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Chapter 2: Q9E (page 497)

Which of the following wave functions satisfies the wave equation, Eq. (15.12)? (a)\(y\left( {x,t} \right) = Acos\left( {kx + \omega t} \right)\) ; (b)\(y\left( {x,t} \right) = Asin\left( {kx + \omega t} \right)\) ; (c) \(y\left( {x,t} \right) = A\left( {coskx + cos\omega t} \right)\) (d) For the wave of part (b), write the equations for the transverse velocity and transverse acceleration of a particle at point x.

Short Answer

Expert verified

(a) Wave equation Eq. (15.12) is satisfied, if \(\nu = \frac{\omega }{k}\)

Step by step solution

01

Given data

\(y\left( {x,t} \right) = Acos\left( {kx + \omega t} \right)\)

02

Concept/ Formula used

Wave equation involves second partial derivatives of wave function.

\(\frac{{{\partial ^2}y\left( {x,t} \right)}}{{\partial {x^2}}} = \frac{1}{{{\nu ^2}}}\frac{{{\partial ^2}y\left( {x,t} \right)}}{{\partial {t^2}}}\) . . . (15.12)

Where,\(\nu \)is wave speed.

03

Verification for wave function

(a)

\(\begin{aligned}{l}\frac{{{\partial ^2}y\left( {x,t} \right)}}{{\partial {x^2}}} = - A{k^2}\cos \left( {kx + \omega t} \right)\\\frac{{{\partial ^2}y\left( {x,t} \right)}}{{\partial {t^2}}} = - A{\omega ^2}\cos \left( {kx + \omega t} \right)\\\end{aligned}\)\(\)

Wave equation Eq. (15.12) is satisfied, if \(\nu = \frac{\omega }{k}\)

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Most popular questions from this chapter

Speed of Propagation vs. Particle Speed. (a) Show that Eq. (15.3) may be written as

\(y\left( {x,t} \right) = Acos\left[ {\frac{{2\pi }}{\lambda }\left( {x - vt} \right)} \right]\)

(b) Use \(y\left( {x,t} \right)\) to find an expression for the transverse velocity \({v_y}\)of a particle in the string on which the wave travels. (c) Find the maximum speed of a particle of the string. Under what circumstances is this equal to the propagation speed \(v\) ? Less than\(v\)? Greater than\(v\)?

Is it possible to have a longitudinal wave on a stretched string? Why or why not? Is it possible to have a transverse wave on a steel rod? Again, why or why not? If your answer is yes in either case, explain how you would create such a wave.

Two loudspeakers, A and B (Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speakers emit sound waves that travel directly from the speaker to point Q. What is the lowest frequency for which (a) constructive interference occurs at point Q; (b) destructive interference occurs at point Q?

If you stretch a rubber band and pluck it, you hear a (somewhat) musical tone. How does the frequency of this tone change as you stretch the rubber band further? (Try it!) Does this agree with Eq. (15.35) for a string fixed at both ends? Explain.

A strong string of mass 3.00 g and length 2.20 m is tied to supports at each end and is vibrating in its fundamental mode. The maximum transverse speed of a point at the middle of the string is \(9.00\;{\rm{m/s}}\). The tension in the string is 330 N.

(a) What is the amplitude of the standing wave at its antinode?

(b) What is the magnitude of the maximum transverse acceleration of a point at the antinode?
See all solutions

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