Download TFY 4170 - Fysikk 2 Justin Wells

TFY 4170 - Fysikk 2
Justin Wells
Forelesning 7: Bølgefysikk
Lasers, wave on a string, wave in a rod and sound waves
Mansfield & O’Sullivan: 12.14, 12.15, 12.16, 12.17, 12.18
Waves
!
!
!
!
!
!
!
Wave phenomena
Wave equation in one
dimension
Energy, power and
intensity of waves
Plane waves and
dispersing waves. Huygens
principle. Reflection and
refraction (brytning).
Interference and
diffraction. Young’s double
slit, many waves.
Diffraction in crystals, Xray, neutron and electron
diffraction.
Standing waves,
resonance.
!
!
!
!
!
!
Doppler effect (classic and
relativistic).
Lasers and coherent
waves.
The wave equation.
Mechanical waves and
sound waves.
Electromagnetic waves,
Maxwell’s equations,
polarisation.
Wave packets and
envelopes, group velocity,
dispersion.
Fourier analysis,
bandwidth.
Question:
Is light coherent or incoherent?
Electromagnetic waves
Electromagnetic waves can be generated by accelerating charges.
A typical radio-wave:
transmitter
receiver
Electric
potential
accelerating
charges
The wavelength of the radio wave is typically around the same
magnitude as the transmitter. The receiver is also similar size to the
transmitter in order to be sensitive to the same wavelength.
Electromagnetic waves
Longwave transmitters and receivers are larger than shortwave
transmitters/receivers.
The transmitter generates electromagnetic waves of a particular
wavelength and propagation direction - and the receiver measures the
intensity.
Visible light is an electromagnetic wave with wavelength around 400-700
nm. It is difficult to make transmitters and receivers on this scale.
Electromagnetic waves can be created by heating up gas
At higher temperature, individual atoms have higher kinetic energy and
collide more often. This creates more acceleration and more
electromagnetic radiation.
These collisions create a continuum of waves with a broad range if
wavelengths. (i.e. not coherent)
Electromagnetic waves
Intensity
red
blue
Radiation from gasses contain a broad range of wavelengths and
phases..... it is called ‘incoherent’
Interference effects are therefore not seen*.
(*) there are some exceptions!
The waves are spherical and propagate in all directions. The intensity/
power is therefore following the “inverse square” relationship.
Electromagnetic waves
We will later look at gas discharge: this is a completely different process
involving excited energy levels and generates light at specific
wavelengths. We need quantum mechanics to describe this properly.
intensity
red
blue
These waves are also incoherent because they are formed at ‘random’
times. i.e. they have a broad range of phases.
Lasers
It is possible to make coherent light using a LASER:
Light Amplification by Stimulated Emission of Radiation.
Gas-laser:
mirror
Laser emission
In a laser, standing waves of a particular wavelength are generated using:
These electromagnetic waves can ‘stimulate’ the production of new waves
with the same wavelength (frequency) and phase.
Lasers
Lasers are very intense.
! Lasers are monochromatic with a
characteristic frequency (depending on a
particular atomic excitation).
! Lasers emit plane waves with little
dispersion and little loss of intensity.
! Laser light is polarised.
! Laser light is coherent.
!
Wave equation:
The equation for a wave which propagates in the positive x-direction can
be written as:
We will now show that this wave is a solution of the wave equation:
And, we will show how various physical phenomena can be described by
this equation.
We differentiate our expression for y(x,t) with respect to position and time
(x and t):
Wave equation:
We see that:
The wave therefore satisfies:
This result can be generalised for 3-dimensional waves:
where ξ(x,y,z;t) is the position and time dependent displacement.
Wave equation: general 1D solution
We will now show that there is a general solution for a 1dimensional wave-equation:
We see what happens when we use this trial solution:
The simple plane wave is therefore just a special form of the general
solution:
Question:
What is the physics behind guitar
tuning?
Waves on an elastic string
Wave displacement:
y
Tsinθ(x+Δx)
B
Δs
T=constant
A
θ
T
Tsinθ(x)
Δx
x
Waves on an elastic string
For small displacements we can use:
The total force in the y-direction is therefore:
@ 2 y(x)
Ty = T x
@x2
Newton’s 2nd law gives:
The wave equation for an elastic string is therefore:
Waves on an elastic string
The wave equation can be written as:
The wave velocity (and therefore frequency/period) depends on the
tension in the string, and the mass per unit length.
Standing waves occur when:
The frequency of such a standing wave is therefore given by:
And the lowest frequency is:
Waves on an elastic string
The lowest frequency is:
And it is proportional to the length.
The frequency increases when the string tension
is increased.
The frequency increases when the string
density (mass per unit length) decreases
These properties are utilised when building
and tuning musical instruments.
Longitudinal waves in a rod:
We will look at a rod with cross-sectional area = A
When a wave propagates, the material in the rod is temporarily displaced
The point P move to P’ and the point Q moves to Q’.
Displacement of P:
Displacement of Q:
Longitudinal waves in a rod:
The length increase of the portion PQ of the rod is:
The relative length increase is known as the ‘strain’ and is:
Young’s modulus:
Longitudinal waves in a rod:
The force on the left-hand end of an element of length Δx is therefore:
The corresponding force at the other end is:
The total force on the length-element Δx is therefore
Newton’s 2nd law gives:
Longitudinal waves in a rod:
The velocity depends on Young’s modulus
and the mass per unit length (density) of
the rod.
! The wave equation is 1-dimensional and
the solution is a 1-dimensional plane
wave.
!
Sound waves in an elastic medium:
We will now look at a sound wave which propagates through a gas:
Displacement of P:
Displacement of Q:
The relative change in the volume is:
Sound waves in an elastic medium:
The density changes quickly as the sound wave propagates through:
The adiabatic compressibility is:
The pressure change over a volume element is therefore:
Net pressure change is:
Sound waves in an elastic medium:
The pressure difference is:
The net force on the volume element is:
Newton’s 2nd law gives:
The wave equation of the sound wave in an elastic medium is therefore:
Repetition – forelesning 7
!
!
!
!
!
Light is generally incoherent
Lasers create coherent monochromatic light:
The wave equation for a 1-dimensional plane
wave is:
Waves on a string and longitudinal waves in
elastic media are all described by this
relationship.
For every system, the wave velocity is
dependent on the physical properties of the
medium (i.e. density, elasticity, etc)