Hmwk_11_Solutions

Hmwk_11_Solutions - Physics 221 Fall 2008 Homework #11...

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1 Physics 221 Fall 2008 Homework #11 Solutions Ch. 15 Due Tues, Nov 11, 2008 11.1 Consider a sinusoidal mechanical wave traveling in the positive x -direction in a medium. (a) Describe two features of a mechanical medium that are required in order for waves to propagate in the medium. First, there must be a restoring force present that tends to return to its equilibrium position an element of the medium that has been displaced from its equilibrium position. Second, the medium must have inertia (mass) so that a particle in the medium does not respond instantaneously to a net force on it. (b) Explain the difference between a longitudinal wave and a transverse wave in a mechanical medium. What property of a wave do the terms “longitudinal” and “transverse” refer to? A longitudinal wave is one in which the particles in the medium move parallel to the axis of motion of the wave as the wave passes through it, whereas a transverse wave is one in which the particles in the medium move along an axis perpendicular to the direction of motion of the wave. These terms refer to the polarization of the wave. (c) What is the “wave equation”? What do we mean by the phrase “a wave function satisfies the wave equation”? What are the requirements on a wave moving in one dimension so that the wave function of the wave satisfies the wave equation? The wave equation describes the time dependence of the motion of the particles in a medium as a function of the spatial dependence of the displacement of the particles from their equilibrium positions. A wave function satisfies the wave equation if, when the wave function is substituted into the wave equation and is then simplified, an identity results. In order to satisfy the wave equation the wave must move in one dimension with constant speed and shape. (d) Can two separate transverse sinusoidal waves propagate along a stretched string at the same time? Why or why not? Yes. A wave function can consist of the superposition of any number of sinusoidal wave functions, and that wave function also satisfies the wave equation. This is known as the principle of superposition. (e) What is the definition of the “dispersion relation” for propagation of sinusoidal waves in a medium? What is the difference between a dispersive medium versus a nondispersive medium with regard to propagation of sinusoidal waves in the medium? The dispersion relation for a medium gives the angular frequency ± of a sinusoidal traveling wave in the medium as a function of the wave number k of the wave. In a dispersive medium, the dispersion relation is nonlinear, and the speed v = ± ( k )/ k of a sinusoidal wave depends on its wave number, whereas in a nondispersive medium, the dispersion relation is a proportional relation and hence the wave speed does not depend on the wave number of the wave.
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2 (f) How does a transverse traveling wave on a stretched string transport energy in its direction of motion? What type(s) of energy is being transported? Where does the energy come from?
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This note was uploaded on 01/28/2010 for the course PHYSICS 221 taught by Professor Johnson during the Fall '06 term at Iowa State.

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Hmwk_11_Solutions - Physics 221 Fall 2008 Homework #11...

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