Doppler Effect

Physics 202Professor Lee

CarknerLecture 11

PAL #10 Music How much would your eardrum move

from a tuning fork sound? Example: f = 440 Hz, = 90 dB

= (10 dB) log (I/I0)

I = I0 10(/10)

I = I = 1X10-3 W/m2

We need to relate I to sm:I = ½ v2sm

2

sm = Air density = = 1.21 kg/m3

Velocity of sound = v = 343 m/s

PAL #10 Music (cont.)

sm = (I/(½ v(2f)2))½

sm = (1X10-3/(½)(1.21)(343)(2440)2)½

sm = Even the loudest sounds only produce

very small motions What if the distance is doubled?

Since I = Ps/4r2, then but sm => √I, so The displacement is ½ as great

The Doppler Effect

If there is any relative motion between the two, the frequency of sound detected will differ from the frequency of sound emitted

Stationary Source

Moving Source

How Does the Frequency Change?

If the source and the detector are moving closer together the frequency increases

If the source and the detector are moving further apart the frequency decreases

Doppler Effect

Doppler Effect and Velocity

The greater the change the larger the velocity

Let us consider separately the situations where either the source or the detector is moving and the other is not

Stationary Source, Moving Detector

In general f = v/ but if the detector is moving then the effective velocity is v+vD and the new frequency is:

but =v/f so,

If the detector is moving away from the

source than the sign is negative

Moving Source, Stationary Detector

In general = v/f but if the source is moving the wavelengths are smaller by vS/f

’ = v/f - vS /f

f’ = v / (v/f - vS/f)

The the source is moving away from the detector then the sign is positive

General Doppler Effect We can combine the last two equations and

produce the general Doppler effect formula:f’ = f ( v±vD / v±vS )

What sign should be used?

For motion toward the sign should be chosen to increase f

Remember that the speed of sound (v) will often be 343 m/s

The Sound Barrier A moving source of sound will produce

wavefronts that are closer together than normal

At the speed of sound the wavefronts are all pushed together and form a shockwave called the Mach cone

This is dangerous because passing through the shockwave makes the plane hard to control

Doppler Effect for Light

However, at low speeds (u<<c, where u is the relative velocity between source and detector) the equations reduce to the classical form:

f’ = f (1 ± u/c)

u = () c c, the speed of light in vacuum, is constant (3 X 108 m/s)

Spectral Line Shifts

When we observe a spectrum of a object, we compare the observed wavelengths to standard ones

For objects moving away from us the spectral lines move to larger wavelengths

For objects moving towards us the spectral lines move to shorter wavelengths

Red Shifted Spectrum

Expansion of the Universe

All galaxies are moving away from all others

In the past, everything in the universe must have been much closer together

Summary: Sound Waves

Sound waves are longitudinal or pressure waves

The medium oscillates in the direction of travel

The speed of sound depends on the density and the bulk modulus (compressibility ) of the medium:

v = (B/)½

Summary: Wave Equations

The equations for the amplitude and pressure of a sound wave are:

s = sm cos (kx-t)

p = pm sin (kx-t)

pm = (v) sm

Waves from two sources will interfere based on the path length difference between the sources and detector

L = m (fully constructive)L = (m+½) (fully destructive)

Summary: Intensity and Music The intensity of sound falls off with a inverse

square law:I = Ps/4r2

I =½v2sm2

The sound level is: = (10 dB) log (I0/I)

Harmonic frequencies of a pipef = nv/2L (open at 2 ends)f = nv/4L (open at 1 end)

Beat frequency = fbeat = f1 - f2

Summary: Doppler Effect Relative motion together produces

an increase in frequency Relative motion apart produces a

decrease in frequencyf’ = f ( v±vD / v±vS )

For light:u = () c