Lecture 4 Wave Refraction and Diffraction Waves in Two Dimensions

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Lecture 4 Wave Refraction and Diffraction Waves in Two Dimensions
Lecture 4 Wave Refraction and Diffraction Waves in Two Dimensions Doppler Effect and Shock Waves Instructor: David Kirkby

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 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

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 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 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
(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 than the original rope, this is essentially same As leaving the end free (a whip) Physics of Music, Lecture 4, D. Kirkby

??? ??? (1) (2) (3) Negative reflection No transmission No reflection
Transmitted unchanged (2) ??? Positive reflection No transmission (3) Physics of Music, Lecture 4, D. Kirkby

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

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

R = -1 T = 0 -1 < R < 0 0 < T < 1 R = 0 T = 1
Physics of Music, Lecture 4, D. Kirkby

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
Plane waves are really just one-dimensional waves since the disturbances at different places are all in parallel directions: direction of travel Plane waves are a mathematical idealization since they require an infinitely long source. Physics of Music, Lecture 4, D. Kirkby

Circular waves look like plane waves up close (another limiting case)
Circular waves (or spherical waves) originate from a point source and spread out in circles (or spheres): Circular waves look like plane waves up close (another limiting case) Physics of Music, Lecture 4, D. Kirkby

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
See also these visualizations of multipole sources. Physics of Music, Lecture 4, D. Kirkby

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

List the variables in this problem:
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 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

Another complex effect occurs when light on the inside of a circular (or cylindrical) object. Try this demo: The resulting pileup of reflected rays produces a characteristic shape called a caustic curve. Physics of Music, Lecture 4, D. Kirkby

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

First, what happens if the toy hits the grass head on (normal incidence) ?
Both wheels enter the grass and slow down at the same time. The toy does not change direction. When leaving the grass, the toy speeds up but again does not change direction. Physics of Music, Lecture 4, D. Kirkby

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 faster than the other, the toy will rotate. Physics of Music, Lecture 4, D. Kirkby

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 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 not parallel? Physics of Music, Lecture 4, D. Kirkby

with elevation (due to increasing temperature or winds).
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 increasing temperature or winds). Physics of Music, Lecture 4, D. Kirkby

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
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 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 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

Try this demo to see diffraction of high frequency sound produced by a “tweeter” speaker:
Physics of Music, Lecture 4, D. Kirkby

Reflection and refraction can both be thought of as limiting cases of diffraction: we can approximate a smooth interface with many point-like sources: Java%20Applets/ntnujava/propagation/propagation.html Physics of Music, Lecture 4, D. Kirkby

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 example is generating sound at a constant frequency. The apparent change of pitch is entirely due to the source’s motion. Christian Doppler ( ) Physics of Music, Lecture 4, D. Kirkby

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: Physics of Music, Lecture 4, D. Kirkby

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 later that shock waves occur when playing a brass instrument. Physics of Music, Lecture 4, D. Kirkby

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 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 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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