/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Two waves in one string are desc... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 18: Problem 1

Two waves in one string are described by the wave functions $$y_{1}=3.0 \cos (4.0 x-1.6 t)$$ and $$ y_{2}=4.0 \sin (5.0 x-2.0 t)$$ where \(y\) and \(x\) are in centimeters and \(t\) is in seconds. Find the superposition of the waves \(y_{1}+y_{2}\) at the points (a) \(x=1.00\), \(t=1.00,\) (b) \(x=1.00, t=0.500,\) and (c) \(x=0.500, t=0\) (Remember that the arguments of the trigonometric functions are in radians.)

Short Answer

Expert verified
The superposition of the waves at (a) \(x=1.00, t=1.00\), (b) \(x=1.00, t=0.500\), and (c) \(x=0.500, t=0\) are calculated by substituting the respective x and t values into both wave functions, simplifying, and adding the resultant displacements.

Step by step solution

01

Understand Superposition

The superposition of two wave functions is simply the sum of the functions. The resulting wave at any point in time and space is found by adding the displacements of the individual waves at that point.
02

Calculate Superposition at (a) x=1.00, t=1.00

Substitute the values of x and t into the two wave functions: \(y_1 = 3.0 \cos(4.0 \times 1.00 - 1.6 \times 1.00)\) and \(y_2 = 4.0 \sin(5.0 \times 1.00 - 2.0 \times 1.00)\), and then add them together to find the total displacement at that point.
03

Calculate Superposition at (b) x=1.00, t=0.500

Again, substitute the values into the wave functions, \(y_1 = 3.0 \cos(4.0 \times 1.00 - 1.6 \times 0.500)\) and \(y_2 = 4.0 \sin(5.0 \times 1.00 - 2.0 \times 0.500)\), and sum them to find the total displacement.
04

Calculate Superposition at (c) x=0.500, t=0

Substitute x=0.500 and t=0 into both functions, \(y_1 = 3.0 \cos(4.0 \times 0.500 - 1.6 \times 0)\) and \(y_2 = 4.0 \sin(5.0 \times 0.500 - 2.0 \times 0)\), to find the displacement at that point.
05

Simplify and Add the Functions

Use a calculator to compute the cosine and sine for the given values, simplify each term, and then add them together to get the total superposition for each case.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Mechanical Waves
Understanding mechanical waves is essential in physics, as they are disturbances that travel through a medium, such as air, water, or solids. These vibrations transfer energy from one location to another without the actual movement of matter as a whole.

For example, when you speak, your vocal cords create a mechanical wave that travels through the air, allowing others to hear what you say. The two main types of mechanical waves are longitudinal waves, where particles oscillate parallel to the direction of energy transfer, and transverse waves, with particles moving perpendicular to the direction of wave travel.

In our exercise, we are dealing with transverse waves on a string, characterized by their wavelength, frequency, amplitude, and speed. Each wave function describes how a point on the string moves over time. By understanding how to combine these wave functions, students can predict the pattern resulting from their interference鈥攁 fundamental aspect of wave dynamics.
Wave Function
A wave function is a mathematical description of the wave at any given time and space coordinates. Typically, it dictates the wave's displacement concerning a position and time. For mechanical waves on a string, the wave function can be written in terms of trigonometric functions of sine or cosine, often including variables that represent time (t), position (x), wavelength, and frequency.

Interpreting Wave Functions

In our example, we have two wave functions, each representing individual waves traveling on the same string. When these two waves meet, they superpose, meaning their displacements add together at each point along the string. This principle is known as the superposition principle鈥攁 crucial concept in wave mechanics. By carefully analyzing the wave functions given in this exercise, students learn to visualize the combined effect of interacting waves, which is pivotal in many branches of physics and engineering.
Trigonometric Functions
Trigonometric functions like sine and cosine play a significant role in describing wave motions. These functions repeat in a regular and predictable pattern called a 'cycle,' making them perfect for modeling the periodic nature of waves.

In practical terms, these functions allow us to calculate precise points on a wave, such as the crest, trough, or equilibrium, based on the wave's amplitude, frequency, and phase constant. When dealing with wave functions in physics, remember that the arguments of trigonometric functions are always in radians, not degrees.

For students analyzing wave behavior, familiarity with trigonometric functions is paramount since they're used to find the position and velocity of points on the wave at any given time鈥攍ike in the textbook example provided. Moreover, by mastering trigonometric identities, students can simplify the addition or subtraction of wave functions with ease, enhancing their ability to tackle complex wave interactions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Two waves in a long string have wave functions given by $$y_{1}=(0.0150 \mathrm{m}) \cos \left(\frac{x}{2}-40 t\right)$$ and $$y_{2}=(0.0150 \mathrm{m}) \cos \left(\frac{x}{2}+40 t\right)$$ where \(y_{1}, y_{2},\) and \(x\) are in meters and \(t\) is in seconds. (a) Determine the positions of the nodes of the resulting standing wave. (b) What is the maximum transverse position of an element of the string at the position \(x=0.400 \mathrm{m}\) ?

Two speakers are driven in phase by a common oscillator at 800 \(\mathrm{Hz}\) and face each other at a distance of 1.25 \(\mathrm{m}\) . Locate the points along a line joining the two speakers where relative minima of sound pressure amplitude would be expected. (Use \(v=343 \mathrm{m} / \mathrm{s} . )\)

Two traveling sinusoidal waves are described by the wave functions $$y_{1}=(5.00 \mathrm{m}) \sin [\pi(4.00 x-1200 t)]$$ and $$y_{2}=(5.00 \mathrm{m}) \sin [\pi(4.00 x-1200 t-0.250)]$$ where \(x, y_{1},\) and \(y_{2}\) are in meters and \(t\) is in seconds. (a) What is the amplitude of the resultant wave? (b) What is the frequency of the resultant wave?

A standing-wave pattern is observed in a thin wire with a length of 3.00 \(\mathrm{m}\) . The equation of the wave is $$y=(0.002 \mathrm{m}) \sin (\pi x) \cos (100 \pi t)$$ where \(x\) is in meters and \(t\) is in seconds. (a) How many loops does this pattern exhibit? (b) What is the fundamental frequency of vibration of the wire? (c) What If? If the original frequency is held constant and the tension in the wire is increased by a factor of \(9,\) how many loops are present in the new pattern?

Review problem. 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?
See all solutions

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Read Explanation

Scientific Method Physics

Read Explanation

Physics of Motion

Read Explanation

Electromagnetism

Read Explanation

Fluids

Read Explanation

Waves Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.