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Textbook error? Incorrect derivation?

  1. Sep 9, 2007 #1
    1. The problem statement, all variables and given/known data

    Hey,

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

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

    [tex]
    {v_{s}(x, t)} = -{\omega}{A_{s}}sin({kx}-{{\omega}{t}}+{\phi})
    [/tex]

    Where,

    [tex]
    {s(x, t)} = {{A_{s}}{cos({kx} - {{\omega}{t}} + {\phi})}}
    [/tex]

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

    [tex]
    {v_{s}(x, t)} = +{\omega}{A_{s}}sin({kx}-{{\omega}{t}}+{\phi})
    [/tex]

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

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

    [tex]
    {v_{s}(x, t)} = +{\omega}{A_{s}}sin({kx}-{{\omega}{t}}+{\phi})
    [/tex]

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

    Any help is appreciated.

    Thanks,

    -PFStudent
     
    Last edited: Sep 9, 2007
  2. jcsd
  3. Sep 9, 2007 #2

    learningphysics

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    Yes you're right.
     
  4. Sep 9, 2007 #3

    G01

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    Just wondering, what book are you using PFStudent?
     
  5. Sep 9, 2007 #4

    Doc Al

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    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?
     
  6. Sep 9, 2007 #5
    :surprised

    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.

    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.

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

    [tex]
    {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]
    [/tex]

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

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

    [tex]
    {v_{s}(x, t)} = +{\omega}{A_{s}}sin({kx}-{{\omega}{t}}+{\phi})
    [/tex]

    -PFStudent
     
    Last edited: Sep 9, 2007
  7. Sep 9, 2007 #6

    G01

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    Surprising. I didn't suspect HRW. I guess even the best of texts can't be perfect.:smile:
     
  8. Sep 9, 2007 #7

    Doc Al

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    It must be that whippersnapper Walker messing things up. I have an old edition and the derivation is correct.
     
  9. Sep 9, 2007 #8

    D H

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    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.
     
  10. Sep 9, 2007 #9
    Yea, I was not sure either between that text and Zemansky's those are the two best introductory (calculus based) textbooks to physics.

    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).

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

    Thanks,

    -PFStudent
     
    Last edited: Sep 9, 2007
  11. Sep 9, 2007 #10

    Doc Al

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    That's exactly what I think happened. :wink:
     
  12. Sep 9, 2007 #11

    D H

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    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.
     
    Last edited: Sep 9, 2007
  13. Sep 9, 2007 #12

    Doc Al

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    I hear you, brother. I have that edition, but actually used an earlier one as an undergrad.
     
  14. Sep 9, 2007 #13

    D H

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    Mine is copyright 1966. You used something even older? Ouch. Your flatulence must be on the verge of fossilizing.
     
  15. Sep 9, 2007 #14

    Doc Al

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    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!
     
  16. Sep 9, 2007 #15

    D H

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    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).
     
    Last edited: Sep 9, 2007
  17. Sep 9, 2007 #16

    Doc Al

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    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.
     
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