Modern Physics
Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
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Chapter 6, Problem 46E

(a)

To determine

To sketch:

Group velocity versus wave number.

(b)

To determine

Group velocity and wavelength

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A real wave function is defined on the half-axis: [0≤x≤00) as y(x) = A(x/xo)e-x/xo where xo is a given constant with the dimension of length. a) Plot this function in the dimensionless variables and find the constant A. b) Present the normalized wave function in the dimensional variables. Hint: introduce the dimensionless variables = x/xo and Y(5) = Y(5)/A.
The phase velocity v of gravity waves in a liquid of depth h is given by the formula v = tanh kh, k where g is the acceleration of free fall and k = 2w/2 is the wavenumber, a being the wavelength. Sketch the dispersion relation for such waves, and show that the group velocity is always between v2 and v. Find the phase and group velocities for gravity waves of frequency 1 Hz in a liquid of depth 0.1 m.
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