Wave Phenomena Physics 15c

Transcription

1 Wae Phenomena Physics 15c Lecture 14 Spherical Waes (H&L Chapter 7) Doppler Effect, Shock Waes (H&L Chapter 8)

2 What We Did Last Time! Discussed waes in - and 3-dimensions! Wae equation and normal modes easily extended ξ ξ ξ ξ t x y z = c w + + = c w! Their forms satisfy isotropy and relatiity! Studied boundary conditions in -D and 3-D! Rectangular drum, Chladni plate, sound in a room! Natural extension of the 1-D problems such as guitar strings and open-ended pipes ξ ξ i( k x+ k y+ k z ωt) i( kx ωt) x y z = e = e Plane waes

3 Goals For Today! Spherical Waes! Multi-dimensional waes from a small source! We know it spreads out Exactly how?! Doppler Shift! Waes generated by a moing source! What if the obserer (listener) is moing?! Shock Waes! When the source moes faster than the waes

4 Waes From a Point Source! Plane waes were easy to handle! But how do we make them?! We need, e.g., an infinitely large, flat speaker! Most waes are generated by a small object! Size of the source << Distance of transmission! Voice from a person s mouth! Radio signals from a cellular phone! Light from the Sun! We can approximate them as a point source! How do we describe waes spreading from a point?

5 Spherical Waes! Consider 3-D waes expanding from! Assume it is isotropic: No dependence on the direction ξ = ξ (,) rt! Non-dispersie wae equation is ξ(,) rt ξ ξ ξ = c (,) w + + c = w ξ r t t x y z Work on this term first x= y = z = 0 r

6 Spherical Waes ξ(,) rt r x x x r ξ = x ξ = x r r r x x = x + y + z = = x x x + y + z r 1 x ξ x ξ = 3 + Next: add up x, y, and z terms r r r r r 3 x + y + z ξ x + y + z ξ r r r r r ξ (,) rt = + 3 ξ ξ = + r r r ( ξ) 1 = ( rξ ) r r r ξ r ξ = ξ r ξ + = + r r r r r

7 Wae Equation ξ (,) rt 1 c r r t t r r = w ( ξ (,)) t! But this is a 1-D wae equation for rξ(r, t)!! We know the solutions already e ξ (,) rt = A i( kx± ωt) r ( rξ(,) r t ) = c ( rξ(,) r t ) w rξ (,) r t Ae! Sinusoidal waes with amplitude decreasing as 1/r! Either expanding from or concentrating to the origin r ω = ck i( kx± ωt) = w

8 Wae Intensity! Intensity = energy flow density! How much energy per unit area is being transmitted! Always proportional to (amplitude)! For spherical waes, intensity falls off as 1/r! If you integrate oer a sphere at r = R, total energy flow is 4π R C = const. R Energy conseration R

9 Spherical Waes s. Normal Modes! Spherical waes and normal modes don t look related ξ (,) rt e i( kx± ωt) i( kx ωt) = r ξ ( xyzt,,, ) = i( kx ωt)! Try adding up e for a gien k π 0 0 π e π e ik i( krcos θ ωt) iωt + 1 iωt ikrs = π e e ds = π e = 1 1 e ikr e r sinθdθdφ ikrs i( kr ωt) i( kr ωt) s cosθ e Integrate oer all direction of k Combination of expanding and shrinking waes Of course they are r θ k

10 Spherical Harmonics! Our spherical waes are isotropic! Real waes generated by small source are often directional! Speakers emit stronger sound forward than to the sides! Een a round one like this "! Right way of dealing with this is iωt ξ = e j ( kr) Y ( θφ, ) n n Bessel function Spherical harmonics function! This gies a complete set of solutions in polar coordinates! n = 0 corresponds to our isotropic solution! You ll learn this in QM for hydrogen atoms

11 Doppler Shift! In 1845, Christian Doppler did an experiment! He hired a freight train and the trumpet section of the Vienna orchestra for two days! Half of the players got on the train and played an E b! The other half did the same in the station! Musicians could tell the difference between the notes

12 Moing Source! Source of the sound moes at s! Original frequency is f 0! Sound waes traels at c w! We follow two peaks of the sound waes! They are generated T = 1 f0 apart! Train moes T in the meantime s ( cw s) T! Arrial times at the listener differ by cw s T Turn into cw f = f0 c frequency c! The distance between two rings is w w s s T s c ( ) w t+ T ct w forward + backward

13 Moing Obserer! Now we moe the obserer at o! Sound source is at rest! Follow two peaks again! Spacing between them is ct! To the obserer, sound seem to approach with elocity cw o! Arrial times differ by cw T cw Turn into o c frequency f = w moing away + approaching w c w o f 0 o ct w

