Physics 307 Homework 7

Due Tuesday, 12 November, at 5 PM

Progress report due 6 November before class

This homework assignment is the first of several parts. I encourage you to try to finish

this assignment ahead of time and begin the next one before the deadline of this one. This

assignment requires you to write some code; the next assignment will involve using that code

to study some physics.

Your “report” for this assignment will just consist of keeping us up-to-date on your progress;

next Thursday we’d like to see where you are, and will come around and check on you.

Please start early. I am giving you two weeks to do this assignment in order to accommo-

date anyone who may have some difficulty. This will be a bit more challenging to code than

your previous one; please don’t put it off until a few days before it’s due. We’ll be doing a

check-in in a week for a grade; you’ll be given a grade based on how well your code works

at that point. We don’t expect it to be perfect, only that you’re making good progress.

In this assignment, you will model a stretched string (as in a guitar or violin) using a lattice

elasticity model, in which point masses (“nodes”) are connected by Hooke’s-law springs. Fo

eference, the force law on any given node i (given in more detail in class) is:

~̇vxi =

1

m

[

kr̂i,i−1x(ri,i−1 − r0) + kr̂i,i+1(ri,i+1 − r0)

]

(1)

~̇vyi =

1

m

[

kr̂i,i−1y(ri,i−1 − r0) + kr̂i,i+1(ri,i+1 − r0)

]

(2)

where

~ri,j is a vector pointing from the i’th node to the j’th one, and is thus equal to (xj − xi)̂i+

(yj − yi)ĵ, k is the spring constant of a single spring, r0 is its unstretched length, and m is

the mass of each node.

A lattice simulation of a stretched string has the following parameters:

• The number of nodes in the simulation N + 1 (or the number of springs N)

• The linear mass density of the unstretched string µ

• The unstretched length of the string L0

• The tension T applied to it to stretch it, which gives you the stretched length L

1

• The stiffness α (Young’s modulus times cross-sectional area)

1. Determine values of m, k, and r0 (the “microscopic” quantities of interest to you

simulation) as a function of N,µ, L0 and α (the “macroscopic” quantities).

2. Write a program that simulates the vi

ations of a stretched string of uniform density

and elasticity, as we will discuss in class. (Notes will be posted on the website, too.)

You should be able to easily change (by setting variables) some crucial parameters of

your simulation: N,µ, L0, and α.

You should use the leapfrog solver for your simulation, although it might be useful to

code it using Euler-Cromer first as a learning exercise.

3. Determine the initial conditions (all of the xi’s; all the y-coordinates and velocities will

e zero) for a stretched but unexcited string with tension T applied to it. You will

first need to determine the stretched length L′, then put your N masses at equidistant

points between x = 0 and x = L′. Implement these initial conditions, and verify that

the string doesn’t move.

4. Now stretch your string with sufficient tension to stretch it by 25-100% of its un-

stretched length, and excite it somehow. You can do this by applying some external

force to it at some instant in time, or by choosing initial values for the yi’s. Verify

that it moves realistically. A simple initial condition that you can use is a Gaussian

deformation, in which

y(x) = exp

[

A

−(x− xc)2

σ2

]

where xc is the center of the excitation, σ is its width, A is its amplitude, and σ and

xc are chosen so that the Gaussian dies down before you get to the ends of the string.

You may use any other initial conditions you like for testing.

5. Play around with your initial conditions, along with N , the stretch amount, µ, L, and

so on. Discuss anything interesting that you find. Note that your behavior ought to

e relatively independent of N . Your report should contain descriptions of all the stuff

you did and what happened.

6. Modify your program to track conservation of energy (recall that the elastic potential

energy in each spring is U = 1

2

k(r − r0)2) and verify that energy is approximately

conserved.

2

Physics 307 Homework 8

First two parts due the Thursday after Thanksgiving; the final part is due

at the end of the term.

In this project you will study the properties of your vi

ating string. You have nothing

(much) to code this time, but you have quite a few things to simulate, study, and discuss.

A common exercise in physics is to study a simplified model of a vi

ating string where we

assume that the amplitude is small and the tension is large; this gives rise to the familia

ehavior we will discuss in class. For your project, you will examine this behavior, and

discover the limits of its validity.

As a reminder, some properties of the vi

ating string in this limit:

• Oscillations are a superposition of normal modes: y(x, t = 0) = A sin

(

nπx

L

)

, where L

is the stretched length

• Waves propagate along the string at a rate v =

√

T

µ

• The n’th normal mode has a wavelength λn = 2Ln , giving it a period Tn =

λn

v

= 2L

n

√

µ

T

• These properties don’t depend on the amplitude, in the limit of small amplitude.

