Let v(x,t) = u(x+ct) and show that

• jpcjr
In summary: So, the final equation would be v_x = u_w. However, I understand that this might not be what you were asking for. So, if you could clarify what you were specifically looking for, that would be much appreciated!Welcome to PF,
jpcjr
1. Homework Statement

NOTE: I realize the following is a partial differential equation, but I believe the answer to my question is a straight forward multi-variate calculus question.

Let v(x,t) = u(x+ct,t) and show that v(x,t) solves the following...

ut = κ uxx ; (-∞<x<∞)

u(x,0) = $\phi$(x)

NOTE: You may disregard the following, if necessary for you to answer this question:

u(x,0) = $\phi$(x)

and simply show that v(x,t) solves the following...

ut = κ uxx

ALTERNATIVELY, you may help me by commenting on the correctness of my work below.

2. Homework Equations

See above and below...

3. The Attempt at a Solution

v(x,t) = u(x+ct,t)

v'(x,t) = c u$_{x}$(x+ct,t) + u$_{t}$(x+ct,t)

v$_{x}$(x,t) = c u$_{x}$(x+ct,t)

v$_{xx}$(x,t) = c$^{2}$ u$_{xx}$(x+ct,t)

v$_{t}$(x,t) = u$_{t}$(x+ct,t)

Any help on any part will be GREATLY appreciated!

Welcome to PF,

jpcjr said:
1. Homework Statement

NOTE: I realize the following is a partial differential equation, but I believe the answer to my question is a straight forward multi-variate calculus question.

Let v(x,t) = u(x+ct,t) and show that v(x,t) solves the following...

ut = κ uxx ; (-∞<x<∞)

Just before we start, are you sure it was not meant to be simply v(x,t) = u(x+ct)? Because I know that a function in which the temporal and spatial dependence are related in this way will satisfy a wave equation, which is given by utt = κuxx. This is different from the PDE that you gave. But anyway, taking what you wrote at face value:

jpcjr said:
ALTERNATIVELY, you may help me by commenting on the correctness of my work below.

Homework Equations

See above and below...

The Attempt at a Solution

v(x,t) = u(x+ct,t)

v'(x,t) = c u$_{x}$(x+ct,t) + u$_{t}$(x+ct,t)

v$_{x}$(x,t) = c u$_{x}$(x+ct,t)

v$_{xx}$(x,t) = c$^{2}$ u$_{xx}$(x+ct,t)

v$_{t}$(x,t) = u$_{t}$(x+ct,t)

First of all, I don't think that v' (v "prime") is meaningful here. In single-variable calculus, a prime symbol typically means "derivative with respect to the argument." But this function has two arguments. Since it is a multi-variable function, any derivative is going to be a partial derivative, and you must explicitly specify which variable you are differentiating with respect to. As for your attempt at partial derivatives, my approach would be to write:

u = u(w,t) where w = x+ct

then,

$$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial w}\frac{\partial w}{\partial x}$$$$= \frac{\partial u}{\partial w}$$

Et cetera for the other derivatives

I am sure it is:

Let v(x,t) = u(x+ct,t)

That would mean the following is incorrect, right?

v.sub.x(x,t) = c u.sub.x(x+ct,t)

and should have been...

v.sub.x(x,t) = u.sub.x(x+ct,t)

What I was saying was that v_x = u_w, with underscores denoting subscripts.

1. What is the purpose of defining v(x,t) as u(x+ct)?

The purpose of defining v(x,t) as u(x+ct) is to study the behavior of a wave u(x,t) as it travels with a constant speed c. It allows us to see how the wave changes in time at a fixed position x, or how it changes in space at a fixed time t.

2. How does the substitution of v(x,t) = u(x+ct) affect the wave equation?

Substituting v(x,t) = u(x+ct) into the wave equation results in a simpler form, as the partial derivatives with respect to x and t can be replaced by a single derivative with respect to the new variable x+ct. This makes the equation easier to solve and analyze.

3. Can v(x,t) be used to study any type of wave?

Yes, v(x,t) can be used to study any type of wave as long as it satisfies the wave equation. This includes electromagnetic waves, sound waves, water waves, and more.

4. How does the substitution of v(x,t) = u(x+ct) affect the boundary conditions of the wave equation?

The substitution of v(x,t) = u(x+ct) does not affect the boundary conditions of the wave equation. The boundary conditions still need to be satisfied for both u(x,t) and v(x,t).

5. Can v(x,t) be used to study waves in three dimensions?

Yes, v(x,t) can be used to study waves in three dimensions. The substitution v(x,t) = u(x+ct) can be extended to three dimensions by considering v(x,y,z,t) = u(x+ct, y+ct, z+ct). This allows for the study of waves in all three dimensions with a constant speed c.

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