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Lecture 6

• 2D/3D and and Intensity • Doppler effect for (i) mechanical waves e.g. sound (ii) EM waves Example • A 2.31 kg rope is stretched between supports 10.4 m apart. If one end of the rope is tweaked, how long will it take for the resulting disturbance to reach the other end? Assume that the tension in the rope is 42.8 N. 2D/3D waves

• 2D circular waves: wavefronts (lines locating crests), small section appear as straight lines far away • 3D spherical waves...appear as planes far away, described by D(x, t) (same at every point in yz plane) 2D/3D waves

D(r, t)=A(r) sin (kr ωt + φ0) with A(r) decreasing with−r

Phase and phase difference

phase, φ = kx ωt + φ0 D(x, t)=A sin φ − • wavefronts are surfaces of same displacement constant phase

∆x phase diﬀerence, ∆φ =2π λ ∆φ =2π between adjacent wavefronts (separated by λ) Sound waves

vsound in air at 20◦ = 343 m/s (larger in liquid/solid) • human ears: 20 Hz to 20 k Hz

• ultrasound: > 20 k Hz Electromagnetic (EM) waves • oscillations of EM ﬁeld, can travel in vacuum e.g. light from stars v = c =3 108 m/s in vacuum light × ( vsound) • "visible : 400 nm (violet/blue to 700 nm (orange/red)

λ f f ! sound ⇒ light # sound • EM spectrum: visible + higher (UV/X rays) + lower frequencies (IR/micro/ waves) • index of refraction (light slowed down): in vacuum n = = c speed of light in material v does not change (e.g., f = c = f = vmat. vac. λvac. mat. λmat. sound hitting water): λmat. !< λvac. " ! " ⇒ Power and Intensity • Power is rate of transfer of energy by wave • Brightness/loudness depends also on receiving power:

P intensity, I = a = power-to-area ratio (SI units: W/m2)

• Uniform spherical wave Psource I = 4πr2 (from energy conservation: total energy crossing wavefront is same) 2 I1 r2 = 2 I2 r1

I A2 ∝ 1 2 (energy of oscillations E = 2 kA ) Decibels (for wide range of human hearing)

β = 10 dB log I (dimensionless) I0 ! " 12 2 10− W / m (threshold) • β =0 at threshold • increasing by 10 dB I increases by a factor of 10 Example • The ``planet’’ Pluto’s average distance from the sun is 5 . 9 10 . 12 m . Calculate the sun’s intensity at the distance× of Pluto, assuming the sun radiates a total power of 4 .0 10 26 W. × Doppler effect • relative motion between observer and wave source modiﬁes frequency e.g. pitch of ambulance siren drops as it goes past moving source: Pablo detects ( λ ,f ) , • − − Nancy detects ( λ + ,f + ) vs. ( λ 0 ,f 0 ) if source at rest Doppler effect:derivation • motion of wave crest (once leaves source) governed by medium (not affected by source moving) wave crests bunched up in front/ λ+ < λ0 < λ + speed v f+ >f0 >f stretched out behind: − ⇒ −

In time t =3T , source moves 3vsT , wave (crest 0) moves 3vT 3 wave crests in ⇒ 3vT 3vsT ...= 3λ ∓ ± v f = ± λ ±v = v v ∓ s Doppler effect for moving source

Doppler effect: moving observer • not same as source moving: motion relative to medium (not just source vs. observer) matters... Doppler effect for EM waves

• no medium: use Einstein’s theory of relativity

1+vs/c λred = 1 v /c − s (receding! source: longer wavelength, red shift) 1 v /c λ = − s blue 1+vs/c (approaching! source: shorter wavelength, blue shift) where vs is the speed of the source relative to the observer