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Chapter 16: Problem 82

Two sinusoidal waves of the same wavelength travel in the same direction along a stretched string. For wave \(1, y_{m}=3.0 \mathrm{~mm}\) and \(\phi=\) \(0 ;\) for wave \(2, y_{m}=5.0 \mathrm{~mm}\) and \(\phi=70^{\circ} .\) What are the (a) amplitude and (b) phase constant of the resultant wave?

Short Answer

Expert verified
The amplitude is approximately 6.65 mm, and the phase constant is approximately 45掳.

Step by step solution

01

Understand the given values

We have two sinusoidal waves: - Wave 1 with amplitude \( y_{m1}=3.0 \, \text{mm} \) and phase \(\phi_1=0\).- Wave 2 with amplitude \( y_{m2}=5.0 \, \text{mm} \) and phase \(\phi_2=70^{\circ}\).They travel in the same direction along a stretched string. We need to find the amplitude and phase constant of the resultant wave.
02

Use the principle of superposition

According to the principle of superposition, the resultant wave is the sum of the two individual waves. For two waves \( y_1 = y_{m1} \sin(kx - \omega t + \phi_1) \) and \( y_2 = y_{m2} \sin(kx - \omega t + \phi_2) \), the resultant wave can be given as \( y = y_{m1} \sin(kx - \omega t) + y_{m2} \sin(kx - \omega t + 70^{\circ}) \).
03

Calculate amplitude of resultant wave

The amplitude of the resultant wave is \[ y_m = \sqrt{y_{m1}^2 + y_{m2}^2 + 2y_{m1}y_{m2}\cos(\phi_2 - \phi_1)} \]Substituting the given values:\[ y_m = \sqrt{3.0^2 + 5.0^2 + 2 \times 3.0 \times 5.0 \times \cos(70^{\circ})} \]Calculate the values:\( \cos(70^{\circ}) \approx 0.342 \).Substitute into the equation:\[ y_m = \sqrt{9 + 25 + 30 \times 0.342} \]\[ y_m = \sqrt{9 + 25 + 10.26} \]\[ y_m = \sqrt{44.26} \approx 6.65 \text{ mm} \].
04

Calculate phase constant of resultant wave

The phase constant \( \phi \) of the resultant wave can be found using \[ \tan \phi = \frac{y_{m2} \sin \phi_2}{y_{m1} + y_{m2} \cos \phi_2} \]Substitute the given values:\[ \tan \phi = \frac{5.0 \times \sin 70^{\circ}}{3.0 + 5.0 \times \cos 70^{\circ}} \]Calculate the trigonometric values:\( \sin(70^{\circ}) \approx 0.94 \) and \( \cos(70^{\circ}) \approx 0.342 \).Substitute back:\[ \tan \phi = \frac{5.0 \times 0.94}{3.0 + 5.0 \times 0.342} \]\[ \tan \phi = \frac{4.7}{4.71} \]\[ \phi = \tan^{-1}(1.0) \approx 45^{\circ} \].

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Key Concepts

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

Superposition Principle
The Superposition Principle is a fundamental concept in wave mechanics. It allows us to predict the behavior of two or more overlapping waves. When waves meet, they don't physically alter each other. Instead, they combine temporary based on their amplitudes and phases.
The principle states that the resultant displacement at any point is the sum of the displacements from each individual wave at that point. For instance, if two waves, each described by their sinusoidal functions, overlap, their combined effect is a new wave which is their mathematical sum.
  • This principle helps in understanding complex wave phenomena.
  • It's applicable in various scenarios like sound waves, light waves, and other electromagnetic waves.
In our exercise, this principle was used to combine two sinusoidal waves traveling along a string, resulting in a new wave with a unique amplitude and phase constant.
Wave Amplitude
Wave Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. It's essentially the 'height' of the wave when represented on a graph.
In wave physics, amplitude is directly related to the energy carried by the wave. The larger the amplitude, the more energy the wave carries.
  • Mathematically, amplitude is a crucial parameter in wave functions: it determines the peak value of oscillating quantities.
  • In our problem, wave 1 has an amplitude of 3 mm, and wave 2 has 5 mm.
To find the amplitude of the resultant wave, we employ the formula: \[ y_m = \sqrt{y_{m1}^2 + y_{m2}^2 + 2y_{m1}y_{m2}\cos(\phi_2 - \phi_1)} \]This formula relates to how two wave amplitudes combine depending on their phase differences.
Phase Constant
The Phase Constant is a term that describes the initial angle of a wave when it begins. It's an essential component in wave equations, impacting where the wave starts on its journey.
Phase constant is crucial when multiple waves interfere, as it dictates how waves align when they meet.
  • In the exercise at hand, wave 1 has a phase of 0掳, while wave 2 has a phase of 70掳.
  • To find the resultant wave's phase, we use the formula: \[ \tan \phi = \frac{y_{m2} \sin \phi_2}{y_{m1} + y_{m2} \cos \phi_2} \]
Calculating this provides the phase shift necessary to describe the interference pattern of the combined waves, explaining concepts like constructive and destructive interference.
Trigonometry in Waves
Trigonometry is an indispensable tool in understanding wave behavior. Waves are periodic by nature and are often described using trigonometric functions like sine and cosine.
In our problem, trigonometric identities help in calculating combined wave properties, such as amplitude and phase. The sine and cosine of angles between waves play critical roles in determining how these waves interact.
  • The cosine function helps to account for differences in phases between waves, especially for calculating amplitude.
  • The tangent function is used to find the phase constant of the resultant wave from interfering waves.
Using trigonometry, we understand wave interactions better, from simple oscillations to complex interference patterns.

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Most popular questions from this chapter

The following two waves are sent in opposite directions on a horizontal string so as to create a standing wave in a vertical plane: $$ \begin{aligned} &y_{1}(x, t)=(6.00 \mathrm{~mm}) \sin (4.00 \pi x-400 \pi t) \\ &y_{2}(x, t)=(6.00 \mathrm{~mm}) \sin (4.00 \pi x+400 \pi t) \end{aligned} $$ with \(x\) in meters and \(t\) in seconds. An antinode is located at point \(A\). In the time interval that point takes to move from maximum upward displacement to maximum downward displacement, how far does each wave move along the string?

A standing wave results from the sum of two transverse traveling waves given by $$ y_{1}=0.050 \cos (\pi x-4 \pi t) $$ and $$ y_{2}=0.050 \cos (\pi x+4 \pi t) $$ where \(x, y_{1}\), and \(y_{2}\) are in meters and \(t\) is in seconds. (a) What is the smallest positive value of \(x\) that corresponds to a node? Beginning at \(t=0\), what is the value of the (b) first, (c) second, and (d) third time the particle at \(x=0\) has zero velocity?

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 \(y_{m}\) ?

Three sinusoidal waves of the same frequency travel along a string in the positive direction of an \(x\) axis. Their amplitudes are \(y_{1}, y_{1} / 2\), and \(y_{1} / 3\), and their phase constants are \(0, \pi / 2\), and \(\pi\), respectively. What are the (a) amplitude and (b) phase constant of the resultant wave? (c) Plot the wave form of the resultant wave at \(t=0\), and discuss its behavior as \(t\) increases.

A nylon guitar string has a linear density of \(7.20 \mathrm{~g} / \mathrm{m}\) and is under a tension of \(150 \mathrm{~N}\). The fixed supports are distance \(D=90.0 \mathrm{~cm}\) apart. The string is oscillating in the standing wave pattern shown in Fig. 16-39. Calculate the (a) speed, (b) wavelength, and (c) frequency of the traveling waves whose superposition gives this standing wave.
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