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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)