Part 1: Verifying your code and testing the ideal behavior (due Tuesday, Novem-

er 27)

1. Modify your initial conditions to simulate any normal mode of your choice. Then

modify your code to print out the period of your string; you can determine the half-

period by looking for points at which the velocity at an antinode changes sign. (“If it

was going up before, and now it’s going down, then we’re at a half-period” – you did a

similar thing for the pendulum.) Note that this is only valid if you have behavior that’s

elatively close to the vi

ation of a normal mode; otherwise, the idea of “period” is

more complicated.

2. Recall that our goal was to simulate a vi

ating string using only N and the macro-

scopic quantities T , α, µ, and L0. All microscopic quantities (the node mass m, the

individual spring constant k′, and the equili

ium radius r0 should be determined from

these.

You need to verify that you are co

ectly implementing the relationship between the

macroscopic quantities and the microscopic quantities. You can do this by simulating

1

something well-behaved and verifying that the period doesn’t depend on N . Use

enough tension to stretch your string to XXXXXXXXXX× its starting length, and then simulate

the fundamental mode (n = 1) at a small amplitude. Verify that the period doesn’t

depend (much) on N . This is one check that everything is working properly.

3. Now, adjust the tension, linear density, unstretched length, and verify that the period

has the expected dependence on them. Remember, this expected dependence is only

valid in the “well-behaved” realm, where T is large and the amplitude is small.

IMPORTANT NOTE: The linear mass density µ that appears in the expression fo

the wave speed, and the quantity L that you use to compute the wave speed, are the

values after you stretch the string. You may handle this in any way you choose; one

is to compute a stretched density µ′ after you determine the stretch length that gives

the specified tension.

Part 2: Superpositions of normal modes (also due Tuesday, November 27

as well)

As you know, the vi

ations of a string (in the ideal case) can be characterized by a

superposition of normal modes.

4. You can also excite the string in more physical ways and study its behavior. Modify

your initial conditions so that you give the string a Gaussian “bump” in either velocity

or position in the beginning. Specifically,

yi = Ae

−(xi−xc)2/σ2

where xc is the center of the bump, and σ is its width; A is an amplitude. (Make sure

xc

σ, or you’ll be moving one of the endpoints significantly.) You will probably

want to use a small amplitude to avoid blatantly nonlinear behavior.

If your small amplitude makes the vi

ations too small to see easily, you can always

multiply the y-coordinates by some factor when you display them, just to exaggerate

the motions to make them more visible.

This lets you excite the string with either a “soft” impulse, like a fingertip, or a “sharp”

impulse, like a guitar pick. You can also do this at different points on the string, from

the center to near the edge.

Try different sorts of excitation and qualitatively examine the character of the vi

a-

tions produced. Mathematically, you would use a Fourier transform to determine the

amplitudes of the different normal modes, but you don’t need to do that here; it’s

enough to look at the overall character: “little fast wiggles” means that there is lots

of energy in the higher normal modes, while “slow big waves” means that there is lots

of energy in the lower modes.

5. Acoustically, an excitation that involves mostly lower normal modes sounds “darker”

or more “mellow”; an excitation that involves significant energy in the higher normal

2

modes sounds “

ight” or “tinny”. Consider the difference between exciting a guita

string with a pick (which applies a force to a very small region of the string) and

a finger (which applies a force to a larger region of the string). Simulate something

co

esponding to each of these and look at the oscillation of your string. Does the

observed behavior match the expected sound?

Part 3: Testing the nonlinear behavior you can’t do with pen and pape

(due Friday, December 7, or shortly thereafter)

Note: This part may be modified in the next few weeks.

6. Now we’re going to deviate from the limit of small amplitude (i.e. “what you learned

in pencil-and-paper physics class”) and look at the non-ideal behavior. Simulate a few

different normal modes at a range of amplitudes (ranging up to A ' L), and also vary

the tension. Discuss the validity of the “normal mode” idea outside of the limit of

small amplitude: does your simulation now exhibit independent normal modes, each

with a definite frequency?

