Homework Answers
This is the graph for the first part of the wave, ie. wave with only 1 PHz frequency.
This is the second part, i.e. wave with 1.05 PHz frequency.
We can show here that these are out of phase in the following diagram:
Here are the two independent waves drawn on same graph, showing how these are out of phase.
Now, we can show the total wave u(0,t) by following diagram:
We can clearly see that a wavepacket appears. The envelope is formed by meeting the top most points together, and then meeting the lower most points.
This pattern will keep on repeating at much larger time intervals.
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Problem 3: The following problem helps you to understand the nature of wave packets and light...
370_3_1 Problem 1: The first two problems in this assignment will help you to understand how...
370_3_1
Problem 1: The first two problems in this assignment will help you to understand how a planar mirror resonator (Fabry-Perot resonator) works. Consider two monochromatic waves travelling in z-direction, one wave traveling forward , u,(3,t) = a cos(@t- k z), the other backward, u,(z,t), both having the same amplitude. Determine the total real-valued wave u(z,t)=u,(z,t)+uz(z,t) proceeding in two different ways: (i) using only the real-valued wave functions for the forward and backward travelling waves together with the well-known cosine...
Wave Packets (a) Create a wave by adding two different waves by using Matlab, but different...
Wave Packets (a) Create a wave by adding two different waves by using Matlab, but different combination than those we used in the class. Paste the output graph (and also the code). Any two waves can be chosen, but explain the difference compared to the original wave packet we obtained in the class. You can modify the code I used in the class, or build something from scratch. (b) Add as much waves as you want (minimum:3), using Matlab. Explain...
Problem 1: This problem is just a reminder of material covered earlier in class. It is...
Problem 1: This problem is just a reminder of material covered earlier in class. It is meant as a preparation for a problem on quantum mechanics, Problem 2 below. Consider the Helmholtz equation (5.3-16) SU )+ (2x U4)=0 for the case of an ideal (lossless) planar mirror resonator with vacuum between two metallic mirrors. Assume the mirrors to be located at -d/2 and +d/2, and the wave amplitude to be zero at the boundaries. Determine all possible solutions , wave...
I need help with d) please help thank you Question 1 Wave motion appears in all branches of physics. In the lectures we...
I need help with d) please help thank you
Question 1 Wave motion appears in all branches of physics. In the lectures we considered the solution of the advection equation, a first-order hyperbolic PDE. Here we consider the solution of the wave equation: c2 where c >0 is constant. , We assume all variables have been non-dimensionalised. (a) Eq. (1) has the general solution (d'Alembert, 1747): u(x,t) F(x -ct) +G(x ct), where F and G are arbitrary functions. Consider the...
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the s...
parts a,b, c
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the string is initially at rest. The initial boundary value problem (IBVP) is written as u(0,t) -u(1,t) u(x,0) = f(x), 0 ut (z, 0-0, 0 < x < 1. The solution of this IBVP using the method of separation of variables is given by n-l a) Find the coefficients bn. b) Show that this wave function can be written as the...
(1 point) A mass m = 4 kg is attached to both a spring with spring...
(1 point) A mass m = 4 kg is attached to both a spring with spring constant k = 325 N/m and a dash-pot with damping constant c=4N s/m. The mass is started in motion with initial position Xo = 1 m and initial velocity vo = 9 m/s. Determine the position function z(t) in meters. x(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form x(t)...
2. Consider the following initial value problem for the wave equation, modeling a vi- brating string...
2. Consider the following initial value problem for the wave equation, modeling a vi- brating string with fixed endpoints. au = 922 u u(t,0) = u(t, 7) = 0 u(0,x) = 8 sin(x) sin(2x) sin(3x) (Ou(0,2) = 9 sin(6x) (a) What is the length L of the string? What is the value of the constant c= T/p? (b) Write down the solution of this initial value problem. (Hint: You might find the following identities helpful.)! cos(a + b) = cos...
Can you do (b) and (c) only thank you PrOBleM: SoLuTiONS To THE WAvE EQuATION a)...
Can you do (b) and (c) only thank you
PrOBleM: SoLuTiONS To THE WAvE EQuATION a) By direct substitution determine which of the following functions satisfy the wave equation 1. g(z, t)-A cos(kr - wt) where A, k, w are positive constants 2. h(z,t)-Ae-(kz-wt)2 where A, k, ω are positive constants 3. p(x, t) A sinh(kx-wt) where A, k,w are positive constants 4. q(z, t) - Ae(atut) where A,a, w are positive constants 5. An arbitrary function: f(x, t) -...
Real Analysis II (Please do this only if you are sure) ********************** *********************** I am also providing the convex set definition And key details from my book which surely helps 1...
Real Analysis II
(Please do this only if you are sure)
**********************
***********************
I am also providing the convex set definition
And key details from my book which surely helps
11. Show that K is a convex set by directly applying the definition. Sketch K in the cases n= 1, 2, 3. is a basis for E. This is the n-parallelepiped spanned by vı, vertex 1% with 0 as a Definition. Let K E". Then K is a convex set...
PrOBleM: SoLuTiONS To THE WAvE EQuATION a) By direct substitution determine which of the following functions...
PrOBleM: SoLuTiONS To THE WAvE EQuATION a) By direct substitution determine which of the following functions satisfy the wave equation 1. g(z, t)-A cos(kr - wt) where A, k, w are positive constants 2. h(z,t)-Ae-(kz-wt)2 where A, k, ω are positive constants 3. p(x, t) A sinh(kx-wt) where A, k,w are positive constants 4. q(z, t) - Ae(atut) where A,a, w are positive constants 5. An arbitrary function: f(x, t) - f(kx -wt) where k and w are positive constants....
370_3_1
Problem 1: The first two problems in this assignment will help you to understand how a planar mirror resonator (Fabry-Perot resonator) works. Consider two monochromatic waves travelling in z-direction, one wave traveling forward , u,(3,t) = a cos(@t- k z), the other backward, u,(z,t), both having the same amplitude. Determine the total real-valued wave u(z,t)=u,(z,t)+uz(z,t) proceeding in two different ways: (i) using only the real-valued wave functions for the forward and backward travelling waves together with the well-known cosine...
Problem 1: This problem is just a reminder of material covered earlier in class. It is meant as a preparation for a problem on quantum mechanics, Problem 2 below. Consider the Helmholtz equation (5.3-16) SU )+ (2x U4)=0 for the case of an ideal (lossless) planar mirror resonator with vacuum between two metallic mirrors. Assume the mirrors to be located at -d/2 and +d/2, and the wave amplitude to be zero at the boundaries. Determine all possible solutions , wave...
I need help with d) please help thank you
Question 1 Wave motion appears in all branches of physics. In the lectures we considered the solution of the advection equation, a first-order hyperbolic PDE. Here we consider the solution of the wave equation: c2 where c >0 is constant. , We assume all variables have been non-dimensionalised. (a) Eq. (1) has the general solution (d'Alembert, 1747): u(x,t) F(x -ct) +G(x ct), where F and G are arbitrary functions. Consider the...
parts a,b, c
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the string is initially at rest. The initial boundary value problem (IBVP) is written as u(0,t) -u(1,t) u(x,0) = f(x), 0 ut (z, 0-0, 0 < x < 1. The solution of this IBVP using the method of separation of variables is given by n-l a) Find the coefficients bn. b) Show that this wave function can be written as the...
(1 point) A mass m = 4 kg is attached to both a spring with spring constant k = 325 N/m and a dash-pot with damping constant c=4N s/m. The mass is started in motion with initial position Xo = 1 m and initial velocity vo = 9 m/s. Determine the position function z(t) in meters. x(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form x(t)...
2. Consider the following initial value problem for the wave equation, modeling a vi- brating string with fixed endpoints. au = 922 u u(t,0) = u(t, 7) = 0 u(0,x) = 8 sin(x) sin(2x) sin(3x) (Ou(0,2) = 9 sin(6x) (a) What is the length L of the string? What is the value of the constant c= T/p? (b) Write down the solution of this initial value problem. (Hint: You might find the following identities helpful.)! cos(a + b) = cos...
Can you do (b) and (c) only thank you
PrOBleM: SoLuTiONS To THE WAvE EQuATION a) By direct substitution determine which of the following functions satisfy the wave equation 1. g(z, t)-A cos(kr - wt) where A, k, w are positive constants 2. h(z,t)-Ae-(kz-wt)2 where A, k, ω are positive constants 3. p(x, t) A sinh(kx-wt) where A, k,w are positive constants 4. q(z, t) - Ae(atut) where A,a, w are positive constants 5. An arbitrary function: f(x, t) -...
Real Analysis II
(Please do this only if you are sure)
**********************
***********************
I am also providing the convex set definition
And key details from my book which surely helps
11. Show that K is a convex set by directly applying the definition. Sketch K in the cases n= 1, 2, 3. is a basis for E. This is the n-parallelepiped spanned by vı, vertex 1% with 0 as a Definition. Let K E". Then K is a convex set...
PrOBleM: SoLuTiONS To THE WAvE EQuATION a) By direct substitution determine which of the following functions satisfy the wave equation 1. g(z, t)-A cos(kr - wt) where A, k, w are positive constants 2. h(z,t)-Ae-(kz-wt)2 where A, k, ω are positive constants 3. p(x, t) A sinh(kx-wt) where A, k,w are positive constants 4. q(z, t) - Ae(atut) where A,a, w are positive constants 5. An arbitrary function: f(x, t) - f(kx -wt) where k and w are positive constants....
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