Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
A guitar string is clamped at both ends. For the purpose of this problem we may consider it to have a length, L = x, and a wave speed, c = 1. Show that, u(x, t) = sin(x) cos(t) – } sin(3r) cos(3t) + sin(5x) cos(5t) (a) Satisfies both boundary conditions (clamped at both ends) (b) Is a solution to the wave equation (c) (Intermediate) Satisfies the initial condition if the guitar string is plucked - that is that (x,0) = 0

A guitar string is clamped at both ends. For the purpose of this problem we may consider it to have a length, L = x, and a wave speed, c = 1. Show that, u(x, t) = sin(x) cos(t) – } sin(3r) cos(3t) + sin(5x) cos(5t) (a) Satisfies both boundary conditions (clamped at both ends) (b) Is a solution to the wave equation (c) (Intermediate) Satisfies the initial condition if the guitar string is plucked - that is that (x,0) = 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 77E
See similar textbooks
icon
Related questions
Question
100%

pls all if possible urgent pls

A guitar string is clamped at both ends. For the purpose of this problem we may consider it to have a length, L = x, and a wave speed, c = 1. Show that, u(x, t) = sin(x) cos(t) – sin(3r) cos(3t) + sin(5x) cos(5t) (a) Satisfies both boundary conditions (clamped at both ends) (b) Is a solution to the wave equation (c) (Intermediate) Satisfies the initial condition if the guitar string is plucked - that is that (x, 0) = 0 The "period", T, of the solution is the shortest time it takes for the guitar string to return to its original shape, i.e., u(x, KT) = ...= u(x, 27') = u(x, T') = u(x, 0) (d) The period for our string is, T = 27. Show this by demonstrating that, u(r, T) = u(x, 0) (e) If we change the wave speed to c = 2, our solution becomes, u(x, t) = sin(x) cos(2t) – sin(3r) cos(6t) + sin(5x) cos(10t) What is the period now? – note, you may want to check the argument used to apply clamped conditions u(0, t) = u(x,t) = 0
Transcribed Image Text:A guitar string is clamped at both ends. For the purpose of this problem we may consider it to have a length, L = x, and a wave speed, c = 1. Show that, u(x, t) = sin(x) cos(t) – sin(3r) cos(3t) + sin(5x) cos(5t) (a) Satisfies both boundary conditions (clamped at both ends) (b) Is a solution to the wave equation (c) (Intermediate) Satisfies the initial condition if the guitar string is plucked - that is that (x, 0) = 0 The "period", T, of the solution is the shortest time it takes for the guitar string to return to its original shape, i.e., u(x, KT) = ...= u(x, 27') = u(x, T') = u(x, 0) (d) The period for our string is, T = 27. Show this by demonstrating that, u(r, T) = u(x, 0) (e) If we change the wave speed to c = 2, our solution becomes, u(x, t) = sin(x) cos(2t) – sin(3r) cos(6t) + sin(5x) cos(10t) What is the period now? – note, you may want to check the argument used to apply clamped conditions u(0, t) = u(x,t) = 0
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
SEE MORE TEXTBOOKS