Superposition and Standing Waves A standing wave occurs when two travelling waves with the same wavelength and wave speed pass through each other in opposite directions. Mathematically the first wave is written as: Y, (x, t) = A cos(kx - wt) and the second wave: Y,(x, t) = A cos(kx + wt) where the wave number, k = 27/2, the natural frequency w = 27/T, and A and T are the wavelength and period of the waves, respectively. The positive and negative signs of w show that the waves are travelling in opposite directions. Demonstrate that the sum of these two waves, Ysum = Y1 + Y2, results in a sinusoidal wave, Y(x) with a wavelength A and an amplitude that is twice that of each wave individually. Furthermore, as time advances, the amplitude is oscillating between +A and -A with a period, T. Repeat the summation, only now assume the other harmonic wave form: Y, (x, t) = A sin(kx – wt) and the second wave: Y,(x, t) = A sin(kx + wt) The following trigonometric identities may be helpful: sin(a + B) = sin a cos ß + cos a sin ß cos(a + B) = sin a cos ß – cos a sin ß sin(-8) = – sin 0 cos(-8) = cos e
Superposition and Standing Waves A standing wave occurs when two travelling waves with the same wavelength and wave speed pass through each other in opposite directions. Mathematically the first wave is written as: Y, (x, t) = A cos(kx - wt) and the second wave: Y,(x, t) = A cos(kx + wt) where the wave number, k = 27/2, the natural frequency w = 27/T, and A and T are the wavelength and period of the waves, respectively. The positive and negative signs of w show that the waves are travelling in opposite directions. Demonstrate that the sum of these two waves, Ysum = Y1 + Y2, results in a sinusoidal wave, Y(x) with a wavelength A and an amplitude that is twice that of each wave individually. Furthermore, as time advances, the amplitude is oscillating between +A and -A with a period, T. Repeat the summation, only now assume the other harmonic wave form: Y, (x, t) = A sin(kx – wt) and the second wave: Y,(x, t) = A sin(kx + wt) The following trigonometric identities may be helpful: sin(a + B) = sin a cos ß + cos a sin ß cos(a + B) = sin a cos ß – cos a sin ß sin(-8) = – sin 0 cos(-8) = cos e
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
Related questions
Question
![**Superposition and Standing Waves**
A standing wave occurs when two traveling waves with the same wavelength and wave speed pass through each other in opposite directions.
Mathematically the first wave is written as:
\[ Y_1(x, t) = A \cos(kx - \omega t) \]
and the second wave:
\[ Y_2(x, t) = A \cos(kx + \omega t) \]
where the wave number, \( k = \frac{2\pi}{\lambda} \), the natural frequency \( \omega = \frac{2\pi}{T} \), and \( \lambda \) and \( T \) are the wavelength and period of the waves, respectively. The positive and negative signs of \( \omega \) show that the waves are traveling in opposite directions.
Demonstrate that the sum of these two waves, \( Y_{\text{sum}} = Y_1 + Y_2 \), results in a sinusoidal wave, \( Y(x) \) with a wavelength \( \lambda \) and an amplitude that is twice that of each wave individually. Furthermore, as time advances, the amplitude is oscillating between +A and -A with a period, T.
Repeat the summation, only now assume the other harmonic wave form:
\[ Y_1(x, t) = A \sin(kx - \omega t) \]
and the second wave:
\[ Y_2(x, t) = A \sin(kx + \omega t) \]
The following trigonometric identities may be helpful:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
\[ \cos(\alpha + \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \]
\[ \sin(-\theta) = -\sin \theta \quad \cos(-\theta) = \cos \theta \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe799f64b-1f0b-4ff1-9ee3-5090ddf07729%2Fb70d6ff6-8ae6-41b7-a7ad-8b6a094882d3%2Ffrul0ij_processed.png&w=3840&q=75)
Transcribed Image Text:**Superposition and Standing Waves**
A standing wave occurs when two traveling waves with the same wavelength and wave speed pass through each other in opposite directions.
Mathematically the first wave is written as:
\[ Y_1(x, t) = A \cos(kx - \omega t) \]
and the second wave:
\[ Y_2(x, t) = A \cos(kx + \omega t) \]
where the wave number, \( k = \frac{2\pi}{\lambda} \), the natural frequency \( \omega = \frac{2\pi}{T} \), and \( \lambda \) and \( T \) are the wavelength and period of the waves, respectively. The positive and negative signs of \( \omega \) show that the waves are traveling in opposite directions.
Demonstrate that the sum of these two waves, \( Y_{\text{sum}} = Y_1 + Y_2 \), results in a sinusoidal wave, \( Y(x) \) with a wavelength \( \lambda \) and an amplitude that is twice that of each wave individually. Furthermore, as time advances, the amplitude is oscillating between +A and -A with a period, T.
Repeat the summation, only now assume the other harmonic wave form:
\[ Y_1(x, t) = A \sin(kx - \omega t) \]
and the second wave:
\[ Y_2(x, t) = A \sin(kx + \omega t) \]
The following trigonometric identities may be helpful:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
\[ \cos(\alpha + \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \]
\[ \sin(-\theta) = -\sin \theta \quad \cos(-\theta) = \cos \theta \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON