Textbook error? Incorrect derivation?

[PFStudent]

Homework Statement

Hey,

In my physics textbook the derivation for the transverse velocity, ${v_{s}}$ of a sound wave is given as,

$${v_{s}(x, t)} = {\frac{\partial}{\partial{t}}}{[s(x, t)]}$$

$${v_{s}(x, t)} = -{\omega}{A_{s}}sin({kx}-{{\omega}{t}}+{\phi})$$

Where,

$${s(x, t)} = {{A_{s}}{cos({kx} - {{\omega}{t}} + {\phi})}}$$

I think the book made an error because the transverse velocity of the sound wave really should be,

$${v_{s}(x, t)} = +{\omega}{A_{s}}sin({kx}-{{\omega}{t}}+{\phi})$$

The reason it comes out to positive is because the partial derivative with respect to $t$ of $cos\theta$, should be $-sin\theta$ and then (by the chain rule) the derivative of ${-}{\omega}{t}$ should be ${-}{\omega}$, therefore the two negatives should cancel each other out.

Resulting in a positive (one) coefficient for the function,

$${v_{s}(x, t)} = +{\omega}{A_{s}}sin({kx}-{{\omega}{t}}+{\phi})$$

So is the book wrong then, since they had a negative sign in front of the omega?

Any help is appreciated.

Thanks,

-PFStudent

 

[learningphysics]

Yes you're right.

 

[G01]

Just wondering, what book are you using PFStudent?

 

[Doc Al]

So is the book wrong then, since they had a negative sign in front of the omega?

Yep, looks like the book is wrong. For a sanity check, sketch the wave at time t = 0. For points just to the right of x = 0, which way is s changing with time? For points just to the left?

 

[PFStudent]

Yes you're right.

Well I am glad I got that figured out.

Yea, I compared the same section against Serway's physics book and they had a positive (one) coefficient, so I was not sure.

Thanks for the check learningphysics.

Just wondering, what book are you using PFStudent?

The textbook I am using is,

The Fundamental of Physics 7th (Extended) Edition by [Halliday / Resnick / Walker] and the error is on page 455, Chapter 17-6.

Hopefully, that error will get taken care of in the next edition.

Yep, looks like the book is wrong. For a sanity check, sketch the wave at time t = 0. For points just to the right of x = 0, which way is s changing with time? For points just to the left?

Well, the transverse displacement function for the sound wave is given by,

$${s(x, t)} = {{A_{s}}{cos({kx} - {{\omega}{t}} + {\phi})}},{\textcolor[rgb]{1.00,1.00,1.00}{.}}{\textcolor[rgb]{1.00,1.00,1.00}{.}}{\textcolor[rgb]{1.00,1.00,1.00}{.}}{\textcolor[rgb]{1.00,1.00,1.00}{.}}{\textcolor[rgb]{1.00,1.00,1.00}{.}}[+x{\textcolor[rgb]{1.00,1.00,1.00}{.}}propagation]$$

So, because this is a cosine curve, the points just to the right of $x = 0$ at $t = 0$, that is $s$ should move in the negative (downward) direction.

So, is it then because $s$ moves in the negative (downward) direction, then the coefficient of omega must be negative? As below,

$${v_{s}(x, t)} = +{\omega}{A_{s}}sin({kx}-{{\omega}{t}}+{\phi})$$

-PFStudent

 

[G01]

Surprising. I didn't suspect HRW. I guess even the best of texts can't be perfect.

 

[Doc Al]

The textbook I am using is,

The Fundamental of Physics 7th (Extended) Edition by [Halliday / Resnick / Walker] and the error is on page 455, Chapter 17-6.

It must be that whippersnapper Walker messing things up. I have an old edition and the derivation is correct.

 

[D H]

I have a very old edition of Halliday and Resnick that does not involve "that whippersnapper Walker". They use sine rather than cosine for the zeroth derivative. Somewhere along the line they must have switched to cosine for the zeroth derivative and forgotten to adjust the sign of the first derivative accordingly.

 

[PFStudent]

Surprising. I didn't suspect HRW. I guess even the best of texts can't be perfect.

Yea, I was not sure either between that text and Zemansky's those are the two best introductory (calculus based) textbooks to physics.

It must be that whippersnapper Walker messing things up. I have an old edition and the derivation is correct.

Yea, my professor goes on and on how like his "Bible" to introductory physics is the first edition of Fundemanetals of Physics which he says is way more rigorous and advanced (than the 7th edition).

I have a very old edition of Halliday and Resnick that does not involve "that whippersnapper Walker". They use sine rather than cosine for the zeroth derivative. Somewhere along the line they must have switched to cosine for the zeroth derivative and forgotten to adjust the sign of the first derivative accordingly.

Oh, that might explain it. Thanks for the info.

Thanks,

-PFStudent

 

[Doc Al]

I have a very old edition of Halliday and Resnick that does not involve "that whippersnapper Walker". They use sine rather than cosine for the zeroth derivative. Somewhere along the line they must have switched to cosine for the zeroth derivative and forgotten to adjust the sign of the first derivative accordingly.

That's exactly what I think happened.

 

[D H]

d on how like his "Bible" to introductory physics is the first edition of Fundemanetals of Physics which he says is way more rigorous and advanced (than the 7th edition).

This thread makes me realize how ancient my flatuelence is becoming. My book is plain old Physics, Halliday and Resnick, Third Edition. It predates and is thus more fundamental than Fundamentals of Physics, First Edition.

 

[Doc Al]

This thread makes me realize how ancient my flatuelence is becoming. My book is plain old Physics, Halliday and Resnick, Third Edition. It predates and is thus more fundamental than Fundamentals of Physics, First Edition.

I hear you, brother. I have that edition, but actually used an earlier one as an undergrad.

 

[D H]

Mine is copyright 1966. You used something even older? Ouch. Your flatulence must be on the verge of fossilizing.

 

[Doc Al]

Your flatulence must be on the verge of fossilizing.

I thank you and bow in your general direction.

I'm pretty sure I used the 2nd edition, which was copyright 1966. (But god knows where it is*.) I have the new-fangled 3rd edition right here, which claims to be published in 1977. Why can't they leave well enough alone?

* ah... now I remember: it's in my office!

 

[D H]

Acck! I have a first edition, second printing! Library of Congress Catalog Card Number 66-11527. My flatuence is beyond fossilization.

Edited to add:
To the books credit, I still open it on occasion. It has survived over a dozen moves and a flood (some pages are stuck together, and no, don't go anywhere with than information).

 

[Doc Al]

Acck! I have a first edition, second printing! Library of Congress Catalog Card Number 66-11527. My flatuence is beyond fossilization.

One thousand pardons, oh ancient and feeble one!

I'll have to check when I get to the office. I see no point in trying to "remember", since I can barely remember how to get home.