Lecture 4 Wave Refraction and Diffraction Waves in Two Dimensions Doppler Effect and Shock Waves Instructor: David Kirkby ([email protected])

Lecture 4Wave Refraction and

DiffractionWaves in Two DimensionsDoppler Effect and Shock

Waves

Instructor: David Kirkby ([email protected])

Physics of Music, Lecture 4, D. Kirkby 2

MiscellaneousProblem set #2 is due at the beginning of class next Thursday (Oct 17).

Homework grading policy: all-or-nothing / partial credit?

Do we need to order more textbooks?

Physics of Music, Lecture 4, D. Kirkby 3

Review of Lecture 3We listened to the sound produced by a Simple Harmonic Motion (SHM) and identified two key features that it is missing to be called a “musical” sound:

•Envelope•Complex vibrations

By comparing a few examples of musical sounds, we found two common features:

•Damping•Transients

Damping is due to dissipation and is described by an exponential decay law.

Physics of Music, Lecture 4, D. Kirkby 4

Transients usually occur at beginning of a sound and are due to explosive initial disturbances that push a sound-producing body beyond its linear response regime.

The Principle of Superposition (PoS) allows us to understand the complex vibrations of a linear system as the combined result of many simple vibrations (modes).

Reflection is a universal feature of wave motion. Reflected waves may or may not be flipped, depending on the boundary conditions.

Physics of Music, Lecture 4, D. Kirkby 5

Wave RefractionWhat if, instead of fixing the end of the rope, we attach another rope of different thickness?

Physics of Music, Lecture 4, D. Kirkby 6

Limiting CasesA powerful and general approach to understanding a complex physical system is to identify its limiting cases.

A limiting case is when some parameter of the system is taken to an extreme value. Limiting cases are often equivalent to simpler systems that are already well understood.

With the limiting cases identified and understood, you can now think of the general case in terms of how to fill in the gaps between limiting cases.

Physics of Music, Lecture 4, D. Kirkby 7

Limiting Cases for Rope Waves(1) If the extra rope is heavy enough, thisIs essentially the same as fixing the end.

(2) If the extra rope is the same as the original rope, the wave passes through the join unaffected.

(3) If the extra rope is much lighter thanthe original rope, this is essentially sameAs leaving the end free (a whip)

Physics of Music, Lecture 4, D. Kirkby 8

No reflection

Transmitted unchanged

Negative reflection

No transmission

Positive reflection

No transmission

???

???

(1)

(2)

(3)

Physics of Music, Lecture 4, D. Kirkby 9

Try this demonstration (or this one) to find out.

What did you learn?

If the extra rope is heavier (slower) than the original rope, the reflected pulse is a negative of the original pulse and smaller.

If the extra rope is lighter (faster) than the original rope, the reflected pulse is the same as the original but smaller.

Physics of Music, Lecture 4, D. Kirkby 10

In addition to the reflected wave, we find a transmitted wave.

The process of transmitting a wave through an interface where the wave speed changes is called refraction.

The refracted wave is always a smaller version of the original pulse (it is never flipped to be a negative pulse).

Physics of Music, Lecture 4, D. Kirkby 11

Reflection & RefractionReflection and refraction are complementary processes that both occur at the boundary between two different media.

The reflection coefficient R measures the amplitude of the reflected wave compared with the incident wave. A negative coefficient indicates a negative reflection.

The transmission coefficient T measures the amplitude of the transmitted wave. It is always positive.

The incident wave is converted entirely into transmitted and reflected waves: T - R = 1

Physics of Music, Lecture 4, D. Kirkby 12

R = -1

T = 0

-1 < R < 0

0 < T < 1

R = 0

T = 1

0 < R < 1

1 < T < 2

R = +1

T = 2

Physics of Music, Lecture 4, D. Kirkby 13

Into the Second DimensionUntil now, we have only considered one-dimensional wave (even when we looked at two-dimensional representations such as the air particles).

How are things different in two dimensions?

The main difference is that you can travel in more than one direction.

Physics of Music, Lecture 4, D. Kirkby 14

Special Cases of 2D SourcesPlane waves are really just one-dimensional waves since the disturbances at different places are all in parallel directions:

Plane waves are a mathematical idealization since theyrequire an infinitely long source.

directionof travel

Physics of Music, Lecture 4, D. Kirkby 15

Circular waves (or spherical waves) originate from a point source and spread out in circles (or spheres):

Circular waves look likeplane waves up close

(another limiting case)

Physics of Music, Lecture 4, D. Kirkby 16

Principle of Superposition in 2DThe PoS holds just as well in any number of dimensions.

Example:

Disturbance A = circular wave centered at (-1,0)

Disturbance B = circular wave centered at (+1,0)

What does the combined wave motion look like?

Physics of Music, Lecture 4, D. Kirkby 17

Visualization of Waves in 2D

http://physics.okstate.edu/hauenst/class/ph2414/suppl/waves2/int.html

See also these visualizations of multipole sources.

Physics of Music, Lecture 4, D. Kirkby 18

Reflection & Refraction in 2DWhen we considered reflection & refraction of transverse waves on a rope, we were only considering one-dimensional propagation at the interface between two media.

A pulse reaching a one-dimensional interface can either bounce back (reflect) and/or keep going (refract).

In two dimensions, a wave can also change its direction of propagation…

Physics of Music, Lecture 4, D. Kirkby 19

What determines any change of a wave’s direction of propagation at an interface?

List the variables in this problem:•The angle at which the incident wave hits the interface

•The wave’s speed before the interface•The wave’s speed after the interface

v(before) v(after)

Physics of Music, Lecture 4, D. Kirkby 20

Reflection in 2DReflection from a “smooth” surface is specular: the angle of incidence equals the angle of reflection.

