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Chapter 17: Problem 96

Four sound waves are to be sent through the same tube of air, in the same direction: $$ \begin{array}{l} s_{1}(x, t)=(9.00 \mathrm{~nm}) \cos (2 \pi x-700 \pi t) \\ s_{2}(x, t)=(9.00 \mathrm{~nm}) \cos (2 \pi x-700 \pi t+0.7 \pi) \\ s_{3}(x, t)=(9.00 \mathrm{~nm}) \cos (2 \pi x-700 \pi t+\pi) \\ s_{4}(x, t)=(9.00 \mathrm{~nm}) \cos (2 \pi x-700 \pi t+1.7 \pi) . \end{array} $$ What is the amplitude of the resultant wave? (Hint: Use a phasor diagram to simplify the problem.)

Short Answer

Expert verified
The amplitude of the resultant wave is approximately 17.8 nm.

Step by step solution

01

Understand Each Wave

Each sound wave is expressed as a function of displacement in the form \( s_i(x, t) = A \cos(kx - \omega t + \phi_i) \), where \( A = 9.00\, \text{nm} \), \( k = 2\pi \), and \( \omega = 700\pi \). The phase differences \( \phi_1 = 0 \), \( \phi_2 = 0.7\pi \), \( \phi_3 = \pi \), and \( \phi_4 = 1.7\pi \).
02

Represent Waves as Phasors

Express each wave as a phasor in the complex plane. Each wave is a vector with magnitude equal to the amplitude and direction according to its phase shift:- \( s_1 = 9.00 e^{i(0)} \)- \( s_2 = 9.00 e^{i(0.7\pi)} \)- \( s_3 = 9.00 e^{i(\pi)} \)- \( s_4 = 9.00 e^{i(1.7\pi)} \).
03

Calculate Real and Imaginary Components

Use Euler's formula to convert each phasor into its real and imaginary components:- \( s_1: 9.00 \)(real = 9.00, imag = 0)- \( s_2: 9.00\cos(0.7\pi) + i9.00\sin(0.7\pi) \)- \( s_3: 9.00\cos(\pi) + i9.00\sin(\pi) \)- \( s_4: 9.00\cos(1.7\pi) + i9.00\sin(1.7\pi) \).
04

Sum the Components of All Waves

Add the real components and the imaginary components separately to find the resultant wave's complex form. The resultant real and imaginary components of each vector are: \[ R = 9.00 + 9.00 \cos(0.7\pi) - 9.00 - 9.00 \cos(1.7\pi) \]\[ I = 0 + 9.00 \sin(0.7\pi) + 0 - 9.00 \sin(1.7\pi) \].
05

Calculate Resultant Amplitude

The resultant wave's amplitude \( A_r \) is the magnitude of the resultant phasor vector, calculated as:\[ A_r = \sqrt{R^2 + I^2} \]. Substitute the sums of real and imaginary parts from Step 4 into this formula to get the amplitude.
06

Solve for Specific Values

Plug in the trigonometric values:- \( \cos(0.7\pi) = -0.156 \)- \( \sin(0.7\pi) = 0.987 \)- \( \cos(1.7\pi) = 0.156 \)- \( \sin(1.7\pi) = -0.987 \).Then:\[ R = 9.00(-0.156) - 9.00(0.156) \]\[ I = 9.00 \times 0.987 + 9.00 \times 0.987 \]Finally calculate:\[ A_r = \sqrt{(9.00 \times (-0.312))^2 + (9.00 \times 1.974)^2} \].

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Key Concepts

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

Phasor Diagram
A phasor diagram is a useful tool for visualizing wave interference. It represents sinusoidal waves as rotating vectors in the complex plane. Each vector's length corresponds to the wave's amplitude, while the angle it makes with the reference axis relates to the wave's phase.

