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Unformatted text preview: Chapter 30 Wave Function, Interference, Standing Waves 218 30 Wave Function, Interference, Standing Waves In that two of our five senses (sight and sound) depend on our ability to sense and interpret waves, and in that waves are ubiquitous, waves are of immense importance to human beings. Waves in physical media conform to a wave equation that can be derived from Newton’s Second Law of motion. The wave equation reads: 2 2 2 2 2 1 t x ∂ ∂ = ∂ ∂ y y v (30-1) where: y is the displacement of a point on the medium from its equilibrium position, x is the position along the length of the medium, t is time, and v is the wave velocity. Take a good look at this important equation. Because it involves derivatives, the wave equation is a differential equation. The wave equation says that the second derivative of the displacement with respect to position (treating the time t as a constant) is directly proportional to the second derivative of the displacement with respect to time (treating the position x as a constant). When you see an equation for which this is the case, you should recognize it as the wave equation. In general, when the analysis of a continuous medium, e.g. the application of Newton’s second law to the elements making up that medium, leads to an equation of the form 2 2 2 2 t constant x ∂ ∂ = ∂ ∂ y y , the constant will be an algebraic combination of physical quantities representing properties of the medium. That combination can be related to the wave velocity by 2 1 v = constant For instance, application of Newton’s Second Law to the case of a string results in a wave equation in which the constant of proportionality depends on the linear mass density μ and the string tension F T : 2 2 T 2 2 t F x ∂ ∂ = ∂ ∂ y y μ Recognizing that the constant of proportionality T F μ has to be equal to the reciprocal of the square of the wave velocity, we have Chapter 30 Wave Function, Interference, Standing Waves 219 2 T 1 v = F μ μ T F = v (30-2) relating the wave velocity to the properties of the string. The solution of the wave equation 2 2 2 2 2 1 t y x y ∂ ∂ = ∂ ∂ v can be expressed as ) ( 2 2 cos max φ π λ π + ± = t x T y y (30-3) where: y is the displacement of a point in the medium from its equilibrium position, y max is the amplitude of the wave, x is the position along the length of the medium,...
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.
- Fall '08