Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

(a) Shallow water is non-dispersive; 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

ψ(x,t)=Aei(px-Et)

wherep is the momentum, and E=p2/2mis 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.

Short Answer

Expert verified

(a) It is proved that the wave velocity of deep water waves is twice the group velocityas vp=2vg.

(b) The group velocity and the wave velocity iskm and the group velocity corresponds to the classical speed.

Step by step solution

01

Expression forthe wave velocity for deepwater wave and phase velocity: 

Write the expression forthe wave velocity for deepwater waves.

vp=αλ …… (1)

Here, αis a constant and λis the wavelength.

Write the expression for phase velocity.

vp=ωk …… (2)

Hereω, is the angular velocity and kis the wave number.

02

Determine the relation between wave velocity and group velocity:

(a)

Equate equations (1) and (2).

…… (3)

ωk=αλω=kαλ …… (3)

Write the relation between the wavenumber in terms of wavelength.

k=2πλλ=2πk

Substituteλ=2πk in equation (3).

ω=kα2πkω=2παk

Write the equation for the group velocity.

vg=dωdk

Substitute iω=2παkn the above equation.

vg=ddk(2παk)vg=2παddk(k)vg=2πα12(k)12

On further solving, the above equation becomes,

vg=2πα21kvg=α22πkvg=α2λvg=vp2

Therefore, it is proved that the wave velocity of deep water waves is twice the group velocity.

03

 Step 3: Determine the wave velocity and group velocity:

(b)

From the given problem, the equation is given as:

ψ(x,t)=Aei(pxEt)/ …… (4)

Write the spatial representation of a wave.

ψ(x,t)=Aei(kxωt) …… (5)

Equate equations (4) and (5).

Aei(pxEt)/=Aei(kxωt)i(pxEt)=i(kxωt)k=pω=E

Write the expression for the wave velocity.

vp=ωk

Substituteω=E in the above expression.

vp=(E/)p/vp=Epvp=p2/2mpvp=k2m …… (6)

Write the expression for the group velocity.

vg=dωdk

Substituteω=E in the above expression.

vg=ddk(E)=ddk(p22m)=ddk(2k22m)

On further solving, the above equation becomes,

vg=2mddk(k2)vg=2m(2k)vg=km ……. (7)

From equations (6) and (7).

vp=k2mvp=vg2

Hence, the phase velocity is half of the group velocity.

Group velocity corresponds to the classical speed of the particle. This can be understood as follows.

Write the expression for the classical speed.

vc=pm

Substitutep=k in the above expression.

vc=kmvc=vg

Hence, the from the equation the wave velocity do not represent the classical speed of the particle.

Therefore, the group velocity and the wave velocity is kmand the group velocity corresponds to the classical speed.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the resonant cavity produced by closing off the two ends of a rectangular wave guide, at z=0and at z=d, making a perfectly conducting empty box. Show that the resonant frequencies for both TE and TM modes are given by

role="math" localid="1657446745988" ωlmn=cπ(ld)2+(ma)2+(nb)2(9.204)

For integers l, m, and n. Find the associated electric and magnetic fields

A microwave antenna radiating at 10GHzis to be protected from the environment by a plastic shield of dielectric constant2.5. . What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use Eq. 9.199.]

Light from an aquarium goes from water (n=43)through a plane of glass (n=32)into the air (n=1). Assuming its a monochromatic plane wave and that it strikes the glass at normal incidence, find the minimum and maximum transmission coefficients (Eq. 9.199). You can see the fish clearly; how well can it see you?

Find all elements of the Maxwell stress tensor for a monochromatic plane wave traveling in the z direction and linearly polarized in the x direction (Eq. 9.48). Does your answer make sense? (Remember that -Trepresents the momentum flux density.) How is the momentum flux density related to the energy density, in this case?

In the complex notation there is a clever device for finding the time average of a product. Suppose f(r,t)=Acos(k×r-ωt+δa)and g(r,t)=Bcos(k×r-ωt+δb). Show that <fg>=(1/2)Re(fg~), where the star denotes complex conjugation. [Note that this only works if the two waves have the same k andω, but they need not have the same amplitude or phase.] For example,

<u>=14Re(ε0E~×E~+1μ0B~×B~)and<S>=12μ0Re(E~×B).~ and .

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free