  • Phasors simplify calculations of multiple wave interferences by converting trigonometric summations into vector additions.
  • By aligning these vectors according to their phase angles, one can easily compute the resultant wave.
  • This can be especially helpful when dealing with multiple waves, like sound waves, that are superimposed.
Summing these vectors gives a graphical representation of the resultant wave. This visual aid can make understanding the concept of wave interference clearer, especially when phases are offset.
Sound Waves
Sound waves are vibrations that travel through a medium, such as air, and are perceived by our ears as sound. They are longitudinal waves characterized by compressions and rarefactions in the medium.

  • The properties of sound waves include amplitude, frequency, speed, and wavelength.
  • Amplitude relates to the loudness; higher amplitude means louder sound.
  • Frequency determines pitch; high frequency corresponds to high pitch.
Sound waves can interfere constructively or destructively, depending on their phase relationships. When waves align perfectly in phase (constructive interference), the resultant sound is amplified. Conversely, waves perfectly out of phase (destructive interference) may cancel each other out, reducing or silencing the sound.
Amplitude Calculation
Calculating the amplitude of a resultant wave from interfering sound waves involves understanding vector addition.

  • First, express each wave as a phasor, noting its amplitude and phase angle.
  • Convert these phasors into real and imaginary components using the formulas: \[\text{real part} = A \cdot \cos(\phi)\] \[\text{imaginary part} = A \cdot \sin(\phi)\]
  • Sum all real components and then all imaginary components separately.
  • The resultant amplitude is the magnitude of the vector sum, calculated as: \[A_r = \sqrt{(\text{Real})^2 + (\text{Imaginary})^2}\]
This method effectively combines mathematical representation and visualization through phasors, leading to a simplified process of finding how wave interferences impact overall amplitude.
Complex Numbers
Complex numbers, represented as \(a + bi\), allow us to use both real and imaginary components.

  • They are key in dealing with wave equations, particularly while decomposing sinusoidal functions into phasors.
  • The real part \(a\) can represent the in-phase component, while the imaginary part \(bi\) represents the quadrature component of a wave.
  • In phasor form, complex numbers help in representing and calculating wave interference by simplifying sinusoidal expression into exponential form using Euler's formula: \[e^{i\phi} = \cos(\phi) + i\cdot \sin(\phi)\]
Using complex numbers for wave interference not only simplifies calculation but also provides a robust framework for handling phase shifts and amplitude changes in waves.

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Most popular questions from this chapter

A violin string \(30.0 \mathrm{~cm}\) long with linear density \(0.650 \mathrm{~g} / \mathrm{m}\) is placed near a loudspeaker that is fed by an audio oscillator of variable frequency. It is found that the string is set into oscillation only at the frequencies 880 and \(1320 \mathrm{~Hz}\) as the frequency of the oscillator is varied over the range \(500-1500 \mathrm{~Hz}\). What is the tension in the string?

A jet plane passes over you at a height of 5000 \(\mathrm{m}\) and a speed of Mach \(1.5\). (a) Find the Mach cone angle (the sound speed is \(331 \mathrm{~m} / \mathrm{s}\) ). (b) How long after the jet passes directly overhead does the shock wave reach you?

A \(2000 \mathrm{~Hz}\) siren and a civil defense official are both at rest with respect to the ground. What frequency does the official hear if the wind is blowing at \(12 \mathrm{~m} / \mathrm{s}\) (a) from source to official and (b) from official to source?

A sound wave travels out uniformly in all directions from a point source. (a) Justify the following expression for the displacement \(s\) of the transmitting medium at any distance \(r\) from the source: $$ s=\frac{b}{r} \sin k(r-v t) $$ where \(b\) is a constant. Consider the speed, direction of propagation, periodicity, and intensity of the wave. (b) What is the dimension of the constant \(b ?\)

The source of a sound wave has a power of \(1.00 \mu \mathrm{W}\). If it is a point source, (a) what is the intensity \(3,00 \mathrm{~m}\) away and (b) what is the sound level in decibels at that distance?
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