Transverse waves. Waves. Wave motion. Electromagnetic Spectrum EM waves are transverse.

Transcription

1 Transerse waes Physics Enhanceent Prograe for Gifted Students The Hong Kong Acadey for Gifted Education and, HKBU Waes. Mechanical waes e.g. water waes, sound waes, seisic waes, strings in usical instruents. Electroagnetic waes light (ultraiolet, isible, infrared), crowaes, radio waes, teleision waes, X rays 3. Matter (=quantu) waes electrons, protons, other fundaental particles, atos and olecules 4. Graity waes neer obsered! Electroagnetic Spectru EM waes are transerse. () Transerse wae Wae otion Waefronts up down otion Sending a transerse wae along a string. Each eleent of the string ibrates at right angles to the propagation direction of the wae. () Longitudinal wae Rays Waefronts represented by ripples Rays direction of wae otion, perpendicular () to the waefronts 3 side to side otion Sending a longitudinal wae along a spring. Each eleent of the spring ibrates parallel to the propagation direction of the wae. Plane wae Spherical wae Waefronts represented by planes, spaced one waelength apart. Rays direction of wae otion, to the waefronts, not obserable! 4

2 Extra inforation Waes with both longitudinal and transerse otions Water waes In a water wae, all particles trael in clockwise circles. (see the yellow dots) Rayleigh surface waes a type of waes during earthquake, oing in elliptical paths particles at the surface trace out a counter clockwise ellipse; while particles at a depth could trace out clockwise ellipses (see the yellow dots) Water waes Rayleigh surface waes 5 Transerse waes Transerse waes the displaceent of a point on the string is perpendicular () to the direction of the traelling wae up down otion A single pulse is sent along a stretched string. A typical string eleent (see the dot) oes up and then down once as the pulse passes. eleent s otion wae s direction transerse wae ideal wae for (true for EM and atter waes, approxiate for echanical waes) A sine wae is sent along the string. A typical string eleent oes up and then down continuously as the wae passes. eleent s otion wae s direction 6 Longitudinal waes Equation of wae propagation Longitudinal waes the displaceent of a particle is parallel ( // ) to the direction of trael of the wae side to side otion of the particle A sound wae is set up in an air filled pipe by oing a piston back and forth. The oscillation of an eleent of the air is back and forth as well. eleent s otion // wae s direction longitudinal wae Suppose that at tie t = 0, a traelling wae has the for y x,0 y sinkx, where y is the aplitude and k is the waenuber. At tie t, the traelling wae will hae the sae for, except that it is displaced along the positie x direction by a displaceent t, where is the wae speed. Hence the displaceent at position x and tie t is gien by, sin y x t y k x t This is usually written as for sinusoidal waes yx, t y sinkxt, where k or. k phase Wae equation a function of position and tie which gies the height of the wae at any position x and any tie t t 7 8

3 Waenuber k Angular frequency ω Suppose that at t = 0, a traelling wae has the for At x = 0, the wae function becoes y( x,0) y sinkx y(0, t) y sint Since the waefor repeats itself when displaced by one waelength (λ), ysin kx ysin k x ysin kxk Thus, k =, which gies the waenuber k related to waelength (i.e. distance for one wae cycle) k is also called angular waenuber. Siilar to angular frequency T related to period (i.e. tie for one wae cycle) Since the waefor repeats itself when delayed by one period (T), ysint ysin tt ysin tt Thus, ωt =, which gies the angular frequency Unit: rad/s T, and the frequency f. T Unit: /s = Hertz = Hz 9 0 Wae speed ν Wae equation Since k and T, the wae speed is A wae traelling towards positie x direction is described by f k T The wae traels by a distance of one waelength in one period. Since y(x, t) = y sin(kx t), the peak of the traelling wae is described by kx t In general, any point on the waefor, as the wae oes in space and tie, is described by: kxt constant yxt (, ) ysin k x t ysin kx t A point on the waefor, as the wae oes in space and tie, is described by kx t = constant. A wae traelling towards negatie x direction is described by yxt (, ) ysink x t ysin kx t A point on the waefor, as the wae oes in space and tie, is described by kx + t = constant.

