case, only waves which ae motionless at the ends of the sting can pesist. Thee ae nodes, whee the amplitude is always zeo, and antinodes, whee the amplitude eaches the maximum value.
Standing Waves; Resonance l n The fequencies of the standing waves on a paticula sting ae called esonant fequencies. They ae also efeed to as the fundamental and hamonics. The wavelengths and fequencies of standing waves ae:
Example: Piano sting. A piano sting is.0 m long and has a mass of 9.00 g. (a) How much tension must the sting be unde if it is to vibate at a fundamental fequency of 3 Hz? (b) What ae the fequencies of the fist fou hamonics? Fundamental fequency: n=. l. 0. 0m v FT f ; v. FT f. F T f. 009. 0 679 N 0 3. 0 Fequencies of the fist 4 hamonics n : n : n 3: n 4 : f f f f 3 4 3Hz f 3 f 4 f 6Hz 393Hz 54Hz :
Example: Wave foms. Two waves taveling in opposite diections on a sting fixed at x = 0 ae descibed by the functions D = (0.0 m)sin(.0x 4.0t) and D = (0.0m)sin(.0x + 4.0t) (whee x is in m, t is in s), and they poduce a standing wave patten. Detemine (a) the function fo the standing wave, (b) the maximum amplitude at x = 0.45 m, (c) whee the othe end is fixed (x > 0), (d) the maximum amplitude, and whee it occus. x,t D D 0. sinx 4t 0. sinx t 0. sin x 4t sinx 4t 0. sinxcos 4t 0. 4 sinxcos 4t D 4 D x,t M ax Amplitude: A 0. 4m Occus when sin x 0. 4sin0. 9cos4t 0. 33cos4t ; Amplitude 0. m at x 0. 45m : D x,t 33 k 3. 4m; l n n 4. 57m n,, 3,,... x n, x n 4
I-clicke question 30- If the amplitude of a simple hamonic oscillato is doubled, which of the following quantities will change the most? A) fequency B) peiod C) maximum speed D) maximum acceleation E) total mechanical enegy
I-clicke question 30- Pendulum A has a length of L A and peiod of T A. Pendulum B has a length of L B and peiod of T B. L B = 4L A. Compae thei peiods. A) T B = 4T A B) T B = T A C) T B = T A D) T B = T B / E) T A = 6T B
I-clicke question 30-3 Conside a wave on a sting moving to the ight, as shown below. What is the diection of the velocity of a paticle at the point labeled A? A A) B) C) D) E) zeo
I-clicke question 30-4 Conside a wave on a sting moving to the ight, as shown below. What is the diection of the velocity of a paticle at the point labeled B? A) B) C) D) E) zeo B