14 Moing Source And Obserer! Moe both the source and the obserer! Combine the preious results cw o f = f0 c w s! Define direction of + s and + o as the direction of the sound csound obserer f = f0 Doppler shift for sound c sound source s o

15 Moing at an Angle! What if s and o are not parallel to the sound?! Just take the component that is parallel to the sound cw ocosθo f = f0 c cosθ w s s θs s θo o Pretty straightforward

16 Medium! Why does it matter who (source/obserer) is moing?! Wae (sound) elocity is constant relatie to the medium! Medium (air) gies an absolute reference of elocity! We must measure s and o relatie to the air! If there is wind (air is moing), you must add its effect! Example: moing obserer + constant wind Source Wind w Obserer o f = c + w o w c w + w f 0

17 Electromagnetic Waes! EM waes (light, radio waes, etc.) is special! There is no medium " No absolute reference! Whether source moes or obserer moes shouldn t matter! That s what s relatie about Relatiity! Suppose source and obserer are approaching Moing source! We can consider it in two ways f Stationary obserer! The rule must be different Stationary source = c c+ f0 Not the same f = f c c 0 Moing obserer

18 Relatiistic Doppler Shift! Keyword: time dilation! From the obserer s point of iew! Source is moing at, and I am not! Source s clock runs slower by! Which makes frequency smaller by ( ) 1 c! Multiply this with the usual Doppler shift 1 c c f = f = f c c c Doppler shift of light

19 Relatiistic Doppler Shift! From the source s point of iew! Obserer is moing at, and I am not! Obserer s clock runs slower by the factor! Which makes him think that the frequency is higher by! Multiply this with the usual Doppler shift 1 c+ c+ f = f = f c c 1 c ( ) ! Who (source/obserer) is moing is irreleant c ( c ) 1 c Same conclusion

20 Relatiistic Doppler Shift! Time dilation occurs not only with light! One should always take it into account! In practice, it matters only when is ery large! Neer an issue with sound (and other) waes! Time dilation does not depend on the direction of θ! For θ = 90 f c c f = 1 f = f = 1 f c c c cosθ c cosθ Time dilation Moing source at angle θ Time dilation only

21 Expanding Unierse! We know the Unierse is expanding! Distant stars are moing away from us! Doppler shift makes their light lower in frequency! Longer in waelength toward red in isible spectrum! Called redshift! Use well-known spectral lines as references! Lyman-α line of hydrogen makes an ideal yardstick

22 Expanding Unierse! Amount of redshift is expressed by z λ c+ = 1= 1 λ c 0 z = = 0.146c z = 3.6 = 0.910c! Highest-redshift quasars reach z = 5

23 Shock Waes! What happens when a source moes at c w?! The source flies together with the sound waes it is making! Waes pile up Shock Wae! Energy accumulate at the waefront! Intensity " s Waefront Bullet flying at 1.01 c w

24 Breaking the Sound Barrier F/A-18 Hornet passing the sound barrier

25 Mach Cone! Once the object is faster than sound, the shock wae turns into a cone! Opening angle θ gien by cw sinθ =! Energy concentrates on the cone surface s 4cT w 3 ct w ct w T s T s! Where the cone touches the ground, obserer hears a (super)sonic boom T s ct w T s! Another reason why supersonic aircrafts are unpopular s

26 Čerenko Effect! Can shock waes exist for light?! Nothing traels faster than light in acuum! In material, light slows down c c n Index of refraction > 1! n = 1.33 for water! Object with > 0.75c makes shock waes of light in water Called Čerenko light! β-ray electrons from nuclear reactors create blue glow Reed Reactor generating 40 kw

27 Čerenko Detector! Čerenko light is often used for detection of high energy particles! Example: DIRC detector in BABAR experiment! Particles fly through rectangular bars of quartz (n = 1.544)! Those with > 0.648c emit light! Angle of light measured " Velocity of the particle! Combine with momentum measurement " Mass! Particle identification

28 How Well Does It Work? pion (m = 0.14 GeV/c ) kaon (m = 0.49 GeV/c )

29 Summary! Studied spherical waes! Wae equation of isotropic waes! Solution e! Intensity decreases with! Doppler shift i( kx±ωt)! Doppler shift for sound and for light! Special Relatiity matters for light! Shock waes and Mach cones! Object flying faster than sound! Čerenko effect for light in material r 1 r ( rξ (,) r t ) = w t c sinθ = f c = c ( ξ (,)) c r r t r sound sound f = obserer source f c+ f c 0 0