Plot amplitude vs. period for several different values of the amplitude at moderate

tension (L/L0 ≈ 1.1), going roughly from small A to A = 0.4L. (Include enough points

that the character is clear.) Does your period give you the expected

Due Tuesday, 12 November, at 5 PM

Progress report due 6 November before class

This homework assignment is the first of several parts. I encourage you to try to finish

this assignment ahead of time and begin the next one before the deadline of this one. This

assignment requires you to write some code; the next assignment will involve using that code

to study some physics.

Your “report” for this assignment will just consist of keeping us up-to-date on your progress;

next Thursday we’d like to see where you are, and will come around and check on you.

Please start early. I am giving you two weeks to do this assignment in order to accommo-

date anyone who may have some difficulty. This will be a bit more challenging to code than

your previous one; please don’t put it off until a few days before it’s due. We’ll be doing a

check-in in a week for a grade; you’ll be given a grade based on how well your code works

at that point. We don’t expect it to be perfect, only that you’re making good progress.

In this assignment, you will model a stretched string (as in a guitar or violin) using a lattice

elasticity model, in which point masses (“nodes”) are connected by Hooke’s-law springs. Fo

eference, the force law on any given node i (given in more detail in class) is:

~̇vxi =

1

m

[

kr̂i,i−1x(ri,i−1 − r0) + kr̂i,i+1(ri,i+1 − r0)

]

(1)

~̇vyi =

1

m

[

kr̂i,i−1y(ri,i−1 − r0) + kr̂i,i+1(ri,i+1 − r0)

]

(2)

where

~ri,j is a vector pointing from the i’th node to the j’th one, and is thus equal to (xj − xi)̂i+

(yj − yi)ĵ, k is the spring constant of a single spring, r0 is its unstretched length, and m is

the mass of each node.

A lattice simulation of a stretched string has the following parameters:

• The number of nodes in the simulation N + 1 (or the number of springs N)

• The linear mass density of the unstretched string µ

• The unstretched length of the string L0

• The tension T applied to it to stretch it, which gives you the stretched length L

1

• The stiffness α (Young’s modulus times cross-sectional area)

1. Determine values of m, k, and r0 (the “microscopic” quantities of interest to you

simulation) as a function of N,µ, L0 and α (the “macroscopic” quantities).

2. Write a program that simulates the vi

ations of a stretched string of uniform density

and elasticity, as we will discuss in class. (Notes will be posted on the website, too.)

You should be able to easily change (by setting variables) some crucial parameters of

your simulation: N,µ, L0, and α.

You should use the leapfrog solver for your simulation, although it might be useful to

code it using Euler-Cromer first as a learning exercise.

3. Determine the initial conditions (all of the xi’s; all the y-coordinates and velocities will

e zero) for a stretched but unexcited string with tension T applied to it. You will

first need to determine the stretched length L′, then put your N masses at equidistant

points between x = 0 and x = L′. Implement these initial conditions, and verify that

the string doesn’t move.

4. Now stretch your string with sufficient tension to stretch it by 25-100% of its un-

stretched length, and excite it somehow. You can do this by applying some external

force to it at some instant in time, or by choosing initial values for the yi’s. Verify

that it moves realistically. A simple initial condition that you can use is a Gaussian

deformation, in which

y(x) = exp

[

A

−(x− xc)2

σ2

]

where xc is the center of the excitation, σ is its width, A is its amplitude, and σ and

xc are chosen so that the Gaussian dies down before you get to the ends of the string.

You may use any other initial conditions you like for testing.

5. Play around with your initial conditions, along with N , the stretch amount, µ, L, and

so on. Discuss anything interesting that you find. Note that your behavior ought to

e relatively independent of N . Your report should contain descriptions of all the stuff

you did and what happened.

6. Modify your program to track conservation of energy (recall that the elastic potential

energy in each spring is U = 1

2

k(r − r0)2) and verify that energy is approximately

conserved.

2

Physics 307 Homework 8

First two parts due the Thursday after Thanksgiving; the final part is due

at the end of the term.

In this project you will study the properties of your vi

ating string. You have nothing

(much) to code this time, but you have quite a few things to simulate, study, and discuss.

A common exercise in physics is to study a simplified model of a vi

ating string where we

assume that the amplitude is small and the tension is large; this gives rise to the familia

ehavior we will discuss in class. For your project, you will examine this behavior, and

discover the limits of its validity.

As a reminder, some properties of the vi

ating string in this limit:

• Oscillations are a superposition of normal modes: y(x, t = 0) = A sin

(

nπx

L

)

, where L

is the stretched length

• Waves propagate along the string at a rate v =

√

T

µ

• The n’th normal mode has a wavelength λn = 2Ln , giving it a period Tn =

λn

v

= 2L

n

√

µ

T

• These properties don’t depend on the amplitude, in the limit of small amplitude.

