Introduction to Partial Differential Equations with MATLAB
The subject of partial differential equations has an unchanging core of material but is constantly expanding and evolving. Introduction to Partial Differential Equations with MATLAB is a careful integration of traditional core topics with modern topics, taking full advantage of the computational power of MATLAB to enhance the learning experience. This advanced text/reference is an introduction to partial differential equations covering the traditional topics within a modern context. To provide an up-to-date treatment, techniques of numerical computation have been included with carefully selected nonlinear topics, including nonlinear first order equations. Each equation studied is placed in the appropriate physical context. The analytical aspects of solutions are discussed in an integrated fashion with extensive examples and exercises, both analytical and computational. The book is excellent for classroom use and can be used for self-study purposes. Topic and Features: • Nonlinear equations including nonlinear conservation laws; • Dispersive wave equations and the Schrodinger equation; • Numerical methods for each core equation including finite difference methods, finite element methods, and the fast Fourier transform; • Extensive use of MATLAB programs in exercise sets. MATLAB m files for numerical and graphics programs available by ftp from this web site. This text/reference is an excellent resources designed to introduce advanced students in mathematics, engineering and sciences to partial differential equations. It is also suitable as a self-study resource for professionals and practitioners.
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In physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats. It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The concept can also be applied to periodic waves of non-sinusoidal shape. The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.
Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.
Examples of wave-like phenomena are sound waves, light, and water waves. A sound wave is a periodic variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are periodic variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary periodically in both lattice position and time.
Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength.
1 Sinusoidal waves
1.1 Standing waves
1.2 Mathematical representation
1.3 General media
1.3.1 Nonuniform media
2 More general waveforms
2.1 Envelope waves
2.2 Wave packets
3 Interference and diffraction
3.1 Double-slit interference
3.2 Single-slit diffraction
3.3 Diffraction-limited resolution
5 See also
7 External links
Sinusoidal wavesIn linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components.
Refraction: when a plane wave encounters a medium in which it has a slower speed, the wavelength decreases, and the direction adjusts accordingly.The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by:
where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency.
In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s. Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about: 3×108 m/s divided by 108 Hz = 3 metres. Visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm (430–750 THz) (for other examples, see electromagnetic spectrum).
For sound waves in air, the speed of sound is 343 m/s (1238 km/h) (at room temperature and atmospheric pressure). The wavelengths of sound frequencies audible to the human ear (20 Hz–20 kHz) are between approximately 17 m and 17 mm, respectively, assuming a typical speed of sound of about 343 m/s. Note that the wavelengths in audible sound are much longer than those in visible light.
Frequency and wavelength can change independently, but only when the speed of the wave changes. For example, when light enters another medium, its speed and wavelength change while its frequency does not; this change of wavelength causes refraction, or a change in propagation direction of waves that encounter the interface between media at an angle.
Sinusoidal standing waves in a box that constrains the end points to be nodes will have an integer number of half wavelengths fitting in the box.Standing waves
A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).A standing wave is an undulatory motion that stays in one place. A