Force and power applied to create a traveling wave

[Miles123K]

Homework Statement

A string of tension T and linear density $\rho$ is driven at $x=0$ and two travelling wave in positive and negative x-direction had been created. Calculate the force and power applied to create the travelling wave.

I am studying chapter 8 of the book the physics of waves by Howard Georgi and I am quite confused. I read the entire chapter but I still don't quite get how to acquire the travelling wave mode of a system. Can someone check my solution to this problem but also explain how to systematically work out the "travelling wave mode" from conditions provided like this question?

Relevant Equations

$\frac {\partial^2 \psi} {\partial t^2} = \frac T \rho \frac {\partial^2 \psi} {\partial x^2}$
$F = - T \frac {\partial \psi} {\partial x}$

Again I am really confused, but I just put the traveling wave as:
$\psi(x,t) = Dcos(kx- \omega t)$ for positive x
$\psi(x,t) = Dcos(kx+ \omega t)$ for positive x
Then I simply differentiated and plugged in $x=0$
$F(t) = - T D k sin(\omega t)$
and from this
$\langle P \rangle = T D^2 k \omega sin^2(\omega t)$

 

[Miles123K]

The string is infinite. My greatest confusion is with the process from $A(t) = D cos(\omega t)$ to the wave function. In this case I think I am just guessing the function instead of working it out. I know how to work a standing wave function from this but not a traveling wave.

 

[haruspex]

Then I simply differentiated

Yes, but on what basis did you choose to differentiate wrt x, not t? What were you expecting it to tell you?

At time t, what angle does the string make at the origin? What force is the tension exerting on the driver of the oscillation?

 

[Miles123K]

Yes, but on what basis did you choose to differentiate wrt x, not t? What were you expecting it to tell you?

At time t, what angle does the string make at the origin? What force is the tension exerting on the driver of the oscillation?

for small $\theta$ $sin(\theta) = tan(\theta) = \frac {\psi (a)-\psi(0)} {a}$ for the limit $a$ approaches zero. The angle and force are illustrated in this image.

 

[haruspex]

View attachment 241313
for small $\theta$ $sin(\theta) = tan(\theta) = \frac {\psi (a)-\psi(0)} {a}$ for the limit $a$ approaches zero. The angle and force are illustrated in this image.

Good, but remember the driving force acts on both sides of the origin.
What is the speed at which the force moves the strings at the origin?

 

[Miles123K]

Good, but remember the driving force acts on both sides of the origin.
What is the speed at which the force moves the strings at the origin?

Yes. I think the wave function on the string is piece-wise and the same force drives both side. (Not very sure) The speed would just be the time derivative of $\psi$

 

[haruspex]

Yes. I think the wave function on the string is piece-wise and the same force drives both side. (Not very sure) The speed would just be the time derivative of $\psi$

Sorry, just realized that apart from the factor of 2 you have the right expression for power as a function of time; it just isn’t clear how you got it.
Not sure if the question wants the average power. That's easy enough to find.
I would take it that D and ω are inputs, so you should express k in terms of ω, ρ and T.

 

[Miles123K]

Sorry, just realized that apart from the factor of 2 you have the right expression for power as a function of time; it just isn’t clear how you got it.
Not sure if the question wants the average power. That's easy enough to find.
I would take it that D and ω are inputs, so you should express k in terms of ω, ρ and T.

⟨P⟩=TD2kωsin2(ωt)

Hey I just realized I put down the function of power not its time average. My answer for power is:
$\langle P \rangle = \frac 1 2 T D^2 k \omega = \frac 1 2 Z D^2 \omega^2$
whereas $Z = \frac k \omega = T \frac 1 v = \sqrt {\rho T}$

What I did is use the definition $P = F v$ and used the expression for $F$ and $v$ as stated above.

I didn't quite get what you meant by a factor of 2. Would u mind explain that?

 

[haruspex]

Hey I just realized I put down the function of power not its time average. My answer for power is:
$\langle P \rangle = \frac 1 2 T D^2 k \omega = \frac 1 2 Z D^2 \omega^2$
whereas $Z = \frac k \omega = T \frac 1 v = \sqrt {\rho T}$

What I did is use the definition $P = F v$ and used the expression for $F$ and $v$ as stated above.

I didn't quite get what you meant by a factor of 2. Would u mind explain that?

It is generating a wave each side of the origin. Same power needed for each.

 

[Miles123K]

It is generating a wave each side of the origin. Same power needed for each.

So does that mean my solution is correct?

 

[haruspex]

So does that mean my solution is correct?

I believe so. It bothers me that D and ω are not given as inputs.

 

[Miles123K]

I believe so. It bothers me that D and ω are not given as inputs.

Ummm are they not? They are provided in the question right? Or do you mean something else?

 

[haruspex]

Ummm are they not? They are provided in the question right? Or do you mean something else?

I based that on the synopsis you typed in post #1, but yes, I see they are given in the attachment.