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Chapter 16: Problem 36

Four waves are to be sent along the same string, in the same direction: $$ \begin{array}{l} y_{1}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t) \\ y_{2}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+0.7 \pi) \\ y_{3}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+\pi) \\ y_{4}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+1.7 \pi) . \end{array} $$ What is the amplitude of the resultant wave?

Short Answer

Expert verified
The amplitude of the resultant wave is 7.24 mm.

Step by step solution

01

Understand Wave Functions

Each wave is described by the function \( y_i(x, t) = A_i \sin(kx - \omega t + \phi_i) \), where \( A_i \) is the amplitude, \( k \) is the wave number, \( \omega \) is the angular frequency, and \( \phi_i \) is the phase constant. All waves have an amplitude of 4.00 mm, a wave number of \( 2\pi \), and an angular frequency of \( 400\pi \).
02

Identify Phase Constants

The phase constants for the waves are: \( \phi_1 = 0 \), \( \phi_2 = 0.7\pi \), \( \phi_3 = \pi \), and \( \phi_4 = 1.7\pi \). These phase constants will affect the interference of the waves.
03

Calculate Resultant Amplitude

The resultant amplitude \( A_r \) of multiple interfering waves can be found using the condition for constructive interference:\[ A_r^2 = (\sum A_i \cos(\phi_i))^2 + (\sum A_i \sin(\phi_i))^2 \]. Here, each \( A_i = 4 \) mm. Calculate the sums of the cosines and sines of the phase constants.
04

Sum the Cosines

Calculate the sum of the cosines:\[ \sum A_i \cos(\phi_i) = 4\cos(0) + 4\cos(0.7\pi) + 4\cos(\pi) + 4\cos(1.7\pi) \]\[ = 4(1) + 4(-0.809) + 4(-1) + 4(-0.809) = -7.236 \text{ mm} \].
05

Sum the Sines

Calculate the sum of the sines:\[ \sum A_i \sin(\phi_i) = 4\sin(0) + 4\sin(0.7\pi) + 4\sin(\pi) + 4\sin(1.7\pi) \]\[ = 4(0) + 4(0.587) + 4(0) + 4(-0.587) = 0 \text{ mm} \].
06

Combine Results for Amplitude

Use the results from Steps 4 and 5 to find \( A_r \):\[ A_r^2 = (-7.236)^2 + 0^2 = 52.385 \]\[ A_r = \sqrt{52.385} = 7.24 \text{ mm} \].

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Key Concepts

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

Wave Amplitude
In wave physics, the amplitude of a wave is a critical parameter, as it determines the height of the wave's peak. The amplitude signifies the maximum displacement of points on a wave, which translates into how strong or intense the wave is. For the waves given in the exercise, they all have the same amplitude of 4.00 mm.
Understanding wave amplitude is crucial because it helps us determine how waves interact when they overlap. Because all waves in this scenario have equal amplitudes, their total effect on the resultant wave amplitude will largely depend on their phase relationships.
Phase Constants
Phase constants define the initial angle, or phase, of a wave at the origin. It's part of the wave equation that influences how waves interfere with each other. In the given problem, each wave has a phase constant: \( \phi_1 = 0 \), \( \phi_2 = 0.7\pi \), \( \phi_3 = \pi \), and \( \phi_4 = 1.7\pi \).
These phase constants shift the wave either forward or backward in time, thereby affecting how the waves align with each other when they overlap. If waves have the same phase constant, they are "in phase," meaning they would constructively interfere with each other.
Constructive Interference
Constructive interference occurs when two or more waves overlap in phase, meaning their peaks align to produce a wave with a larger amplitude. To calculate the resultant amplitude when multiple waves interfere constructively, you sum the individual amplitudes as vectors.
Using the provided phase constants and amplitudes, we compute the resultant amplitude via trigonometric relationships.
  • Sum the cosines of the phase constants weighted by the amplitudes.
  • Sum the sines of the phase constants also weighted by amplitudes.
These sums are then combined to determine the amplitude of the resultant wave.
Wave Function
A wave function describes the displacement of a wave at any given point in space and time. In mathematics, it is expressed as \( y(x, t) = A \sin(kx - \omega t + \phi) \), where \( A \) is amplitude, \( k \) the wave number, \( \omega \) the angular frequency, and \( \phi \) the phase constant.
The wave function is the backbone of wave analysis because it allows us to model the behavior of waves quantitatively. In this exercise, the wave functions provide a structured way to account for how each sinusoidal component contributes to the overall phenomenon, enabling the systematic calculation of the resultant wave effect.

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Most popular questions from this chapter

The speed of electromagnetic waves (which include visible light, radio, and \(x\) rays \()\) in vacuum is \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\). (a) Wavelengths of visible light waves range from about \(400 \mathrm{~nm}\) in the violet to about \(700 \mathrm{~nm}\) in the red. What is the range of frequencies of these waves? (b) The range of frequencies for shortwave radio (for example, FM radio and VHF television) is \(1.5\) to \(300 \mathrm{MHz}\). What is the corresponding wavelength range? (c) X-ray wavelengths range from about \(5.0 \mathrm{~nm}\) to about \(1.0 \times 10^{-2} \mathrm{~nm} .\) What is the frequency range for \(x\) rays?

A wave has an angular frequency of \(110 \mathrm{rad} / \mathrm{s}\) and a wavelength of \(1.80 \mathrm{~m}\). Calculate (a) the angular wave number and (b) the speed of the wave.

Two waves are generated on a string of length \(3.0\) \(\mathrm{m}\) to produce a three-loop standing wave with an amplitude of \(1.0 \mathrm{~cm} .\) The wave speed is \(100 \mathrm{~m} / \mathrm{s}\). Let the equation for one of the waves be of the form \(y(x, t)=y_{m} \sin (k x+\omega t) .\) In the equation for the other wave, what are (a) \(y_{m}\), (b) \(k,(\mathrm{c}) \omega\), and (d) the sign in front of \(\omega\) ?

A string that is stretched between fixed supports separated by \(75.0 \mathrm{~cm}\) has resonant frequencies of 420 and \(315 \mathrm{~Hz}\), with no intermediate resonant frequencies. What are (a) the lowest resonant frequency and (b) the wave speed?

Two sinusoidal waves, identical except for phase, travel in the same direction along a string, producing the net wave \(y^{\prime}(x, t)=(3.0 \mathrm{~mm}) \sin (20 x-4.0 t+0.820 \mathrm{rad})\), with \(x\) in meters and \(t\) in seconds. What are (a) the wavelength \(\lambda\) of the two waves, (b) the phase difference between them, and (c) their amplitude \(y_{m} ?\)
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