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Physics / Sears and Zemansky's University Physics with Modern Physics 13 / Chapter 15 / Problem 45E

Sears and Zemansky's University Physics with Modern Physics | 13th Edition | ISBN: 9780321696861 | Authors: Hugh D. Young; Roger A. Freedman; A. Lewis Ford

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Consider again the rope and traveling wave of the Exercise

Chapter 15, Problem 45E

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QUESTION:

Problem 45E

Consider again the rope and traveling wave of the Exercise seen below. Assume that the ends of the rope are held fixed and that this traveling wave and the reflected wave are traveling in the opposite direction. (a) What is the wave function y(x, t) for the standing wave that is produced? (b) In which harmonic is the standing wave oscillating? (c) What is the frequency of the fundamental oscillation?

Exercise:

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t) = 2.30 mm cos [(6.98 rad/m)x + (742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude, (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope, (g) average power transmitted by the wave.

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QUESTION:

Problem 45E

Consider again the rope and traveling wave of the Exercise seen below. Assume that the ends of the rope are held fixed and that this traveling wave and the reflected wave are traveling in the opposite direction. (a) What is the wave function y(x, t) for the standing wave that is produced? (b) In which harmonic is the standing wave oscillating? (c) What is the frequency of the fundamental oscillation?

Exercise:

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t) = 2.30 mm cos [(6.98 rad/m)x + (742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude, (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope, (g) average power transmitted by the wave.

ANSWER:

Solution 45E

Step 1:

a) Equation for the standing wave, $y(x,t)\ =2.30\ mm\ cos\ [\ (6.98\ rad/m)\ x\ +\ (742\ rad/s)\ t\ \ ]\$

The derivative w.r.t time is,

$dy(x,t)/dt\ =\ -742\ rad/s\ \times{}2.30\ mm\ sin\ [\ (6.98\ rad/m)\ x\ +\ (742\ rad/s)\ t\ \ ]\$

Again taking time derivative,

$d{\ }^{2}y(x,t)/dt{\ }^{2}\ =\ -(742\ rad/s){\ }^{2}\ \times{}2.30\ mm\ cos\ [\ (6.98\ rad/m)\ x\ +\ (742\ rad/s)\ t\ \ ]\$

Or, we can write,

$d{\ }^{2}y(x,t)/dt{\ }^{2}\ +\ (742\ rad/s){\ }^{2}\ y\ (x,t)\ =\ 0\$

Solution for this equation is,

$y(t)\ =\ T{\ }_{0}cos\ [\ (742\ rad/s)\ t\ +\ ϕ\ \ ]\$

Similarly, the derivative w.r.t position is,

$dy(x,t)/dx\ =\ -6.98\ rad/m\ \times{}2.30\ mm\ sin\ [\ (6.98\ rad/m)\ x\ +\ (742\ rad/s)\ t\ \ ]\$

Again taking time derivative,

$d{\ }^{2}y(x,t)/dx{\ }^{2}\ =\ -(6.98\ rad/m){\ }^{2}\ \times{}2.30\ mm\ cos\ [\ (6.98\ rad/m)\ x\ +\ (742\ rad/s)\ t\ \ ]\$

Or, we can write,

$d{\ }^{2}y(x,t)/dx{\ }^{2}\ +\ (6.98\ rad/s){\ }^{2}\ y\ (x,t)\ =\ 0\$

Solution for this equation is,

$y(x)\ =\ X{\ }_{0}cos\ [\ (6.98\ rad/m)\ x\ +\ δ\ \ ]\$

Therefore, the equation, y (x,t) = y(x) y(t)

$y(x,t)\ =X{\ }_{0}T{\ }_{0}cos\ [\ (6.98\ rad/m)\ x\ +\ δ\ \ ]cos\ [\ (742\ rad/s)\ t\ +\ ϕ\ \ ]$

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