4 Exaple A transerse wae traelling along a string is described by yxt (, ) sin 7.x.7t, in which the nuerical constants are in SI units. (a) What is the aplitude of this wae? (b) What are the waelength, period, and frequency of this wae? (c) What is the elocity of this wae? (d) What is the displaceent y at x =.5 c and t = 8.9 s? (e) What is the transerse elocity u of this eleent of the string at the place tie in (d)? (f) What is the transerse acceleration a y at the position and tie in (d)? Answers of Exaple (a) y = = 3.7 (b) (c) (d) (e) (f) c, k 7. k cs 7. y u y cos kx t t cos s T.3 s,.7 yxt (, ) sin u a sin y y kx t y t s assue sinusoidal waes y( xt, ) sin 7.x.7t y k ω f Hz T Section Suary Transerse waes - displaceent of a point wae propagation direction Longitudinal waes - displaceent of a point // wae propagation direction Equation of wae propagation (towards +e x direction) y( xt, ) y sin[ k( x t)] or yxt (, ) y sin( kxt) Equation of wae propagation (towards -e x direction) Waenuber, angular frequency, frequency, wae speed k yxt (, ) y sin( kxt) T f f T k Wae speed on a Stretched String Consider the peak of a wae traelling fro left to right on the stretched string. If we obsere the wae fro a reference frae oing at the wae speed, the peak becoes stationary, but the string oes fro right to left with speed. Find the speed. 5 6

5 Wae speed on a Stretched String Consider a sall segent of length l at the peak. Let be the tension in the string. Vertical coponent of the force on the eleent: both sides sall θ l F sin, where R is the R radius of curature. Mass of the segent: Centripetal acceleration in oing reference frae: l a R Using Newton s second law, F=a, This reduces to l R μ = Δ / Δl = ass per unit length = linear density l R stretching = lengthening causes an elastic restoring force Note that represents the elastic property of the stretched string, and represents its inertial property. 7 Exaple Two strings (String and String ) hae been tied together with a knot and then stretched between two rigid supports. The strings hae linear densities = kg and =.8 04 kg. Their lengths are L = 3 and L =, and String is under a tension of 400 N. Siultaneously, on each string a pulse is sent fro the rigid support end, towards the knot. Which pulse reaches the knot first? Analysis start fro sipler case (single string, constant density, sae tension); speeds, eeting point? Different strings effect due to diff. densities or diff. tensions or diff. lengths? Are tensions different? Approach calculate wae speeds on both strings, then calculate arrial ties at knot. 8 Answer of Exaple L t L L t L s 3 Thus, the pulse on string reaches the knot first. s Energy and Power of a Traelling String Waes Kinetic energy Consider a string eleent of ass d. Kinetic energy: dk du Since y(x, t) = y sin(kx t), y u y cos kx t t μ = ass per Since d = dx, unit length dk dxy cos kx t Rate of kinetic energy transission: dk dx y cos kxt dt dt power 9 0

6 Energy and Power of a Traelling String Waes Using = dx/dt, dk y cos kx t dt Kinetic energy is axiu at the y = 0 position. Energy and Power of a Traelling String Waes Potential energy Potential energy is carried in the string when it is stretched. Stretching is largest when the displaceent has the largest gradient. Hence, the potential energy is also axiu at the y = 0 position This is different fro the haronic oscillator, in which the energy is consered. Consider the extension s of a string eleent. (, ) (, ) s dx y xdx t y xt dx dy y y dx dx dx dx x x slope Energy and Power of a Traelling String Waes Energy and Power of a Traelling String Waes Using power series expansion, ( ( + a ) ~ + ½ a for sall a ) Since = dx/dt and k = / = /, y y s dx x x The potential energy of the string eleent is gien by the work done in extending the string eleent, use work dw = force τ distance Δs y du dw s dx k y cos kx tdx x dx du dk y cos kxt dt dt Mechanical energy (power): Aerage power of transission: de dk du cos y kxt dt dt dt de P y cos kx t dt Rate of potential energy transission:, where represents aeraging oer tie. du dt dx k y cos kxt dt power Since cos (kx t) = /, aerage power: P y 3 4

