General Physics IGeneral Physics I

Lecture 15: Waves: From Lecture 15: Waves: From Sound to LightSound to Light

Prof. WAN, Xin （万歆）

xinwan@zju.edu.cnhttp://zimp.zju.edu.cn/~xinwan/

OutlineOutline

● Sound frequency and sound level● Speed of sound waves● The Doppler effect and shock waves● Measuring the speed of light

− Galileo's attempt− Ole Roemer and the speed of light− James Bradley and the aberration of light

● Does the speed of light depend on the motion of the light source?

Categories of Sound WavesCategories of Sound Waves

Sound IntensitySound Intensity

● We define the intensity I of a wave, or the power per unit area, to be the rate at which the energy being transported by the wave flows through a unit area A perpendicular to the direction of travel of the wave.

amplitude

frequency

density

speed of sound

Definition of Sound LevelDefinition of Sound Level

● Because the range of sound intensities is so wide, it is convenient to use a logarithmic scale, where the sound level is defined by the equation

● Threshold of pain:

(threshold of hearing)

Sound LevelsSound Levels

Speed of Sound in a SolidSpeed of Sound in a Solid

● If a solid bar is struck at one end with a hammer, a longitudinal pulse propagates down the bar with a speed

where Y is the Young’s modulus for the material and the density of material.

Elasticity in LengthElasticity in Length

● Young’s Modulus:

A Wave of AtomsA Wave of Atoms

Collective motion of the atoms oscillating around their equilibrium positions are wave-like excitations, also known as phonons.

Wave is different from oscillation in the sense that the particles oscillate as a function of both time and position.

Model of Monoatomic Crystal Model of Monoatomic Crystal

● One-dimensional example:

un−1 un un+1

a

Equilibrium positions: Xn=na

Deviations from the equilibrium: un=xn−Xn

Interatomic PotentialInteratomic Potential

● We consider nearest-neighbor interactions only.

ϕ(xn+1−xn) = ϕ0 +12K (xn+1−xn−a)

2+ ⋯

= ϕ0 +12K (un+1−un)

2+ ⋯

In the harmonic approximation,

Uharm=

12K∑

n

(un+1−un)2

First derivative vanishes at the equilibrium!

Equations of MotionEquations of Motion

Mun = −dUharm

dun

= K [un+1−un ]−K [un−un−1 ]

= (Ka)ΔuΔ x∣Xn+

a2

−(Ka)ΔuΔ x∣Xn−

a2

For wavelength much greater than a,

M ( ∂2u

∂ t2 )Xn

= (Ka) [ ( ∂u∂x )Xn+a2

−( ∂u∂x )Xn−a2

]= (Ka2

) ( ∂2u

∂x2 )Xn

The Linear Wave EquationThe Linear Wave Equation

∂2u

∂ t2= v2

∂2u

∂x2

The linear wave equation

v=a√ KM

where

The linear wave equation applies in general to various types of waves. For waves on strings, u represents the vertical displacement of the string. For sound waves, u corresponds to displacement of air molecules from equilibrium or variations in either the pressure or the density of the gas through which the sound waves are propagating. In the case of electromagnetic waves, u corresponds to electric or magnetic field components.

Now, ...Now, ...

● Can you show that

are two equivalent forms of the speed of sound in the solid?

v = a√ KM

(macroscopic) (microscopic)

Speed of Sound in a LiquidSpeed of Sound in a Liquid

● The speed of all mechanical waves follows an expression of the general form

We will, hopefully, come back to this issue in the part of thermodynamics for a complete understanding.

Galilean Transformation Galilean Transformation

The two inertial observers agree on measurements of acceleration.

Moving ObserverMoving Observer

We take the frequency of the source to be f, the wavelength to be , and the speed of sound to be v.

