Physics for Scientists and Engineers with Modern Physics
10th Edition
ISBN: 9781337553292
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 17, Problem 8P
To determine
To verify: The wave function for a standing wave given in the equation is a solution of the general linear wave equation.
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Let y₁(x, t) A₁ cos(k₁x – w₁t) and y2(x, t) = A2 cos(k2x — w2t) be two solutions to the wave
equation with the same wave speed v. Show that the superposed wave function
y(x, t) = y₁(x, t) + y₂(x, t)
is also a solution to the wave equation.
WS19 (rev. 2.5)
Page 71
19.
Waves
Problems
1. The equation of a certain wave on a string is given by
y(x,t) = 0.1 sin{2r(x - 10t)}
with x and y in meters, andt in seconds.
Which way is the wave moving, to the left or to the right? How do you know?
(b) What is the wavelength of the wave? frequency? phase speed?
(c) If the string has a mass per unit length of 0.1 kg/m, what is the tension in the
string?
(d) What is the maximum transverse velocity of the string (that is, the maximum
vertical velocity of particles in the cord)?
[Ans: (b) 1 m; 10 Hz; 10 m/s; (c) 10 N; (d) 2n m/s]
(a)
bow do
Chapter 17 Solutions
Physics for Scientists and Engineers with Modern Physics
Ch. 17.1 - Prob. 17.1QQCh. 17.2 - Consider the waves in Figure 17.8 to be waves on a...Ch. 17.4 - When a standing wave is set up on a string fixed...Ch. 17.6 - Prob. 17.4QQCh. 17.6 - Prob. 17.5QQCh. 17 - Two waves on one string are described by the wave...Ch. 17 - Two pulses of different amplitudes approach each...Ch. 17 - Two wave pulses A and B are moving in opposite...Ch. 17 - Why is the following situation impossible? Two...Ch. 17 - Two pulses traveling on the same string are...
Ch. 17 - Two identical loudspeakers 10.0 m apart are driven...Ch. 17 - Two sinusoidal waves on a string are defined by...Ch. 17 - Prob. 8PCh. 17 - Prob. 9PCh. 17 - Prob. 10PCh. 17 - Prob. 11PCh. 17 - Prob. 12PCh. 17 - A string that is 30.0 cm long and has a mass per...Ch. 17 - Prob. 14PCh. 17 - Review. A sphere of mass M = 1.00 kg is supported...Ch. 17 - Prob. 16PCh. 17 - Prob. 17PCh. 17 - Prob. 18PCh. 17 - Prob. 19PCh. 17 - Prob. 20PCh. 17 - The fundamental frequency of an open organ pipe...Ch. 17 - Ever since seeing Figure 16.22 in the previous...Ch. 17 - An air column in a glass tube is open at one end...Ch. 17 - Prob. 24PCh. 17 - Prob. 25PCh. 17 - Prob. 26PCh. 17 - As shown in Figure P17.27, water is pumped into a...Ch. 17 - As shown in Figure P17.27, water is pumped into a...Ch. 17 - Prob. 29PCh. 17 - Prob. 30PCh. 17 - Prob. 31PCh. 17 - Prob. 32PCh. 17 - Prob. 33PCh. 17 - Prob. 34APCh. 17 - Prob. 35APCh. 17 - A 2.00-m-long wire having a mass of 0.100 kg is...Ch. 17 - Prob. 37APCh. 17 - Prob. 38APCh. 17 - Prob. 39APCh. 17 - Review. For the arrangement shown in Figure...Ch. 17 - Prob. 41APCh. 17 - Two speakers are driven by the same oscillator of...Ch. 17 - Prob. 43APCh. 17 - Prob. 44APCh. 17 - Prob. 45APCh. 17 - Prob. 46APCh. 17 - Review. A 12.0-kg object hangs in equilibrium from...Ch. 17 - Review. An object of mass m hangs in equilibrium...Ch. 17 - Prob. 49APCh. 17 - Prob. 50CP
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- A copper wire has a density of =8920 kg/m3, a radius of 1.20 mm, and a length L. The wire is held under a tension of 10.00 N. Transverse waves are sent down the wire. (a) What is the linear mass density of the wire? (b) What is the speed of the waves through the wire?arrow_forwardTwo sinusoidal waves are moving through a medium in the positive x-direction, both having amplitudes of 7.00 cm, a wave number of k=3.00 m-1, an angular frequency of =2.50 s-1, and a period of 6.00 s, but one has a phase shift of an angle =12 rad. What is the height of the resultant wave at a time t=2.00 s and a position x=0.53 m?arrow_forwardVerify by direct substitution that the wave function for a standing wave given by: y= (2A cos kx) cos wt is a solution of the general linear wave equation: 1 a?y ax?arrow_forward
- Show that a standing wave given by the equation: y (x, t) = A sin (kx) sin (ωt) satisfies the wave equation, verify that: v0 = ω / k; shows that the standing wave also satisfies the equation of harmonic oscillator: ∂2y(x,t)/∂t2 = −ω2y(x,t), interpret this result.arrow_forwardQuestion 19 A string hanging from a shelf supports aarrow_forwardAn E&M wave has an e-field given by E(x,t) - -(4.950V/m)k cos(ky + (1.38 x 101 rud/)t) Which of the following is correct for B(x, t) and the direction of propagation? (c - 3x10 m/s) B(7, t) - -(0.165 NA)i cos ((4.6 x 10° rad/m)y + (1.38 x 1015 rad/)e),y B(x, t) - (33.0 nT)j cos (2.3 x 10*rad/m)y + (2.78 x 101 rad/s)e). - d/m)y + (1.38 x 10 rad 1. I. rad/).-y B(x, t) = (11.5 aT)t cos ((4.6 x 10 rad/mly + (1.38 x 10 rad (1.38 x 101 rad, /,). -y. B(x, t) - (16.5 nT)t cos ((4.6 x 10* rad II. IV. IV None of the abovearrow_forward
- A transverse wave traveling along an x axis has the fornm given by (16-18) y =y," sin(kx ± ω1 + φ). Figure 16-8a gives the displacement of string elements as a function of , al at time0. Figure 16-8h gives the displacements of the element at x 0 as a function oft. Find the values of the quantities shown in Eq. 16-18, including the correct choice of sign. (min) 10 20 -10 -20 -9 *レ b)arrow_forwardConsider the one-dimensional wave equation Pu u 0<<1, t20 u(0, t) = u(1, t) = 0 u(x, 0) = 3 sin(27x) du (x, 0) = x – x². Use CTCS scheme with Ar = 0.1, At = 0,05 to estimate the value of the displacement u(0.9,0.15).arrow_forwardExample 14-8 depicts the following scenario. Two people relaxing on a deck listen to a songbird sing. One person, only 1.66 m from the bird, hears the sound with an intensity of 2.86×10−6 W/m^2. A bird-watcher is hoping to add the white-throated sparrow to her "life list" of species. How far could she be from the bird described in example 14-8 and still be able to hear it? Assume no reflections or absorption of the sparrow's sound.arrow_forward
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