7 Energy and Power of a Traelling String Waes This result can be interpreted in the following way. Consider the front of a propagating wae along a string. In a tie dt, a string eleent of length dx = dt is set into a siple haronic otion. Its elocity aplitude is y. Aerage power: d = dx de d y Energy of the string eleent: fro E = ½ ν de dx P y y dt dt as before Exaple 3 A string has a linear density of 55 g/ and is stretched with a tension of 45 N. A wae whose frequency f and aplitude y are 0 Hz and 8.5, respectiely, is traelling along the string. At what aerage rate is the wae transporting energy along the string? f ( )(0) rads Siply applying the equations s 0.55 (0.55)(9.58)(754.0) (0.0085) 99.8 W P y 5 6 Section Suary Wae speed of a stretched string: τ elastic property, tension μ inertial property, linear density (i.e. ass per unit length) Kinetic energy is axiu at the y = 0 position. Potential energy is also axiu at the y = 0 position. Aerage power: P y Principle of Superposition of Waes Oerlapping waes algebraically add to produce a resultant wae. Note: not deriatie, only su of waes y'( x, t) y ( x, t) y ( x, t) Oerlapping waes do not in any way alter each other. True for sall aplitudes, but tall waes change each other. e.g. crashing water waes! 7 8

8 Interference of Waes Suppose we send two sinusoidal waes of the sae waelength and aplitude in the sae direction along a stretched string. y (, ) sin x t y kxt y (, ) sin x t y kxt is called the phase difference or phase shift between the two waes. Cobined displaceent: y'( x, t) y sin kx t y sin kx t delay Interference of Waes Using the trigonoetric identity, we obtain sin sin sin cos y'( x, t) y cos sin kxt aplitude as before phase shift The resultant wae () is also a traelling wae in the sae direction () has a phase constant of / (3) has an aplitude of y = y cos(/) 9 30 Interference of Waes Interference of Waes double height Fully constructie interference If = 0, y'( x, t) y sin kxt See fig. (d) Fully destructie interference If =, y ( x, t) 0 (zero aplitude) See fig. (e) If is between 0 and or between and, Interediate interference the aplitude is interediate. See fig. (f) (axiu aplitude) 3 3

9 Exaple 4 Two identical sinusoidal waes, oing in the sae direction along a stretched string, interfere with each other. The aplitude y of each wae is 9.8, and the phase difference between the is 00. (a) What is the aplitude y of the resultant wae due to the interference, and what is the type of this interference? (b) What phase difference, in radians and waelengths, will gie the resultant wae an aplitude of 4.9? Answers to Exaple 4 (a) (b) o 00 y' ycos ()(9.8 )cos 3 y' y cos 4.9 ()(9.8) cos cos two possible solutions rad.6 rad ()(9.8) = +.6 rad: The second wae leads (traels ahead of) the first wae. =.6 rad: The second wae lags (traels behind) the first wae..636 In waelengths, the phase difference is 0.4 waelength Standing Waes A Two opposite traeling waes add to each other, foring a standing wae: wae Standing Waes Superposition of two waes of equal waelength and equal aplitude, traelling in opposite directions A wae wae su antinodes A su nodes A su = A + A 35 36

10 y (, ) sin( ) xt y kx t y (, ) sin( ) xt y kx t Cobined displaceent: Standing Waes Consider two sinusoidal waes of the sae waelength and aplitude traelling in the opposite direction along a stretched string. y '( x, t) y sin( kx t) y sin( kx t) Using the trigonoetric identity, sin sin sin cos we obtain y' ( x, t) [y sin kx]cost x and t are now decoupled! standing wae 37 Standing Waes Properties: () The resultant wae is not a traelling wae, but is a standing wae. e.g. the locations of the axia and inia do not change. () There are positions where the string is peranently at rest. They are called nodes, and are located at sin(kx) = 0 kx n for n 0,,, n x n for n 0,,, k (3) There are positions where the string has the axiu aplitude. They are called antinodes, and are located at sin(kx) = kx for n 0,,, 3 5,,, n The nodes are separated by half waelength. x n for n 0,,, The antinodes are separated by half waelength. 38 Standing Waes Energy in standing waes: - does not trael - exchanges between kinetic energy K and potential energy U start here: K=0 Reflections at a Boundary Fixed end: ()The fixed end becoes a node. ()The reflected wae ibrates in the opposite transerse direction. Free end: ()The free end becoes an antinode. ()The reflected wae ibrates in the sae transerse direction. At the fixed end: Displaceent = 0 wall pulls string down transerse pulse string pulls string down At the free end: Displaceent gradient = 0 See aniation Reflection of Waes in Physics

