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# Speed of longitudinal wave 30 times the speed of a transverse wave?

1. ### PirateFan308

94
1. The problem statement, all variables and given/known data
What must be the stress (F/A) in a stretched wire of a material whose Young's modulus is Y for the speed of longitudinal waves to equal 30 times the speed of transverse waves?

2. Relevant equations
$Y=\frac{Fl_0}{Al}$

$v_L=f\lambda = \sqrt{F/\mu}$

$v_T = \omega A sin(kx-\omega t)$

3. The attempt at a solution
I know that $v_L=30v_T$ but my main problem is that longitudinal velocity remains constant while transverse velocity is dependent on position and time, making it impossible for one to be a multiple of the other unless they are both equal to 0, which cannot be the case. I'm not sure what I'm missing ... thanks!

2. ### darkxponent

385
those are irrevelant equations that u are using....the question says speed of transverse wave not transverse velocity of particle

3. ### darkxponent

385

these equations are not written correct. You havent written the formula for speed of longitudnal wave. The formula written is for the speed of transverse wave instead of longitudnal wave. The formula written for speed of Transverse wave is the formulae for "transverse velocity of particle".

4. ### PirateFan308

94
Is $v_T = \lambda f = \sqrt{F/\mu}$ and $v_L = \frac{Y}{\rho}$ correct?

5. ### emailanmol

297

Hint:
You have missed a square root.

6. ### PirateFan308

94
$v_L = \sqrt{\frac{Y}{\rho}}$??

7. ### PirateFan308

94
So if $v_L=\sqrt{Y/\rho}$ and $v_t=\sqrt{F/\mu}$ then

$\sqrt{Y/\rho}=30\sqrt{F/\mu}$ which is equivalent to $Y=\frac{900F\rho}{\mu}$

$Y=\frac{Fl_0}{Al}$

so $\frac{F}{A}=\frac{Yl}{l_0}=\frac{900Fl\rho}{l_0 \mu}$

but $l\rho = A$ and $\mu = \frac{mass}{length} ~~so~~\mu l_0=mass$

so $\frac{F}{A}=\frac{mass}{900AF} ~~so~~F^2=\frac{mass}{900}$

which doesn't work ...

I also tried it a different way, letting $v_T=fλ$ so that $Y=900v^2\rho$ so then $\frac{F}{A}=\frac{Yl}{l_0} = \frac{900v^2\rho l}{l_0} = \frac{900v^2A^2}{l_0}$ which also doesn't work. Any help would be appreciated! Thanks!!

8. ### emailanmol

297
Skip the step where you have got in Y=FL/AL(0).

You have no way of relating L and L(0).

Also Lp=A is not right.

p=M/V sp Lp=LM/V.
But here L is change in length.
So you can't write V=AL as L is change in length.So this doesn't help.

Y/p=900F/u.

Divide it by A on both sides

Y/Ap=900F/Au.

So F/A=Yu/900Ap

Is there any way you can find out what u/Ap is ?
( Hint:You have applied a relation between u and l in your last post.That might help)

9. ### PirateFan308

94
I got it - thank you!!