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Problem 9.23 (a) Shallow water is nondispersive; waves travel at a speed that is proportional to the square root of the depth. In deep water, however, the waves can't "feel" all the way down to the bottom-they behave as though the depth were pro- portional to λ. (Actually, the distinction between “shallow” and “deep” itself depends on the wavelength: If the depth is less than λ, the water is "shallow"; if it is substantially greater than λ, the water is "deep.") Show that the wave velocity of deep water waves is twice the group velocity. (b) In quantum mechanics, a free particle of mass m traveling in the x direction is described by the wave function (x,t) = Ae(px-Er)/h where p is the momentum, and E = p²/2m is the kinetic energy. Calculate the group velocity and the wave velocity. Which one corresponds to the classical speed of the particle? Note that the wave velocity is half the group velocity.

Problem 9.23 (a) Shallow water is nondispersive; waves travel at a speed that is proportional to the square root of the depth. In deep water, however, the waves can't "feel" all the way down to the bottom-they behave as though the depth were pro- portional to λ. (Actually, the distinction between “shallow” and “deep” itself depends on the wavelength: If the depth is less than λ, the water is "shallow"; if it is substantially greater than λ, the water is "deep.") Show that the wave velocity of deep water waves is twice the group velocity. (b) In quantum mechanics, a free particle of mass m traveling in the x direction is described by the wave function (x,t) = Ae(px-Er)/h where p is the momentum, and E = p²/2m is the kinetic energy. Calculate the group velocity and the wave velocity. Which one corresponds to the classical speed of the particle? Note that the wave velocity is half the group velocity.

University Physics Volume 1
University Physics Volume 1
18th Edition
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:William Moebs, Samuel J. Ling, Jeff Sanny
Chapter17: Sound
Section: Chapter Questions
Problem 30P: Consider a sound wave moving through the air modeled with the equation...
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Problem 9.23 (a) Shallow water is nondispersive; waves travel at a speed that is proportional to the square root of the depth. In deep water, however, the waves can't "feel" all the way down to the bottom-they behave as though the depth were pro- portional to λ. (Actually, the distinction between “shallow” and “deep” itself depends on the wavelength: If the depth is less than λ, the water is "shallow"; if it is substantially greater than λ, the water is "deep.") Show that the wave velocity of deep water waves is twice the group velocity. (b) In quantum mechanics, a free particle of mass m traveling in the x direction is described by the wave function (x,t) = Ae(px-Er)/h where p is the momentum, and E = p²/2m is the kinetic energy. Calculate the group velocity and the wave velocity. Which one corresponds to the classical speed of the particle? Note that the wave velocity is half the group velocity.
Transcribed Image Text:Problem 9.23 (a) Shallow water is nondispersive; waves travel at a speed that is proportional to the square root of the depth. In deep water, however, the waves can't "feel" all the way down to the bottom-they behave as though the depth were pro- portional to λ. (Actually, the distinction between “shallow” and “deep” itself depends on the wavelength: If the depth is less than λ, the water is "shallow"; if it is substantially greater than λ, the water is "deep.") Show that the wave velocity of deep water waves is twice the group velocity. (b) In quantum mechanics, a free particle of mass m traveling in the x direction is described by the wave function (x,t) = Ae(px-Er)/h where p is the momentum, and E = p²/2m is the kinetic energy. Calculate the group velocity and the wave velocity. Which one corresponds to the classical speed of the particle? Note that the wave velocity is half the group velocity.
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