11 Section Suary Standing Waes and Resonance Two sinusoidal waes of the sae waelength and aplitude along a stretched string: Traelling in the sae direction still traelling wae y '( x, t ) y cos sin kx t oscillator oscillator Circular shape: soewhat coplicated, special ath functions loops: Fully constructie interference: = 0 axiu aplitude Fully destructie interference: = zero aplitude Traelling in the opposite direction standing wae, energy does not trael, K.E. P.E. exchanges nodes: always at rest antinodes: ax. aplitude x 3 loops: Rectangular shape: sipler: like two linear patterns superposed y ' ( x, t ) [ y sin kx] cos t n n k x n *Fixed end node out-of-phase reflection 4 loops: for n 0,,,, 5, 6, 7, loops also possible for n 0,,, *Free end antinode in-phase 4 Standing Waes and Resonance 4 Standing Waes and Resonance See Youtube Resonance Phenoena in D on a Plane and Milleniu Bridge Opening. Consider a string with length L stretched between two fixed ends. Case (a): nodes at the ends, antinode in the iddle Boundary condition: nodes at each of the fixed ends. L, When the string is drien by an external force, at a certain frequency the standing wae will fit this boundary condition. L. Resonant frequency: f. L Case (b): 3 nodes and antinodes Then this oscillation ode will be excited. L. The frequency at which the oscillation ode is excited is called the resonant frequency. Resonant frequency: f. L Case (c): 4 nodes and 3 antinodes L See aniation Standing Waes Deo 3, L. 3 Resonant frequency: 43 f 3. L 44

12 Standing Waes and Resonance In general, L n, L n, Resonant frequency: f n L, n = n = n = 3 etc. for n =,, 3, for n =,, 3, fundaental ode, or first haronic second haronic third haronic Exaple 5 A string of ass =.5 g and length L = 0.8 is under tension = 35 N. (a) What is the waelength of the transerse waes producing the standing-wae pattern in Fig. 6-5, and what is the haronic nuber n? (b) What is the frequency f of the transerse waes and of the oscillations of the oing string eleents? (c) What is the axiu agnitude of the transerse elocity u of the eleent oscillating at coordinate x = 0.8? (d) At what point during the eleent s oscillation is the transerse elocity axiu? Answers to Exaple 5 (a) (b) 0.8 L 0.4 Since there are four loops, n = s 0.005/ f Hz 0.4 L = Answers to Exaple 5 (continued) we need the waefor for general location x, not a special location like node or antinode (c) y' ( x, t) [y sin kx]cost where y = y u( x, t) (y sin kxcost) t t sin kxsint y Magnitude: u y sin kx Here, f ( )(806.) At x=0.8, u k 0.4 ( )(806.)(0.00)sin (0.8) 6.6 s 0.4 (d) The transerse elocity is axiu when y =

13 Exaple 6 In the arrangeent of Fig. 8-3, a otor sets the string into otion at a frequency of 0 Hz. The string has a length of L =., and its linear ass density is.6 g/. To what alue ust the tension be adjusted (by increasing the hanging weight) to obtain the pattern of otion haing four loops? this is a resonance ode with 4 loops Answers to Exaple 6 F F = μ equation () fn n ( n,,3,...) = Lf L n /n n equation () To find the tension, we can substitute Eq. () into Eq. () and obtain 4Lfn F n The tension corresponding to n = 4 (for 4 loops) is found to be F 4. 0 Hz kg/ 8.3 N Traelling waes Suary of equations y( xt, ) y sin[ k( x t)] or yxt (, ) y sin( kxt) Waenuber Angular frequency Frequency k f T T Wae elocity Traelling wae (opposite direction) f k y( xt, ) y sin( kx t) Transitted power Interference Standing wae Suary of equations P y y' ycos yxt (, ) [ y sin kx]cost Reflection at fixed end node, oscillation at opposite transerse direction Reflection at free end antinode, oscillation at sae transerse direction Stretched string Vibrating string (fixed ends) f n L 5 5