Measures of Central Tendency: Mean, Median and Mode
Lo7
1.
2. 15-5 INTERFERENCE
The moment that two waves meet while traveling along the
same medium is defined as wave interference. This creates a
new shape of the medium, in that instant of interference, from
the net effect of the two individual waves on the particles.
One dimensional (1D) waves are simpler than two dimensional
(2D) and three dimensional (3D) waves due to the fact that two
1D waves with the same frequency and wavelength have a fixed
phase difference.
This phase difference depends on the change in phase
constant and is not subject to changes due to time or position.
4. Constructive interference occurs when waves with
amplitudes in the same direction overlap
At the point of interference, the waves overlap and
the medium is displaced upward or downward by the
sum of the amplitudes
Waves that have identical frequency and wavelength
are produced by two point sources in phase.
When two sources are at large negative amplitude, a
whole integer of the wavelength will be between
each source and point.
Key Notes
5. MATHEMATICAL CONDITIONS
d1 = path length from source 1 √x2-(y-1)2
d2 = path length from source 2 √x2+(y+1)2
d1=mλ and m = 1,2,3….
d2=nλ and n = 1,2,3….
So,
Δd=d2-d1=(n-m)λ=pλ and p=0, ±1,±2,±3….
Therefore an integer multiple of the wavelength must
be the path difference.
6. SPHERICAL WAVES
Oscillate in time and space
Amplitude is constant over any spherical surface
(centered on source)
Spatial variation (r) is the distance from the source
s (r,t) = smrcos(kr-ωt+ϕ)
The wave front spreads over a larger area as it
propagates and depends on r
As the distance from the source increases, the
amplitude decreases
8. TWO WAVES IN PHASE
The cosine functions of two waves in phase differ by
2π
We neglect ϕ since the waves are in phase and
achieve the equation
(kd2-ωt)-(kd1-ωt)=k(d2-d1)=n2π
Therefore d2-d1=n(2π/k)=nλ and n=0,±1,±2,±3…
9. [Image demonstrating when waves are in phase and out of phase from
Nelson- Physics for Scientists and Engineers pg. 433]
“Two plane waves with the same
wavelength travelling in slightly
different directions. The lines represent
the crest of the waves, the black
circles indicate where the waves are
perfectly in phase, and the open circles
indicate where the waves are exactly
out of phase.”
10. DESTRUCTIVE INTERFERENCE
Destructive Interference occurs when the waves are out
of phase and the amplitudes are in opposite directions
of each other. One path is a whole integer of the
wavelength and the other is ½ an integer of the
wavelength
Δd=d2-d1=[(2n+1)⁄2]λ=(n+½)λ n=0,±1,±2,±3……
Image of Destructive Interference from phsy.uconn.edu
11. At the moment of destructive interference, the net
effect of the maximum amplitude of the individual
waves being in opposite direction cause the wave to
‘cancel out’, but not permanently. It is important to
note that this is only momentarily, in the instant of the
interference, and has no permanent effect on the
particles of the wave. The positive displacement of the
wave ‘pulls’ the particles of the negative displacement
of the wave, and vice versa, resulting in a
displacement of the medium that is smaller than both
of the waves maximum displacements. The
amplitudes do not need to be equal. For example, if
the amplitudes are equal then the waves will cancel
out, but if one wave is +2 units and the other wave is -
1 unit then the resulting wave will be 1 unit.
12. REVIEW QUESTIONS
#1) Bobby and his father go fishing. Waves propagate
in the water from the fishing lines being thrown into
the water. If the white areas demonstrate crests and
the dark areas demonstrate troughs, where in the
image is a) constructive interference occurring and
b) deconstructive interference occurring
14. Question # 2 True or False
a) A sound wave is emitted from a speaker in the middle of
the room. As the wave oscillates away from the source,
the amplitude increases in proportion to the distance that
the wave is away from the speaker.
b) When two waves with equal amplitudes in opposite
directions collide, they will always cancel out.
c) If a wave with the equation s(x,t)=12cos(3t-2π+½π) is
passing through the same medium as a wave with the
equation s(x,t)=6cos(2t-2π+π) constructive interference
will occur and create a momentarily new maximum
amplitude (where x is in m and t is in s) of 18 m.
15. SOLUTIONS #2
a) False; for spherical waves as the spatial distance
from the source increase, the amplitude
decreases. There is also no information to
suggest a connection between proportionality.
b) True
c) False; constructive interference cannot occur if
the waves are not perfectly in phase.
16. QUESTION #3
If the first path length is 5λ and the second is 5.5λ,
does interference occur? If so, what type and
what would ‘n’ be
17. SOLUTION #3
Since the path lengths differ by one-half of a
wavelength, the interference will be destructive.
d1=5λ and d2=5.5λ
=(5+0.5)λ
=(n+½)λ
So n = 5