Part 1: Verifying your code and testing the ideal behavior (due Tuesday, Novem-

er 27)

1. Modify your initial conditions to simulate any normal mode of your choice. Then

modify your code to print out the period of your string; you can determine the half-

period by looking for points at which the velocity at an antinode changes sign. (“If it

was going up before, and now it’s going down, then we’re at a half-period” – you did a

similar thing for the pendulum.) Note that this is only valid if you have behavior that’s

elatively close to the vi

ation of a normal mode; otherwise, the idea of “period” is

more complicated.

2. Recall that our goal was to simulate a vi

ating string using only N and the macro-

scopic quantities T , α, µ, and L0. All microscopic quantities (the node mass m, the

individual spring constant k′, and the equili

ium radius r0 should be determined from

these.

You need to verify that you are co

ectly implementing the relationship between the

macroscopic quantities and the microscopic quantities. You can do this by simulating

1

something well-behaved and verifying that the period doesn’t depend on N . Use

enough tension to stretch your string to XXXXXXXXXX× its starting length, and then simulate

the fundamental mode (n = 1) at a small amplitude. Verify that the period doesn’t

depend (much) on N . This is one check that everything is working properly.

3. Now, adjust the tension, linear density, unstretched length, and verify that the period

has the expected dependence on them. Remember, this expected dependence is only

valid in the “well-behaved” realm, where T is large and the amplitude is small.

IMPORTANT NOTE: The linear mass density µ that appears in the expression fo

the wave speed, and the quantity L that you use to compute the wave speed, are the

values after you stretch the string. You may handle this in any way you choose; one

is to compute a stretched density µ′ after you determine the stretch length that gives

the specified tension.

Part 2: Superpositions of normal modes (also due Tuesday, November 27

as well)

As you know, the vi

ations of a string (in the ideal case) can be characterized by a

superposition of normal modes.

4. You can also excite the string in more physical ways and study its behavior. Modify

your initial conditions so that you give the string a Gaussian “bump” in either velocity

or position in the beginning. Specifically,

yi = Ae

−(xi−xc)2/σ2

where xc is the center of the bump, and σ is its width; A is an amplitude. (Make sure

xc

σ, or you’ll be moving one of the endpoints significantly.) You will probably

want to use a small amplitude to avoid blatantly nonlinear behavior.

If your small amplitude makes the vi

ations too small to see easily, you can always

multiply the y-coordinates by some factor when you display them, just to exaggerate

the motions to make them more visible.

This lets you excite the string with either a “soft” impulse, like a fingertip, or a “sharp”

impulse, like a guitar pick. You can also do this at different points on the string, from

the center to near the edge.

Try different sorts of excitation and qualitatively examine the character of the vi

a-

tions produced. Mathematically, you would use a Fourier transform to determine the

amplitudes of the different normal modes, but you don’t need to do that here; it’s

enough to look at the overall character: “little fast wiggles” means that there is lots

of energy in the higher normal modes, while “slow big waves” means that there is lots

of energy in the lower modes.

5. Acoustically, an excitation that involves mostly lower normal modes sounds “darker”

or more “mellow”; an excitation that involves significant energy in the higher normal

2

modes sounds “

ight” or “tinny”. Consider the difference between exciting a guita

string with a pick (which applies a force to a very small region of the string) and

a finger (which applies a force to a larger region of the string). Simulate something

co

esponding to each of these and look at the oscillation of your string. Does the

observed behavior match the expected sound?

Part 3: Testing the nonlinear behavior you can’t do with pen and pape

(due Friday, December 7, or shortly thereafter)

Note: This part may be modified in the next few weeks.

6. Now we’re going to deviate from the limit of small amplitude (i.e. “what you learned

in pencil-and-paper physics class”) and look at the non-ideal behavior. Simulate a few

different normal modes at a range of amplitudes (ranging up to A ' L), and also vary

the tension. Discuss the validity of the “normal mode” idea outside of the limit of

small amplitude: does your simulation now exhibit independent normal modes, each

with a definite frequency?

Plot amplitude vs. period for several different values of the amplitude at moderate

tension (L/L0 ≈ 1.1), going roughly from small A to A = 0.4L. (Include enough points

that the character is clear.) Does your period give you the expected

Nov 29, 2021

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