Solution Found!
(a) Shallow water is nondispersive; waves travel at a
Chapter 9, Problem 23P(choose chapter or problem)
Problem 23P
(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 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 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
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.
Questions & Answers
QUESTION:
Problem 23P
(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 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 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
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.
ANSWER:
Solution
Step 1 of 3
- We need to prove that wave velocity of deep water waves is twice the group velocity.
Wave velocity is the ratio of angular frequency with the wave number.
..............(1)
In terms of wavelength it can be written as,
And we know that
So ..................(2)
Comparing equation (1) and (2),
Implies,
We know that,
Comparing the above equation with equation implies,