This simple rule still leads to some complex effects.

For example, a distance light source reflected from a sphere has highlights:

Physics of Music, Lecture 4, D. Kirkby 21

Another complex effect occurs when light on the inside of a circular (or cylindrical) object.

Try this demo: http://www.cacr.caltech.edu/~roy/Caustic/

The resulting pileup ofreflected rays producesa characteristic shapecalled a caustic curve.

Physics of Music, Lecture 4, D. Kirkby 22

Refraction in 2D: Toy AnalogyWhat happens when the toy enters the grass (where its wheels will turn slower)?

Physics of Music, Lecture 4, D. Kirkby 23

First, what happens if the toy hits the grass head on (normal incidence) ?

Both wheels enter the grass and slow down at the sametime. The toy does not change direction. When leaving thegrass, the toy speeds up but again does not change direction.

Physics of Music, Lecture 4, D. Kirkby 24

What if the toy enters the grass at an angle, so one wheel hits the grass and slows down before the other?

During the transition period when one wheel turns fasterthan the other, the toy will rotate.

Physics of Music, Lecture 4, D. Kirkby 25

The amount that the toy rotates depends on:•how different the speeds are in and out of the grass,

•how long the wheels are turning at different speeds

The length of time the wheels are turning at different speeds depends on the angle at which the toy approaches the grass.

Waves arriving at an interface where their propagation speed changes undergo exactly the same change in direction.

Physics of Music, Lecture 4, D. Kirkby 26

Refraction: LensesIf a wave passes through a pair of parallel interfaces, it emerges on a path parallel to its original path.

What if the interfaces are notparallel?

Physics of Music, Lecture 4, D. Kirkby 27

The air near the surface of the earth can act as a sound lens if the speed of sound varies with elevation.

A continuous change of the speed with elevation causes the wave directions to be continuously deflected in a smooth curve:

E.g., if speed ofsound increaseswith elevation(due to increasingtemperature orwinds).

Try this demo.

Physics of Music, Lecture 4, D. Kirkby 28

Refraction: PrismsWhat if the speed of wave propagation depends on the frequency?

The frequency of visible light corresponds to its color:

The speed of light in air is almost independent of frequency, but varies in glass. This leads to prism effects:

Physics of Music, Lecture 4, D. Kirkby 29

Refraction: Water WavesOcean waves are often approximately plane waves.

As they approach the shore, the wave speed decreases in shallower water causing the waves to become more parallel with the shoreline:

Physics of Music, Lecture 4, D. Kirkby 30

DiffractionReflection and refraction are universal properties of wave propagation at an interface where the medium changes.

Another universal feature is diffraction.

Diffraction results in waves spreading out from any discontinuity (eg, and edge or isolated point) they find.

Diffraction allows waves to bend around an obstacle.

When you hear someone talking around the corner, you are hearing diffracted sound (and possibly also reflected sound).

Physics of Music, Lecture 4, D. Kirkby 31

Diffraction and WavelengthDiffraction is important for how sound spreads out from a source. The amount of spreading increases when sound passes through a narrow opening (narrow compared to the wavelength)

Physics of Music, Lecture 4, D. Kirkby 32

Try this demo to see diffraction of high frequency sound produced by a “tweeter” speaker:

http://www.silcom.com/~aludwig/images/difdemo.gif

Physics of Music, Lecture 4, D. Kirkby 33

Reflection and refraction can both be thought of as limiting cases of diffraction: we can approximate a smooth interface with many point-like sources:

http://www.physics.gatech.edu/academics/tutorial/phys2121/Java%20Applets/ntnujava/propagation/propagation.html

Physics of Music, Lecture 4, D. Kirkby 34

Another 2D Effect: Doppler EffectThings get more interesting when the source of a wave is moving. This is particularly true for sound waves where a source can easily reach speeds near or even exceeding the speed of sound.

This results in the Doppler effect:

Note that the source in this exampleis generating sound at a constantfrequency. The apparent change ofpitch is entirely due to the source’smotion.

Christian Doppler (1803-1853)

Physics of Music, Lecture 4, D. Kirkby 35

What do we observe?

The sound appears to have a higher frequency as its source approaches, and then a lower frequency as it recedes.

Try changing the source’s speed in this demo and watch what happens to the spacing of the wave crests:

http://www.colorado.edu/physics/2000/applets/doppler.html

Physics of Music, Lecture 4, D. Kirkby 36

Catching Up: Shock WavesThings get even more interesting when a source of sound travels at the speed of sound or faster! This causes a pileup of the wave crests, or shock wave.

Shock waves are important for music also! We will see laterthat shock waves occur when playing a brass instrument.

Physics of Music, Lecture 4, D. Kirkby 37

The Third DimensionReal waves propagate in 3 dimensions, not 1 or 2.

Adding the third dimension gives even more complex patterns, but there is nothing fundamentally new that we cannot describe in 2 dimensions.

We also do not have a good way to visualize wave phenomena in 3 dimensions.

Physics of Music, Lecture 4, D. Kirkby 38

SummaryWe studied how waves propagate through an interface (refraction) where their speed changes, first in one dimension then in two dimensions.

We learned how apply the technique of limiting cases to get a qualitative feel for reflection and refraction at an interface.

We learned about ways to visualize waves in two dimensions and that the Principle of Superposition still holds in two dimensions.

We studied diffraction and the Doppler effect.

Physics of Music, Lecture 4, D. Kirkby 39

Review QuestionsWhat would pulses sent down a rope with 3 segments look like?

Why can you hear someone speaking around the corner of a building? (Does this still work if there are no other buildings nearby to reflect their voice?)

Can you catch up with the sound of your own voice? Can you overtake it? (You can’t do either of these things with light!)