Spherical WavesSpherical Waves

● The wave intensity at a distance r from the source is

● The intensity is proportional to the square of the amplitude. Hence,

Analyze the Moving ObserverAnalyze the Moving Observer

● The speed of the waves relative to the observer is

● The wavelength is unchanged.

Positive vO for observer moving toward source, and

negative vO for observer moving away from source.

Moving SourceMoving Source

During each vibration, which lasts for a time T (the period), the source moves a distance

Analyze Moving SourceAnalyze Moving Source

● For observer A, the wavelength is shortened to

● The frequency heard by observer A is

● For observer B, simply use a negative vS.

Doppler EffectDoppler Effect

● Finally, if both source and observer are in motion, we find the following general relationship for the observed frequency:

f ' =v+vOv−vS

f

The word toward is associated with an increase in observed frequency. The words away from are associated with a decrease in observed frequency.

Echocardiogram (ECG)Echocardiogram (ECG)

Shock WavesShock Waves

Mach number: vs / v

(vS > v )

Measuring the Speed of LightMeasuring the Speed of Light

AliceBob

Speed of light = Time for the light to arrive at Alice

Distance from the flashlight to Alice

The same for Bob.

Galileo's AttemptGalileo's Attempt

Can Galileo and his assistant measured the speed of light in this way? Why?

(Genesis 1:3) And God said, "Let there be light," and there was light.

Measuring the Speed of SoundMeasuring the Speed of Sound

牵牛看见那边的山顶了吗？看到了你说站在那边能听得到我说话吗？…恩 也许能也许不能吧那你过去﹐我站在这里喊如果你听得见﹐那就回答我要我到那边去？是啊

Jeon Ji-Hyun in My Sassy Girl (2001)

Greek Myth: Jupiter and IoGreek Myth: Jupiter and Io

Planet Jupiter and Its Moon IoPlanet Jupiter and Its Moon Io

Io is the innermost of the four Galilean moons of the planet Jupiter. Orbital period ~ 42 hours

Jupiter orbital period ~ 12 years

Galileo spacecraft true-color image of Io.

Galileo was an unmanned spacecraft launched on October 18, 1989 by Space Shuttle Atlantis. Galileo arrived at Jupiter on December 7, 1995.

Earth-Jupiter DistanceEarth-Jupiter Distance

Ole Roemer and the Speed of LightOle Roemer and the Speed of Light

● Jupiter is stationary. 12 years >> 42 hours.

● Roemer found that it takes 11 minutes for light to travel from the Sun to Earth.

● What is the speed of light based on this observation?

Ole Roemer (1644–1710)

Aberration of LightAberration of Light

James Bradley (1693-1762)

Walking in the RainWalking in the Rain

tanθ =vV

θ

Gene Kelly knew this in Singin' in the Rain (1952).

Moving Source or Observer?Moving Source or Observer?

stationary

Speed of light = Time for the light to arrive at Alice

Distance from the flashlight to Alice

But what if Alice is moving relative to the light source?

v

A Good Experiment is NeededA Good Experiment is Needed

● Does the speed of light depend on the motion of the source of light?

The 1964 ExperimentThe 1964 Experiment

Beryllium target

Proton

Gamma rays

B A

Detector locations

31 m apart

T. Alvaeger et al., Phys. Lett. 12, 260 (1964)

The proton-beryllium collisions generated neutral pions with speed exceeding 0.99c (i.e., moving light source), which decayed into gamma rays.

The Speed of Gamma RaysThe Speed of Gamma Rays

● The speed of gamma rays from neutral pions

Separation of A and B

t (production to A) – t (production to B)

= 2.9979 108 m/s

The same as that from a stationary source!

Constancy of the Speed of LightConstancy of the Speed of Light

v stationary

Although Alice move with speed v relative to Bob, she notes the same speed for the light from Bob's flashlight as Bob does.

The ContrastThe Contrast

v stationary

Alice notes a larger speed of the bullet than Bob does.

Next...Next...

● We march to the theory of relativity.