Can somebody help me understand this BVP question?

[John004]

Homework Statement

So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation? At least in problem 2 he says what he wants us to show.

Homework Equations

utt = c²uxx

The Attempt at a Solution

The problem set is in the attachment

 

[DrClaude]

So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation?

No, it means starting from the generic wave equation given in the preamble of the question and showing that the specific equations given in problem 1 actually correspond to the specific physical situation.

 

[John004]

No, it means starting from the generic wave equation given in the preamble of the question and showing that the specific equations given in problem 1 actually correspond to the specific physical situation.

I feel like I'm missing something obvious; I don't see how I can "show" that. The problem describes the physical process and then proceeds to write down the boundary conditions that correspond to that process. I don't get what I'm supposed to do. Am I just supposed to explain in words why the boundary conditions are valid?

 

[DrClaude]

I would go about it as if the equations were not given. Start with the wave equation and discuss the specific physical problem, setting up the parameter $a^2$ and the initial conditions, arriving at the equations given.

 

[John004]

I would go about it as if the equations were not given. Start with the wave equation and discuss the specific physical problem, setting up the parameter $a^2$ and the initial conditions, arriving at the equations given.

So If I understand you correctly, this is how I would go about answering the question.

Suppose that a 1 foot flexible piece of wire is stretched between the points (0,0) and (1,0). The tension in the wire is 10 Ib and the weight of the wire is 0.032 Ib. The parameter "a" in the wave equation is defined as a² = Tension/density; therefore since

mg = 0.032 Ib and δ(linear mass density) = m/L, a² = (10 Ib) (32 ft/s²)(1 ft)/(0.032 Ib) = 10⁴ (ft/s)²

The wave equation then becomes y_tt(x,t) = 10⁴ y_xx(x,t)

At t = 0 the string lies completely on the x-axis but has a velocity of 1 ft/s in the positive y - direction. The wire is under no external forces.

Since the wire is being stretched between the endpoints, that implies that the endpoints are fixed, therefore
y(0,t) = y(L,t) = 0 for t ≥ 0

it was said that the wire lies completely on the x-axis at t = 0, therefore
y(x,0) = 0 for 0 ≤ x ≤ 1
Since the velocity of the wire at t = 0 was 1 ft/s
y_t(x,0) = 1 for 0 < x < 1

So in short, the physical process described above can be summarized in the following way

y_tt(x,t) = 10⁴ y_xx(x,t) for 0 < x < 1 for t > 0
y(0,t) = y(L,t) = 0 for t ≥ 0
y(x,0) = 0 for 0 ≤ x ≤ 1
y_t(x,0) = 1 for 0 < x < 1

 

[Ray Vickson]

So If I understand you correctly, this is how I would go about answering the question.

Suppose that a 1 foot flexible piece of wire is stretched between the points (0,0) and (1,0). The tension in the wire is 10 Ib and the weight of the wire is 0.032 Ib. The parameter "a" in the wave equation is defined as a² = Tension/density; therefore since

mg = 0.032 Ib and δ(linear mass density) = m/L, a² = (10 Ib) (32 ft/s²)(1 ft)/(0.032 Ib) = 10⁴ (ft/s)²

The wave equation then becomes y_tt(x,t) = 10⁴ y_xx(x,t)

At t = 0 the string lies completely on the x-axis but has a velocity of 1 ft/s in the positive y - direction. The wire is under no external forces.

Since the wire is being stretched between the endpoints, that implies that the endpoints are fixed, therefore
y(0,t) = y(L,t) = 0 for t ≥ 0

it was said that the wire lies completely on the x-axis at t = 0, therefore
y(x,0) = 0 for 0 ≤ x ≤ 1
Since the velocity of the wire at t = 0 was 1 ft/s
y_t(x,0) = 1 for 0 < x < 1

So in short, the physical process described above can be summarized in the following way

y_tt(x,t) = 10⁴ y_xx(x,t) for 0 < x < 1 for t > 0
y(0,t) = y(L,t) = 0 for t ≥ 0
y(x,0) = 0 for 0 ≤ x ≤ 1
y_t(x,0) = 1 for 0 < x < 1

Good. You have done exactly what the question asked you to do.

 

[John004]

Good. You have done exactly what the question asked you to do.

The question just seems weird to me, at least the wording does. It feels circular. Same thing with question 2. In this class we haven't gone over how to solve these PDE's yet, so I'm thinking that for question 2 when its asking me to show that the string hangs in the fashion described by the parabolic function, I should just take the appropriate derivatives and substitute back into the wave equation and confirm that the equality holds, correct?

 

[Ray Vickson]

The question just seems weird to me, at least the wording does. It feels circular. Same thing with question 2. In this class we haven't gone over how to solve these PDE's yet, so I'm thinking that for question 2 when its asking me to show that the string hangs in the fashion described by the parabolic function, I should just take the appropriate derivatives and substitute back into the wave equation and confirm that the equality holds, correct?

How can we tell? We do not know the approach taken by your textbook or course notes, so we don't know whether the course (initially, at least) emphasizes things like physical derivations of PDEs for some phenomena, or whether it essentially starts with a PDE and then discusses boundary conditions and the like. I have seen different books on the subject take very different approaches to these questions.