A Shallow Water Is Non Dispersive The Waves Travel At Speed

(a) Shallow water is non dispersive; the 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 proportional 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 3. 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 ψ (x, t) = Aei(px–Et)/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 ha/f the group velocity.

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