Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 6, Problem 46E
(a)
To determine
To sketch:
Group velocity versus wave number.
(b)
To determine
Group velocity and wavelength
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
The angular frequency of the surface waves in a liquid is given in terms of the wave number
k by w = Vgk +Tk³/p, where g is the acceleration due to gravity, p is the density of the
liquid, and T is the surface tension (which gives an upward force on an element of the
surface liquid). Find the phase and group velocities for the limiting cases when the surface
waves have: (a) very large wavelengths and (b) very small wavelengths.
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.
TTX
Consider the wave function V(x, t) = A(cos (A)) e-jot for – 1
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
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- 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.arrow_forwardUsing the linearized theory, calculate the lift and wave-drag coefficients for an infinitely thin flat plate in a Mach 2.6 freestream at angles of attack of(a) α = 5◦ (b) α = 15◦ (c) α = 30◦Compare these approximate results with those from the exact shockexpansion theory. What can you conclude about the accuracy of linearized theory in this case?arrow_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_forward
- The differential equation for transverse vibrations of a string whose density increases linearly from one end to the other is y + (Ax + B)y = 0, where A and B are constants. Find the general solution of this equation in terms of Bessel functions. Hint: Make the change of variable Ax + B = Au.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_forwardQ.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?arrow_forward
- You place your ear onto a steel railroad track and hear the sound of a distant train through the rails Δt = 3.5 seconds before you do through the air. The speed of sound in steel is vs = 6100 m/s, and and the air temperature is 48° C. Find the distance, D, to the train in meters.arrow_forwardA simple harmonic progressive wave of amplitude 2 mm and frequency 500 Hz travels with a velocity of 350 m/s in given medium. Write down the equation of the wave in S.I. units.arrow_forwardThe period of oscillation T of a water surface wave isassumed to be a function of density ρ , wavelength l , depth h , gravity g , and surface tension Y . Rewrite this relationshipin dimensionless form. What results if Y is negligible?Hint: Take l , ρ , and g as repeating variables.arrow_forward
- If p=A*exp(i[kx-ωt]) is a plane wave in a medium with a density of ρa, determine the acoustic particle velocity, u.arrow_forwardThe dispersion equation for Waves is: w2(k)=(A+Bk2)(A+Bk2+Ck3) in the case when k<<B/C calculate phase and group velocities of these waves as function of "k", and plot the graphs of these functionsarrow_forwardWaves on deep water with surface tension T and density p are governed by the dispersion relation: ² = gk +k³, where w is the angular frequency of the wave, k the wavenumber, and g is a constant. Calculate: 1- The phase and group velocities of the waves. 2- The critical wavenumber, ke, when the phase speed of the wave reaches a minimum.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning