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Consider again the rope and traveling wave. Assume that
Chapter 15, Problem 45E(choose chapter or problem)
Consider again the rope and traveling wave. 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. Yon 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.
Questions & Answers
QUESTION:
Consider again the rope and traveling wave. 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. Yon 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 × 2.30 mm sin [ (6.98 rad/m) x + (742 rad/s) t ] Again taking time derivative, 2 2 2 d y(x,t)/dt = (742 rad/s) × 2.30 mm cos [ (6.98 rad/m) x + (742 rad/s) t ] Or, we can write, 2 2 2 d y(x,t)/dt + (742 rad/s) y (x,t) = 0 Solution for this equation is, y(t) = T co0 [ (742 rad/s) t + ] Similarly, the derivative w.r.t position is, dy(x,t)/dx = 6.98 rad/m × 2.30 mm sin [ (6.98 rad/m) x + (742 rad/s) t ] Again taking time derivative, 2 2 2 d y(x,t)/dx = (6.98 rad/m) × 2.30 mm cos [ (6.98 rad/m) x + (742 rad/s) t ] Or, we can write, 2 2 2 d y(x,t)/dx + (6.98 rad/s) y (x,t) = 0 Solution for this equation is, y(x) = X co0 [ (6.98 rad/m) x + ] Therefore, the equation, y (x,t) = y(x) y(t) y(x,t) = X T0cos0[ (6.98 rad/m) x + ]cos [ (742 rad/s) t + ]