Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
expand_more
expand_more
format_list_bulleted
Question
Chapter 6, Problem 47E
(a)
To determine
To calculate:
The phase and group velocities.
(b)
To determine
Spread of the wave as it travels.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
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.
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.
Q.18. Verify the statement in the text that, if the phase velocity is the same for all
wavelengths of a certain wave phenomenon (that is, there is no dispersion), the group and
phase velocities are the same?
Chapter 6 Solutions
Modern Physics
Ch. 6 - Prob. 1CQCh. 6 - Prob. 2CQCh. 6 - Prob. 3CQCh. 6 - Prob. 4CQCh. 6 - Prob. 5CQCh. 6 - Prob. 6CQCh. 6 - Prob. 7CQCh. 6 - Prob. 8CQCh. 6 - Prob. 9CQCh. 6 - Prob. 10CQ
Ch. 6 - The diagram below plots (k) versus wave number for...Ch. 6 - Prob. 12CQCh. 6 - Prob. 13ECh. 6 - Prob. 14ECh. 6 - Prob. 15ECh. 6 - Prob. 16ECh. 6 - Prob. 17ECh. 6 - Prob. 18ECh. 6 - Prob. 19ECh. 6 - Prob. 20ECh. 6 - Prob. 21ECh. 6 - Prob. 22ECh. 6 - Prob. 23ECh. 6 - Prob. 24ECh. 6 - Prob. 25ECh. 6 - Prob. 26ECh. 6 - Prob. 27ECh. 6 - Prob. 28ECh. 6 - Obtain the smoothness conditions at the...Ch. 6 - Prob. 30ECh. 6 - Prob. 31ECh. 6 - Jump to Jupiter The gravitational potential energy...Ch. 6 - Prob. 33ECh. 6 - Obtain equation (618) from (616) and (617).Ch. 6 - Prob. 35ECh. 6 - Prob. 36ECh. 6 - Prob. 37ECh. 6 - Prob. 38ECh. 6 - Prob. 39ECh. 6 - Prob. 40ECh. 6 - Prob. 41ECh. 6 - Prob. 42ECh. 6 - Prob. 43ECh. 6 - Prob. 44ECh. 6 - Prob. 45ECh. 6 - Prob. 46ECh. 6 - Prob. 47ECh. 6 - Prob. 48ECh. 6 - Prob. 49ECh. 6 - Prob. 50ECh. 6 - Prob. 51CECh. 6 - Prob. 52CECh. 6 - Prob. 53CECh. 6 - Prob. 54CECh. 6 - Prob. 56CE
Knowledge Booster
Similar questions
- A plane light wave propagates in a medium characterized by the law of dispersion: v = a + bλ, where a, b are positive constants, and v is the phase velocity. Show that the ratio of the path S traversed by the wave during the time interval t to the duration of this interval is equal to the group velocity.arrow_forwardThe amplitude of a scattered wave is given by 1 S(0) = (21 + 1)exp[id] sin 3i P(cos 0), l=0 where e is the angle of scattering, I is the angular momentum, ik is the incident momentum, and & is the phase shift produced by the central potential that is doing the scattering. The total cross section is oiot = S IS(O)²a2. Show that 47 Otot = FL(21+ 1)sin² § . %3D0arrow_forwardShow that for those waves whose angular frequency o and wave number k obey the dispersion relation k?c = o + constant, the product of the phase and group velocities is equal to c, Vg0 ph = c, where c is the speed of light.arrow_forward
- (a) Write the wave function w(x)= wo•e'kx in the form w(x) function 4(x, t) that corresponds to w(x) written in this form. = a + ib, where a and b are real quantities. (Assume that wo is real.) (b) Write the time-dependent wavearrow_forward7.29 The speed of propagation of a surface wave in a liquid of depth much greater than λ is given by V= gλ + 2TT βλ 2TTY where g = acceleration of gravity, λ = wavelength, p = density, Y = surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves).arrow_forward(a) A 2D hexagonal lattice of lattice spacing a = 3 Å has isotropic speed of sound v = 10³ m/s. Starting from the basic assumptions of the Debye model, show that the Debye frequency OD is 1.27 x 10¹3s-¹, and find the magnitude of the Debye wave vector KD (hint: think about how you will accommodate the correct number of phonon modes). (b) Na (sodium) has a body-centered cubic structure with lattice constant a = 4.225 Åarrow_forward
- if the wave function of a s ystem ucr) == (à) e: from the Periodr=othe r= infinite is a Cons tant magnitude , the mean dimension of this function is equalarrow_forwardFind the phase velocity and group velocity for the following two wave-forms (x and t are measured in SI units) (i) 1(x,t) = cos(4t - 2x) + cos(8t- 4.r). (ii) 2(x,t) Does these waveforms satisfy the wave equation %3D = cos(8t- 6a) + cos (4t – 4.x). %3D v² Ət? where v is a constant independent of frequency. Explain.arrow_forwarda с d Consider electromagnetic waves for which the relation εμ w² - w² C between wave number k and the angular frequency w holds, for some constant parameter wc. (Such waves can actually exist in wave guides.) k(w) = = Give a detailed description of the meaning of the constants e, μ and c which appear in the given expression for k(w). Compute the phase velo up as well as the group velocity Ugr for such waves. Compare the resulting expression for the product uph Ugr of the phase and the group velocity with the phase velocity of plane electromagnetic waves. Expand the expression for the phase and group velocities found in part b to lowest non-trivial order in the region w >>> wc. Interpret the result. What happens with the phase and group velocity if w approaches the specific value wc (but still w > wc)? What happens when w is smaller than we?arrow_forward
- (a) A source of spherical waves of frequency f and pressure amplitude A at 1 meter is a distance d above a plane, rigid boundary. Calculate the amplitude of the pressure on the boundary as a function of d and r. (b) For the same source at the same distance above a plane, pressure release boundary, calculate the normal component of the particle velocity on the boundary. (c) Make plots of (a) as a function of R, the distance along the boundary measured from the point closest to the source, for various values of d. (d) Repeat (c) for (b).arrow_forwardFor vibrational states, the Boltzmann equation can be written as = exp(-AE/kT) where N, and N, are the populations of the lower and higher energy states, respectively, AE is the energy difference between the states, k is Boltzmann's constant, and Tis the temperature in kelvins. For temperatures of 20°C and 40°C, cakulate the ratios of the intensities of the anti-Stokes and Stokes lines for CCI, at (a) 218 cm-, (b) 459 cm", (c) 790 cm-!. For each temperature and Raman shift, calculate how much more intense the Stokes line is compared to the anti-Stokes line.arrow_forwardCalculate the longest wave length that can analyses by a rock salt crystal of spacing d = 2.82 A° in the first order.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
University Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSON
Introduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
Lecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON
College Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
University Physics (14th Edition)
Physics
ISBN:9780133969290
Author:Hugh D. Young, Roger A. Freedman
Publisher:PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:9781107189638
Author:Griffiths, David J., Schroeter, Darrell F.
Publisher:Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:9780321820464
Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...
Physics
ISBN:9780134609034
Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:PEARSON