# Can somebody help me understand this BVP question?

• John004
In summary, the physical process described above can be summarized in the following way: at t = 0 the string lies completely on the x-axis but has a velocity of 1 ft/s in the positive y-direction.
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.

utt = c2uxx

## The Attempt at a Solution

The problem set is in the attachment

#### Attachments

• PDE HW 1.png
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John004 said:
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.

DrClaude said:
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?

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.

DrClaude said:
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 ytt(x,t) = 104 yxx(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
yt(x,0) = 1 for 0 < x < 1

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

ytt(x,t) = 104 yxx(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
yt(x,0) = 1 for 0 < x < 1

John004 said:
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 ytt(x,t) = 104 yxx(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
yt(x,0) = 1 for 0 < x < 1

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

ytt(x,t) = 104 yxx(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
yt(x,0) = 1 for 0 < x < 1

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

Ray Vickson said:
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?

John004 said:
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.

## 1. What is a BVP question?

A BVP (Boundary Value Problem) question is a type of mathematical problem that involves finding a solution to a differential equation subject to specified boundary conditions. In other words, it is a problem that requires finding a function that satisfies both the differential equation and the given boundary conditions.

## 2. How do I approach solving a BVP question?

The first step in solving a BVP question is to understand the given problem and identify the type of differential equation involved. Then, you can use various mathematical techniques such as separation of variables, substitution, or series expansion to solve the equation. Finally, you can use the given boundary conditions to find the specific solution that satisfies both the equation and the conditions.

## 3. What are some common types of BVP questions?

Some common types of BVP questions include the heat equation, the wave equation, and the Laplace equation. These equations are commonly used in physics and engineering to model various physical phenomena such as heat transfer, wave propagation, and electrostatics.

## 4. Can I use software to solve a BVP question?

Yes, there are various software programs that can be used to solve BVP questions, such as MATLAB, Mathematica, and Maple. These programs use numerical methods to find approximate solutions to the given problem. However, it is still important to understand the mathematical concepts and techniques involved in solving BVP questions.

## 5. What are some tips for solving BVP questions?

Some tips for solving BVP questions include: understanding the problem and the given boundary conditions, choosing an appropriate technique to solve the equation, and checking the solution for accuracy. It is also helpful to practice solving various types of BVP questions to improve your problem-solving skills.

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