Lecture 4
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Lecture 4. Wave Refraction and Diffraction Waves in Two Dimensions Doppler Effect and Shock Waves. Instructor: David Kirkby ([email protected]). Miscellaneous. Problem set #2 is due at the beginning of class next Thursday (Oct 17). Homework grading policy: all-or-nothing / partial credit?

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Lecture 4

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Lecture 4

Lecture 4

Wave Refraction and Diffraction

Waves in Two Dimensions

Doppler Effect and Shock Waves

Instructor: David Kirkby ([email protected])


Miscellaneous

Miscellaneous

  • Problem 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


Review of lecture 3

Review of Lecture 3

  • We 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


Lecture 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


Wave refraction

Wave Refraction

  • What if, instead of fixing the end of the rope, we attach another rope of different thickness?

Physics of Music, Lecture 4, D. Kirkby


Limiting cases

Limiting Cases

  • A 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


Limiting cases for rope waves

Limiting Cases for Rope Waves

  • (1) If the extra rope is heavy enough, this

  • Is 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 same

  • As leaving the end free (a whip)

Physics of Music, Lecture 4, D. Kirkby


Lecture 4

Negative reflection

No transmission

No reflection

Transmitted unchanged

Positive reflection

No transmission

(1)

???

(2)

???

(3)

Physics of Music, Lecture 4, D. Kirkby


Lecture 4

  • 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


Lecture 4

  • 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


Reflection refraction

Reflection & Refraction

  • Reflection 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


Lecture 4

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


Into the second dimension

Into the Second Dimension

  • Until 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


Special cases of 2d sources

direction

of travel

Special Cases of 2D Sources

  • Plane 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.

Physics of Music, Lecture 4, D. Kirkby


Lecture 4

  • 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


Principle of superposition in 2d

Principle of Superposition in 2D

  • The 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


Visualization of waves in 2d

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


Reflection refraction in 2d

Reflection & Refraction in 2D

  • When 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


Lecture 4

  • 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


Reflection in 2d

Reflection in 2D

  • Reflection 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


Lecture 4

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

  • 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/

Physics of Music, Lecture 4, D. Kirkby


Refraction in 2d toy analogy

Refraction in 2D: Toy Analogy

  • What happens when the toy enters the grass (where its wheels will turn slower)?

Physics of Music, Lecture 4, D. Kirkby


Lecture 4

  • 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


Lecture 4

  • 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


Lecture 4

  • 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


Refraction lenses

Refraction: Lenses

  • If 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


Lecture 4

  • 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:

Try this demo.

E.g., if speed of

sound increases

with elevation(due to increasingtemperature orwinds).

Physics of Music, Lecture 4, D. Kirkby


Refraction prisms

Refraction: Prisms

  • What 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


Refraction water waves

Refraction: Water Waves

  • Ocean 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


Diffraction

Diffraction

  • Reflection 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


Diffraction and wavelength

Diffraction and Wavelength

  • Diffraction 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


Lecture 4

  • 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


Lecture 4

  • 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


Another 2d effect doppler effect

Another 2D Effect: Doppler Effect

  • Things 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’s

  • motion.

Christian Doppler (1803-1853)

Physics of Music, Lecture 4, D. Kirkby


Lecture 4

  • 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


Catching up shock waves

Catching Up: Shock Waves

  • Things 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


The third dimension

The Third Dimension

  • Real 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


Summary

Summary

  • We 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


Review questions

Review Questions

  • What 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!)

Physics of Music, Lecture 4, D. Kirkby


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