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Chapter 18: Problem 3

Two pulses traveling on the same string are described by $$y_{1}=\frac{5}{(3 x-4 t)^{2}+2} \quad $$ and $$\quad y_{2}=\frac{-5}{(3 x+4 t-6)^{2}+2}$$ (a) In which direction does each pulse travel? (b) At what time do the two cancel everywhere? (c) At what point do the two pulses always cancel?

Short Answer

Expert verified
The first pulse, represented by \(y_{1}\), travels forwards, while the second pulse, represented by \(y_{2}\), travels backwards. The two pulses cancel each other out at time \(t = 1\) second and they always cancel each other at position \(x = 2\) meter.

Step by step solution

01

Determine the direction of each pulse

The direction of the pulse depends on the sign of the coefficient of \(t\) in the denumerator of the fraction. For a forward-travelling wave, \(x - vt\) should be in the denumerator, where \(v\) is the velocity of the wave. This means the sign of the velocity should be the opposite of the sign of the time coefficient. For \(y_{1}\), the coefficient of \(t\) is -4, meaning this pulse is travelling forwards. For \(y_{2}\), the coefficient of \(t\) is 4, hence this pulse is travelling backwards.
02

Determine the time when the two pulses cancel everywhere

For the two pulses to cancel out everywhere, the sum of the displacements \(y_{1}\) and \(y_{2}\) must be zero at a specific time. Therefore, we have to solve the equation \(y_{1} + y_{2} = 0\). That leads to: \[\frac{5}{(3x - 4t)^2 + 2} - \frac{5}{(3x + 4t -6)^2 + 2} = 0\] Upon simplification, \(t = 1\) (since no real solution exists for \(x\)). Therefore, the two pulses cancel each other at \(t = 1\) second.
03

Determine the point where the two pulses always cancel

For the pulses to cancel out at a specific point, \(y_{1} = -y_{2}\) at that point for every time value. This sets up: \[\frac{5}{3x - 4t^2 + 2} = \frac{-5}{3x + 4t -6^2 + 2}\] After cleaning and simplifying, we'll find that \(x = 2\). Therefore, at \(x = 2\) meter, the two pulses always cancel each other

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Direction
In wave mechanics, the direction in which a wave travels is determined by carefully examining the equation describing it. For the given pulses, we found out that their direction relies largely on the coefficient of the time variable, denoted as "t."
If a wave equation is of the form \(x - vt\), where \(v\) is the wave's velocity, the wave moves forward. In this case, \(y_1\) displays a negative time coefficient (-4), indicating that it travels forward. Conversely, \(y_2\) has a positive time coefficient (4), meaning it moves backward.
  • Forward-moving waves indicate a decrease in the phase term: \(x - vt\).
  • Backward-moving waves indicate an increase in the phase term: \(x + vt\).
Understanding the wave's direction is crucial as it affects interactions such as interference and cancellation.
Wave Cancellation
Wave cancellation occurs when two waves meet and result in a decreased or nullified amplitude at specific points or times. In this exercise, cancellation happens when both wave displacements add up to zero.
This is the essence of wave interference, particularly destructive interference, where two waves of equal amplitude and opposite phases meet.
  • To find when they cancel everywhere, we solve \( y_1 + y_2 = 0 \).
  • Simplifying provided us with \( t = 1\) second as the time of cancellation.
This means at one second, the total displacement of the waves is zero, a total cancellation.
The setting happens across all space simultaneously at this precise time, showcasing the phenomenon of wave superposition.
Wave Propagation
Wave propagation refers to the manner in which waves travel through a medium. It's about how the energy carried by the wave moves from one location to another.
In the given problem, it's important to note how these waves, although defined on a single string, exhibit different propagation patterns.
  • Wave \( y_1 \) progresses forward, as stated, moving in the direction of decreasing \(x\) values.
  • In contrast, wave \( y_2 \) exhibits backward propagation, moving in the direction of increasing \(x\) values.
This dual behavior is pivotal when analyzing situations of wave interference, as it affects the timing and locations of the resulting interference patterns.
Careful consideration of propagation ensures that one can predict how interference will occur, both spatially and temporally over the medium.

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Most popular questions from this chapter

A series of pulses, each of amplitude \(0.150 \mathrm{m},\) is sent down a string that is attached to a post at one end. The pulses are reflected at the post and travel back along the string without loss of amplitude. What is the net displacement at a point on the string where two pulses are crossing, (a) if the string is rigidly attached to the post? (b) if the end at which reflection occurs is free to slide up and down?

Standing Waves in Air Columns Note: Unless otherwise specified, assume that the speed of sound in air is \(343 \mathrm{m} / \mathrm{s}\) at \(20^{\circ} \mathrm{C},\) and is described by $$ v=(331 \mathrm{m} / \mathrm{s}) \sqrt{1+\frac{T_{\mathrm{C}}}{273^{\circ}}} $$ at any Celsius temperature \(T_{\mathrm{C}}\). The overall length of a piccolo is \(32.0 \mathrm{cm} .\) The resonating air column vibrates as in a pipe open at both ends. (a) Find the frequency of the lowest note that a piccolo can play, assuming that the speed of sound in air is \(340 \mathrm{m} / \mathrm{s} .\) (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is \(4000 \mathrm{Hz}\), find the distance between adjacent antinodes for this mode of vibration.

A string of length \(L\), mass per unit length \(\mu,\) and tension \(T\) is vibrating at its fundamental frequency. What effect will the following have on the fundamental frequency? (a) The length of the string is doubled, with all other factors held constant. (b) The mass per unit length is doubled, with all other factors held constant. (c) The tension is doubled, with all other factors held constant.

Verify by direct substitution that the wave function for a standing wave given in Equation 18.3 $$ y=2 A \sin k x \cos \omega t $$ is a solution of the general linear wave equation, Equation 16.27: $$ \frac{\partial^{2} y}{\partial x^{2}}=\frac{1}{v^{2}} \frac{\partial^{2} y}{\partial t^{2}} $$

A student holds a tuning fork oscillating at \(256 \mathrm{Hz}\). He walks toward a wall at a constant speed of \(1.33 \mathrm{m} / \mathrm{s}\) (a) What beat frequency does he observe between the tuning fork and its echo? (b) How fast must he walk away from the wall to observe a beat frequency of \(5.00 \mathrm{Hz} ?\)
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