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Chapter 1: Problem 3

Investigate the motion of a string having transverse waves on it specified by the equations \(\eta=A \sin (\kappa x-\omega t)\) and \(\zeta=A \cos (x x-\omega t)\). If the string, instead of being infinitely long, is attached to a wall, so that reflections occur, find what sort of standing-wave pattern exists. Does the motion have a connection with rope jumping \(^{2}\) \(\dagger\) For an analysis of the nonlinear vibration of a string segment with fixed ends, see G. F. Carrier, Quart. Appl. Math., 3: 157 (1945).

Short Answer

Expert verified
Answer: The motion of the rope during rope jumping is similar to a standing wave pattern, with nodes at the ends of the rope where the person is holding it and antinodes created by the peak wavelengths as they oscillate up and down. In rope jumping, avoiding the rope involves jumping over the antinode at the appropriate moment in the wave's oscillation.

Step by step solution

01

Understanding Transverse Waves

Transverse waves are waves that oscillate perpendicular to the direction of energy propagation. The wave travels along the string but causes the individual segments of the string to move up and down. The equations given are for two different types of transverse waves - a sine wave and a cosine wave - both characterized by the amplitude A, wavenumber \(\kappa\), and angular frequency \(\omega\).
02

Examining the Effects of Attaching the String to a Wall

When the string is attached to a wall, reflections occur at its endpoints. Due to the boundary conditions (string's ends being fixed to the wall), the reflected waves will interfere with the incident waves, leading to the formation of standing waves.
03

Finding the Standing Wave Pattern

A standing wave pattern is a result of the superposition of the incident and reflected waves. To find the standing wave \(\Upsilon\), we can add the individual sine and cosine waves, resulting in \(\Upsilon (x,t) = \eta(x,t) + \zeta(x,t)\). The resulting pattern is characterized by points of maximum and minimum amplitude known as antinodes and nodes. At the wall, there will always be a node because the string cannot oscillate at that point. The general equation for a standing wave is \(\Upsilon (x,t) = 2A \sin(\kappa x) \cos(\omega t)\).
04

Investigating the Connection to Rope Jumping

Rope jumping involves a rope attached to two fixed points, with a person hopping over the oscillating waves as the rope is swung. The motion of the rope during rope jumping is similar to the studied standing wave pattern, with nodes at the ends of the rope where the person is holding it while the middle creates antinodes as the peak wavelengths regularly move upwards and then downwards. During rope jumping, avoiding the rope involves jumping over the antinode at the appropriate moment in the wave's oscillation.

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Most popular questions from this chapter

Prove the following trigonometric identities by expressing the left side in complex form: (a) \(\cos (x+y)=\cos x \cos y-\sin x \sin y\) (b) \(\sin (x+y)=\sin x \cos y+\cos x \sin y\) (c) \(\sum_{n=1}^{N} \cos n \theta=\frac{\cos \frac{1}{3}(N+1) \theta \sin \frac{1}{2} N \theta}{\sin \frac{1}{2} \theta}\) (d) \(\sum_{n=1}^{N} \sin n \theta=\frac{\sin \frac{1}{2}(N+1) \theta \sin \frac{1}{2} N \theta}{\sin \frac{1}{2} \theta}\) Many other trigonometric identities can be similarly proved. 1.3.3 Investigate the multiplication and division of two complex numbers. Show that in general the product (quotient) of their real parts does not equal the real part of the product (quotient) of the two numbers.

A long string, for which the transverse wave velocity is \(c\), is given a displacement specified by some function \(\eta=\eta_{0}(x)\) that is localized near the middle of the string. The string is released at \(t=0\) with zero initial velocity. Find the equations for the traveling waves that are produced and make a sketch showing the waves at several instants of time with \(t \geq 0\). Hint: Find two waves traveling in opposite directions that together satisfy the initial conditions.

The two waves \(\eta_{1}=A_{1} \cos (\kappa x-\omega t)\) and \(\eta_{2}=A_{2} \cos (\kappa x+\omega t)\) travel together on a stretched string. Show how the wave amplitude ratio \(A_{1} / A_{2}\) can be found from a measurement of the so-called standing-wave ratio \(S\) of the resulting wave pattern, which is defined as the ratio of the maximum amplitude to the minimum amplitude of the envelope of this pattern. Note that \(S\) and the position and amplitude of the minima (or maxima) wave disturbance uniquely determine the two traveling waves that give rise to the standing-wave pattern.

Show how the solution \((1.5 .5)\) of the wave equation can be transformed into a general traveling-wave solution.

Investigate the multiplication and division of two complex numbers. Show that in general the product (quotient) of their real parts does not equal the real part of the product (quotient) of the two numbers.
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