Lecture 6

• 2D/3D waves

• Sound and light

• Power 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)

Phase and phase difference

• wavefronts are surfaces of same displacement constant phase

2D/3D waves D(r, t) = A(r) sin (kr − ωt + φ0)

with A(r) decreasing with r

phase difference, ∆φ = 2π ∆xλ

∆φ = 2π between adjacent wavefronts(separated by λ)

phase, φ = kx− ωt + φ0

D(x, t) = A sin φ

Sound waves

• human ears: 20 Hz to 20 k Hz

• ultrasound: > 20 k Hz

vsound in air at 20◦ = 343 m/s(larger in liquid/solid)

Electromagnetic (EM) waves• oscillations of EM field, can travel in vacuum

e.g. light from stars

• visible spectrum: 400 nm (violet/blue to 700 nm (orange/red)

• EM spectrum: visible + higher frequencies (UV/X rays) + lower frequencies (IR/micro/radio waves)

• index of refraction (light slowed down):

frequency does not change (e.g., sound wave hitting water):

vlight = c = 3× 108 m/s in vacuum(" vsound)

! λsound ⇒ flight # fsound

n = speed of light in vacuumspeed of light in material = c

v

fvac.

(= c

λvac.

)= fmat.

(= vmat.

λmat.

)

⇒ λmat. < λvac.

Power and Intensity• Power is rate of transfer of energy by wave

• Brightness/loudness depends also on area receiving power:

• Uniform spherical wave

intensity, I = Pa = power-to-area ratio

(SI units: W/m2)

I = Psource4πr2

(from energy conservation:total energy crossing wavefront is same)I1I2

= r22

r21

I ∝ A2

(energy of oscillations E = 12kA2)

Decibels (for wide range of human hearing)

• at threshold

• increasing by 10 dB I increases by a factor of 10

β = 10 dB log(

II0

)(dimensionless)

10−12 W / m2

β = 0

(threshold)

Example• The ``planet’’ Pluto’s average distance from the sun

is . . Calculate the sun’s intensity at the distance of Pluto, assuming the sun radiates a total power of W.

5.9× 1012 m

4.0× 1026

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 , Nancy detects vs. if source at rest

(λ0, f0)(λ−, f−)

(λ+, f+)

Doppler effect:derivation• motion of wave crest (once leaves source)

governed by medium (not affected by source moving) wave crests bunched up in front/stretched out behind:

In time t = 3T ,source moves 3vsT ,wave (crest 0) moves 3vT⇒ 3 wave crests in3vT ∓ 3vsT ...= 3λ±

f± =v

λ±

=v

v ∓ vs

λ+ < λ0 < λ− + speed v ⇒ f+ > f0 > f−

Doppler effect: moving observer

• not same as source moving: motion relative to medium (not just source vs. observer) matters...

Doppler effect for moving source

Doppler effect for EM waves

• no medium: use Einstein’s theory of relativity

λred =√

1+vs/c1−vs/c

(receding source: longer wavelength, red shift)λblue =

√1−vs/c1+vs/c

(approaching source: shorter wavelength, blue shift)where vs is the speed of the source relative to the observer