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Chapter 16: Problem 39

Two sinusoidal waves of the same period, with amplitudes of \(5.0\) and \(7.0 \mathrm{~mm}\), travel in the same direction along a stretched string; they produce a resultant wave with an amplitude of \(9.0 \mathrm{~mm}\). The phase constant of the \(5.0 \mathrm{~mm}\) wave is \(0 .\) What is the phase constant of the \(7.0 \mathrm{~mm}\) wave?

Short Answer

Expert verified
The phase constant of the 7.0 mm wave is approximately 84.26掳.

Step by step solution

01

Understanding the Problem

We are given two waves with the same period but different amplitudes traveling in the same direction. The amplitudes are 5.0 mm and 7.0 mm. The resultant wave has an amplitude of 9.0 mm. The phase constant of the 5.0 mm wave is 0. We need to find the phase constant of the 7.0 mm wave.
02

Reviewing Wave Superposition

When two waves combine, their resultant amplitude depends not only on their individual amplitudes but also on the phase difference between them. The formula for the resultant amplitude is:\[ A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\Delta\phi)} \]where \(A\) is the resultant amplitude, \(A_1\) and \(A_2\) are the amplitudes of the individual waves, and \(\Delta\phi\) is the phase difference.
03

Setting Up the Equation

Substitute the given values into the resultant amplitude equation:\[ 9 = \sqrt{5^2 + 7^2 + 2 \cdot 5 \cdot 7 \cdot \cos(\Delta\phi)} \]
04

Solve for Cosine of Phase Difference

Square both sides and simplify the equation:\[ 81 = 25 + 49 + 70\cos(\Delta\phi) \]\[ 81 = 74 + 70\cos(\Delta\phi) \]\[ 70\cos(\Delta\phi) = 7 \]\[ \cos(\Delta\phi) = \frac{7}{70} \]\[ \cos(\Delta\phi) = 0.1 \]
05

Finding the Phase Constant

The phase constant \(\Delta\phi\) corresponds to the angle whose cosine is 0.1. Thus, we calculate arcsin to find the phase constant:\[ \Delta\phi = \cos^{-1}(0.1) \approx 84.26\degree \]

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Key Concepts

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

Sinusoidal Waves
Sinusoidal waves are a common model for representing wave phenomena because of their distinct oscillatory motion. They are characterized by their smooth, repetitive oscillation. A sinusoidal wave can be described mathematically by a function such as:
  • Position with respect to time: \( y(t) = A \sin(\omega t + \phi) \)
  • Where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.
In the context of physics, sinusoidal waves can represent different types of waves, such as sound waves, light waves, or even water waves.
Sinusoidal waves have specific properties that make them unique:
  • Amplitude (\( A \)): The peak value of the wave; determines the maximum displacement from the rest position.
  • Period (\( T \)): The time it takes for one complete cycle of the wave.
  • Frequency (\( f \)): The number of cycles per unit of time, typically measured in hertz (Hz).
  • Phase constant (\( \phi \)): This determines the wave's initial displacement at \( t = 0 \), influencing the wave's starting point.
These properties can be manipulated to gain insight into the behavior of waves when they interact with each other, such as in superposition.
Phase Constants
Phase constants are crucial in determining the behavior and appearance of sinusoidal waves. They essentially describe how the wave is "shifted" in terms of its position at a given instance in time. This shift can affect how two waves combine when they meet.
  • Definition: The phase constant, often denoted by \( \phi \), is a measure of the initial angle of the sinusoidal function at \( t = 0 \).
  • Impact on Wave: The phase constant alters the starting point of the wave and shifts the wave along the time axis. A wave with a phase constant of zero starts at the origin in a plot of its waveform, while a wave with a phase constant will start elsewhere.
  • Superposition: During wave superposition, the phase difference between two waves is essential. It determines whether the waves will interfere constructively or destructively:
    • Constructive interference occurs when their phase difference is such that peaks and troughs align, creating a larger resultant wave.
    • Destructive interference happens when peaks and troughs misalign, reducing the resultant wave's amplitude.
    Understanding the phase constant helps in predicting how two or more sinusoidal waves will interact, an essential concept in wave physics as evidenced in the original exercise's requirement to find the phase difference between waves.
Resultant Amplitude
Resultant amplitude in the context of wave superposition refers to the amplitude of the new wave created when two or more waves meet and combine. The principle of superposition tells us that the total displacement at any point is the sum of the individual displacements from the waves at that point.
  • Equation: The formula for calculating the resultant amplitude of two sinusoidal waves with amplitudes \( A_1 \) and \( A_2 \) is:\[ A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\Delta\phi)} \]
  • This formula accounts for both the amplitudes of the individual waves and their phase difference \( \Delta\phi \).
  • Significance of \( \Delta\phi \): The phase difference between the waves is crucial. It influences whether the waves will add up constructively (resulting in a larger amplitude) or destructively (potentially reducing the amplitude).
In the example from the original exercise, the calculated resultant amplitude demonstrates the intricacies of wave interaction. By knowing the resultant amplitude and the amplitudes of individual waves, one can deduce the phase difference and better understand the superposed wave's behavior.

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Most popular questions from this chapter

String \(A\) is stretched between two clamps separated by distance \(L\). String \(B\), with the same linear density and under the same tension as string \(A\), is stretched between two clamps separated by distance \(4 L\). Consider the first eight harmonics of string \(B\). For which of these eight harmonics of \(B\) (if any) does the frequency match the frequency of (a) \(A\) 's first harmonic, (b) \(A\) 's second harmonic, and (c) \(A\) 's third harmonic?

A \(1.50 \mathrm{~m}\) wire has a mass of \(8.70 \mathrm{~g}\) and is under a tension of \(120 \mathrm{~N}\). The wire is held rigidly at both ends and set into oscillation. (a) What is the speed of waves on the wire? What is the wavelength of the waves that produce (b) one-loop and (c) twoloop standing waves? What is the frequency of the waves that produce (d) one-loop and (e) two-loop standing waves?

(a) What is the fastest transverse wave that can be sent along a steel wire? For safety reasons, the maximum tensile stress to which steel wires should be subjected is \(7.00 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}\). The density of steel is \(7800 \mathrm{~kg} / \mathrm{m}^{3} .\) (b) Does your answer depend on the diameter of the wire?

When played in a certain manner, the lowest resonant frequency of a certain violin string is concert \(\mathrm{A}(440 \mathrm{~Hz}) .\) What is the frequency of the (a) second and (b) third harmonic of the string?

A generator at one end of a very long string creates a wave given by $$ y=(6.0 \mathrm{~cm}) \cos \frac{\pi}{2}\left[\left(2.00 \mathrm{~m}^{-1}\right) x+\left(8.00 \mathrm{~s}^{-1}\right) t\right] $$ and a generator at the other end creates the wave $$ y=(6.0 \mathrm{~cm}) \cos \frac{\pi}{2}\left[\left(2.00 \mathrm{~m}^{-1}\right) x-\left(8.00 \mathrm{~s}^{-1}\right) t\right] $$ Calculate the (a) frequency, (b) wavelength, and (c) speed of each wave. For \(x \geq 0\), what is the location of the node having the (d) smallest, (e) second smallest, and (f) third smallest value of \(x\) ? For \(x \geq 0\), what is the location of the antinode having the (g) smallest, (h) second smallest, and (i) third smallest value of \(x\) ?
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