WAVE REFLECTION IN 2-DIMENSIONS
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Waves aren’t always confined to one dimension as they are in a spring
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Two and even three dimensional waves are actually common
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With these waves, we focus on the wave fronts which are the entire width of a wave
crest
• A ray is an arrow representing the direction of motion of a wave front in one
particular line
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When a wave front encounters a barrier, it undergoes reflection
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The Law of Reflection states that the “incident wave” (before reflection)
and “reflected wave” strike the barrier at equal angles: the angle of
incidence (θi or θ1) is equal to the angle of reflection (θr or θ2). Where both
of these are measured with respect to normal to the barrier.
WAVE REFRACTION IN 2-DIMENSIONS
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Refraction occurs when a two dimensional wave enters a “new” medium which
supports the wave at a different speed
• NOTE: “new” can include the same medium in a different physical
condition, a water wave for example, may enter water of a
different depth
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When such a wave enters a new medium, the direction of the transmitted wave is
different than the direction of motion of the incident wave
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It turns out that this change in speed results in the ‘bending’ of wave fronts as a wave
moves across a boundary that separates different kinds of media
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This phenomenon was first described mathematically by a Dutch mathematician
named Willebrord Snell (1591 – 1626) and describes the math associated with the
following picture:
• Snell developed his law by using simple geometry and basic math
• All forms of his law can be derived from a basic understanding of simple
(right angle) trigonometric functions and the law of similar triangles
o The first form requires we know the wavelengths (λ) of the incident
and refracted waves
o Please see part I of the derivation provided on next page for an
explanation of why we need to know wavelength!
λ1 sin θ 2 = λ2 sin θ1
λ1 = wavelength of wave in medium 1 (m)
θ1 = angle of incidence (°)
λ2 = wavelength of wave in medium 2 (m)
θ2 = angle of refraction (°)
Example 1:
Suppose a water wave moves from deep water to shallow water. Knowing that the
wavelength of the wave decreases from 30 cm in the deep to 20 cm in the shallow,
calculate the angle of refraction knowing that:
a) it strikes the boundary at an angle of 25° with respect to the boundary
b) it strikes the boundary at an angle of 25° with respect to the normal of the
boundary
• By combining the wave equation with the first form of Snell’s law we can create
another usable form based on different information (ie: speed of wave)
• Please see part II of the derivation provided on the next page for the
mathematical explanation of how the wave equation is linked to the first
form of Snell’s law
v1 sin θ 2 = v2 sin θ1
v1 = speed of wave in medium 1 (m/s)
v2 = speed of wave in medium 2 (m/s)
θ1 = angle of incidence (°)
θ2 = angle of refraction (°)
NOTE: the angles are measured between the wave velocities and the normal to the
boundary separating the two media
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It is worth noting that when a wave enters a faster medium, it bends away from the
normal, and when entering a slower medium, it bends toward the normal.
Example 2:
Suppose a water wave moves from deep water to shallow water. Knowing that the
speed of the wave decreases from 6.0 m/s in the deep to 3.5 m/s in the shallow,
calculate the angle of refraction knowing that:
a) it strikes the boundary at an angle of 25° with respect to the boundary
b) it strikes the boundary at an angle of 25° with respect to the normal of the
boundary
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the last form of Snell’s law applies to situations when light waves refract
• this can be seen as a specialized version of Snell’s law that only works
when considering situations involving the bending of light
o we do this because many technologies involve the bending of light
waves
NOTE:
When light travels from one medium to another, its speed and wavelength
change however…
Its frequency remains constant
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a ‘new’ quantity called index of refraction is introduced into Snell’s law to
perform this customization
o the symbol used to denote index of refraction in mathematical
expressions is: n
o index of refraction (n) is nothing more than a comparison
ƒ it compares the speed of light in a medium to the speed of
light in a vacuum
ƒ it can be calculated using:
n=
c
v
n = index of refraction
c = speed of light in vacuum (300 000 000 m/s)
v = speed of light in medium (m/s)
NOTE: due to the nature of the equation, index of refraction values will always be equal
or greater than 1 (because nothing can travel faster than the speed of light) and
have no units!
o all media that transmit light waves have a unique index of refraction
o these are often printed in textbooks in the form of a table:
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incorporating this idea of index of refraction into Snell’s law yields the following
expression:
• Please see part III of the derivation provided on the next page for the
mathematical explanation of how index of refraction is linked to the second
form of Snell’s law
n1 sin θ1 = n2 sin θ 2
n1 = index of refraction in medium 1
θ1 = angle of incidence (°)
n2 = index of refraction in medium 2
θ2 = angle of refraction (°)
Example 3:
A light beam traveling through air hits a sheet of crown glass at an angle of 30°
relative to the normal. At what angle is it refracted?
SNELL’S LAW DERIVATION
DIFFRACTION OF WAVES
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Diffraction is the bending of a wave around an obstacle
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This may seem counterintuitive, but what we are essentially saying is that waves
cannot be cleanly blocked - the “shadow” of a wave can seal itself up reforming the
wave
• this phenomenon was first explained by Dutch scientist Christiaan Huygens
(1629 – 1695)
o he suggested that a wave front that exists at one instant in time
generates a wave front that exists later on
o a current wave front generates a future wave front
ƒ “every point on a wave front acts as a source of tiny wavelets
that move forward with the same speed as the wave, the
wave front at a later instant is the surface that is tangent to
the wavelets”
DIFFRACTION THROUGH SLIT BARRIERS
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The diffraction of waves through different configurations of slits in a barrier generate
distinct and reproducible patterns
DIFFRACTION THROUGH A SINGLE SLIT BARRIER:
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It is worth noting that the extent to which a wave diffracts (ie: bends) depends on:
1.
the wavelength (longer wavelengths diffract more)
2.
the width of the opening (smaller openings lead to more
diffraction)
If λ/opening is ‘large’ → wide dispersion
If λ/opening is ‘small’ → narrow dispersion
DIFFRACTION THROUGH A DOUBLE SLIT BARRIER:
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The wave fronts that are generated at each opening interfere with each other at
various locations
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Lines of constructive interference occur when 2 crests or 2 troughs meet
• These lines are called antinodal lines
Lines of destructive interference occur when a crest meets a trough
• These lines are called nodal lines
• The lines appear as the calm regions in a ripple tank!
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