#### Transcript applies light

```Some general properties of waves
Summing waves
• The wave equation is linear
• A sum of waves will be the arithmetical sum of the
function representing them – and still a solution of
the wave equation, but warning: energy is
proportional to the square of the amplitude!
Beats
Group velocity and phase velocity
Standing (stationary) waves
Some very basic physics of stringed
instruments……….
2f1
3f1
4f1
The fundamental frequency
determines the pitch of the
note.
the higher harmonics
determine the “colour” or
“timbre” of the note.
(ie why different instruments
sound different)
Fundamental wavelength =
2L
From v = fλ,
f1= v/2L
So, for a string of fixed
length, the pitch is
determined by the wave
velocity on the string…..
The string length on standard violin is
325mm. What tension is required
to tune a steel “A” string (diameter
=0.5mm) to correct pitch
(f=440Hz)?
Density of steel = 8g cm
Changing the “harmonic content”
Plucking a string at a certain
point produces a triangular
waveform, that can be built up
from the fundamental plus the
higher harmonics in varying
proportions.
Plucking the string in a
different place (or even in a
different way) gives a different
waveform and therefore
different contributions from
higher harmonics (see Fourier
analysis)
string plucked here
This makes the sound
different, even though pitch is
the same…………………
Doppler Effect
• The Doppler effect is the apparent change in the
frequency of a wave motion when there is
relative motion between the source of the waves
and the observer.
• The apparent change in frequency f
experienced as a result of the Doppler effect is
known as the Doppler shift.
• The value of the Doppler shift increases as the
relative velocity v between the source and the
observer increases.
• The Doppler effect applies to all forms of waves.
Doppler Effect (moving source)
Suppose the source moves at a steady velocity vs towards a stationary observer.
The source emits sound wave with frequency f.
From the diagram, we can see that the distance
between crests is shortened such that
 '    vs
Since  = c/f and  = 1/f,
we get

c c vs
 
f' f
f

c
1

 f ' (
)f  f
 v s
c  vs
1  c




vs
Doppler Effect (moving observer)
Consider an observer moving with velocity vo toward a stationary source S.
The source emits a sound wave with frequency f and wavelength  = c/f.
The velocity of the sound wave relative to the observer is c + vo.
 f '
c  vo

c  vo

c/ f
vo
 f '  (1  ) f
c
c
Doppler Shift
Consider a source moving towards an
observer, the Doppler shift f is

1
f  f '  f  f 
 v s
1  c


f  1 


1


v
f
s
1  c 
If vs<<c, then we get


 f


f v s

f
c

v s 
   
 
c 
The above equation also applies to a receding source, with vs taking as
negative
The same equation applies for the moving observer (note the limit vs<<c)
Applications of Doppler Effect (Astronomy)
• The velocities of distant galaxies can be determined from
the Doppler shift ( The apparent change in frequency).
• Light from such galaxies is shifted toward lower frequencies,
indicating that the galaxies are moving away from us.
This is called the red shift.
Red
shift
Blue
shift
Red Shift
Hubble’s Law
• Hubble found that (almost) every
galaxy was moving away from us.
• Moreover, the further away it
was, the faster it was moving
away from us.
This line can be described by an equation
which relates the distance to a galaxy to
the recession velocity – Hubble's Law.
This is a plot of some galaxies.
v  H0d
H 0  25 km/s/Mly 1/(14 Gy)
• The x axis is the distance to the galaxy
• The y axis is the speed at which the
galaxy is moving away from us
What happens if vs > c
Cherenkov effect
Huygens’ Principle (conjectured in 1600)
• All points on a given wave front can be taken
as point sources for the production of
spherical secondary waves, called wavelets,
which propagate in the forward direction with
speeds characteristic of waves in that medium
– After some time has elapsed, the new position of
the wave front is the surface tangent to the
wavelets
• Demonstrated by Kirkhhoff in 1882, but
Huygens was missing two points:
– Amplitude varies as f(θ) ~ (1+cosθ)/2
– Phase is anticipated by π/2
In many problems these two points can be
neglected
Huygen’s Construction for a Plane Wave
• At t = 0, the wave front is
indicated by the plane AA’
• The points are
representative sources for
the wavelets
• After the wavelets have
moved a distance cΔt, a
new plane BB’ can be
drawn tangent to the
wavefronts
Huygen’s Construction for a Spherical Wave
• The inner arc represents
part of the spherical wave
• The points are
representative points
where wavelets are
propagated
• The new wavefront is
tangent at each point to
the wavelet
Huygen’s Principle and the Law of Reflection
• The Law of
Reflection can be
derived from
Huygen’s Principle
• AA’ is a wave front of
incident light
• The reflected wave
front is CD
• Triangle ADC is
congruent to triangle
AA’C
• θ1 = θ1’
• This is the Law of
Reflection
Huygen’s Principle and the Law of Refraction
• In time Δt, ray 1 moves
from A to B and ray 2
moves from A’ to C
• From triangles AA’C and
ACB, all the ratios in the
Law of Refraction can be
found
– n1 sin θ1 = n2 sin θ2
Total Internal Reflection
• Total internal reflection can occur
when light attempts to move from a
medium with a high index of
refraction to one with a lower index of
refraction
– Ray 5 shows internal reflection
• A particular angle of incidence
will result in an angle of
refraction of 90°
– This angle of incidence is called the
critical angle
• For angles of incidence greater
than the critical angle, the beam
is entirely reflected at the
boundary
– This ray obeys the Law of Reflection at
the boundary
Optical fibers
• An application of internal
reflection
• Plastic or glass rods are
used to “pipe” light from
one place to another
• Applications include
– medical use of fiber optic
cables for diagnosis and
correction of medical
problems
– Telecommunications
Frequency Between Media
• As light travels from one
medium to another, its
frequency does not
change
– Both the wave speed and
the wavelength do change
– The wavefronts do not pile
up, nor are created or
destroyed at the boundary,
so ƒ must stay the same
Fermat’s principle
```