PHYSICS:F/SCI.+ENGRS-W/WEBASSIGN
PHYSICS:F/SCI.+ENGRS-W/WEBASSIGN
10th Edition
ISBN: 9781337888479
Author: SERWAY
Publisher: CENGAGE L
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Chapter 16, Problem 22P

(a)

To determine

To shows: That the wave function y(x,t)=x2+v2t2 is a solution to the linear wave equation.

(a)

Expert Solution
Check Mark

Explanation of Solution

Any function is a solution of linear wave equation in general if it satisfies the equation completely.

The linear wave equation in general is,

  2yx2=1v22yt2

The given wave function is,

  y(x,t)=x2+v2t2        (I)

Differentiate equation (I) partially with respect to x.

  y(x,t)x=x(x2+v2t2)=2x

Again differentiate partially with respect to x.

  2y(x,t)x2=x(2x)=2        (II)

Differentiate equation (I) partially with respect to t.

  y(x,t)t=t(x2+v2t2)=v2(2t)

Again differentiate partially with respect to t.

  2y(x,t)t2=t{v2(2t)}=2v21v22y(x,t)t2=2=2y(x,t)x2

Conclusion:

Therefore, the wave function y(x,t)=x2+v2t2 is a solution to the linear wave equation.

(b)

To determine

To shows: That the wave function y(x,t)=x2+v2t2 can be written as f(x+vt)+g(xvt) and determine the functional form of f and g.

(b)

Expert Solution
Check Mark

Answer to Problem 22P

The functional form of f is 12(x+vt)2 and the functional form of g is 12(xvt)2.

Explanation of Solution

It can be proved as,

  f(x+vt)+g(xvt)=12(x+vt)2+12(xvt)2=12(x2+v2t2+2xvt)+12(x2+v2t22xvt)=12×2(x2+v2t2)=y(x,t)

Therefore,

The functional form of f=12(x+vt)2

The functional form of g=12(xvt)2

Conclusion:

Therefore, the functional form of f is 12(x+vt)2 and the functional form of g is 12(xvt)2.

(c)

To determine

Repeat part (a) and part (b) for the function y(x,t)=sin(x)cos(vt).

(c)

Expert Solution
Check Mark

Explanation of Solution

Any function is a solution of linear wave equation in general if it satisfies the equation completely.

The given wave function is,

    y=sin(x)cos(vt)        (III)

Differentiate equation (I) partially with respect to x.

    y(x,t)x=x(sin(x)cos(vt))=cos(x)cos(vt)

Again differentiate partially with respect to x.

    2y(x,t)x2=x(cos(x)cos(vt))=sin(x)cos(vt)        (VI)

Differentiate equation (I) partially with respect to t.

     y(x,t)t=t(sin(x)cos(vt))=vsin(x)sin(vt)

Again differentiate partially with respect to t.

    2y(x,t)t2=t{vsin(x)sin(vt)}=v2sin(x)cos(vt)1v22y(x,t)t2=sin(x)cos(vt)=2y(x,t)x2

From the trigonometry,

  sin(x+vt)=sinxcosvt+cosxsinvt        (I)

  sin(xvt)=sinxcosvtcosxsinvt        (II)

Add equation (I) and (II).

  sin(x+vt)+sin(xvt)=sinxcosvt+cosxsinvt+sinxcosvtcosxsinvt=2sinxcosvt12sin(x+vt)+12sin(xvt)=sin(x)cos(vt)

Conclusion:

Therefore, the functional form of f is 12sin(x+vt) and the functional form of g is 12sin(xvt).

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Chapter 16 Solutions

PHYSICS:F/SCI.+ENGRS-W/WEBASSIGN

Ch. 16.9 - An airplane flying with a constant velocity moves...Ch. 16 - A seismographic station receives S and P waves...Ch. 16 - Two points A and B on the surface of the Earth are...Ch. 16 - You are working for a plumber who is laying very...Ch. 16 - You are working on a senior project and are...Ch. 16 - When a particular wire is vibrating with a...Ch. 16 - (a) Plot y versus t at x = 0 for a sinusoidal wave...Ch. 16 - Consider the sinusoidal wave of Example 16.2 with...Ch. 16 - A sinusoidal wave traveling in the negative x...Ch. 16 - (a) Write the expression for y as a function of x...Ch. 16 - Review. The elastic limit of a steel wire is 2.70 ...Ch. 16 - Transverse waves travel with a speed of 20.0 m/s...Ch. 16 - Why is the following situation impossible? An...Ch. 16 - Tension is maintained in a string as in Figure...Ch. 16 - Prob. 14PCh. 16 - Transverse waves are being generated on a rope...Ch. 16 - In a region far from the epicenter of an...Ch. 16 - A long string carries a wave; a 6.00-m segment of...Ch. 16 - A two-dimensional water wave spreads in circular...Ch. 16 - A horizontal string can transmit a maximum power...Ch. 16 - Prob. 20PCh. 16 - Show that the wave function y = eb(x vt) is a...Ch. 16 - Prob. 22PCh. 16 - A sinusoidal sound wave moves through a medium and...Ch. 16 - Earthquakes at fault lines in the Earths crust...Ch. 16 - An experimenter wishes to generate in air a sound...Ch. 16 - A sound wave propagates in air at 27C with...Ch. 16 - Prob. 27PCh. 16 - A rescue plane flies horizontally at a constant...Ch. 16 - The speed of sound in air (in meters per second)...Ch. 16 - A sound wave moves down a cylinder as in Figure...Ch. 16 - The intensity of a sound wave at a fixed distance...Ch. 16 - The intensity of a sound wave at a fixed distance...Ch. 16 - The power output of a certain public-address...Ch. 16 - A fireworks rocket explodes at a height of 100 m...Ch. 16 - You are working at an open-air amphitheater, where...Ch. 16 - Why is the following situation impossible? It is...Ch. 16 - Show that the difference between decibel levels 1...Ch. 16 - Submarine A travels horizontally at 11.0 m/s...Ch. 16 - Prob. 39PCh. 16 - Why is the following situation impossible? At the...Ch. 16 - Review. A block with a speaker bolted to it is...Ch. 16 - Review. A block with a speaker bolted to it is...Ch. 16 - A sinusoidal wave in a rope is described by the...Ch. 16 - The wave is a particular type of pulse that can...Ch. 16 - Some studies suggest that the upper frequency...Ch. 16 - An undersea earthquake or a landslide can produce...Ch. 16 - A sinusoidal wave in a string is described by the...Ch. 16 - A rope of total mass m and length L is suspended...Ch. 16 - A wire of density is tapered so that its...Ch. 16 - Prob. 50APCh. 16 - Prob. 51APCh. 16 - A train whistle (f = 400 Hz) sounds higher or...Ch. 16 - Review. A 150-g glider moves at v1 = 2.30 m/s on...Ch. 16 - Consider the following wave function in SI units:...Ch. 16 - Prob. 55APCh. 16 - Prob. 56APCh. 16 - A string on a musical instrument is held under...Ch. 16 - Assume an object of mass M is suspended from the...Ch. 16 - Equation 16.40 states that at distance r away from...Ch. 16 - In Section 16.7, we derived the speed of sound in...
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