Does Wave Speed on a Wire Change with Temperature Increase?

[Fluxxx]

Homework Statement

A wire is strung tightly between two immovable posts. Decide whether the speed of the transverse wave on this wire would increase, decrease, or remain the same when the temperature increases. Ignore any change in the mass per unit length of the wire.

Homework Equations

$$v=\sqrt{\frac{F}{m/L}}$$
$$\Delta L=\alpha L_{0} \Delta T$$

The Attempt at a Solution

In the second equation we see that as the temperature increases the length expands. In the first equation, bigger length would imply that the speed increases. But the answer given in the textbook is "Decreases". Why?

 

[sergiokapone]

Mass per unit length of the wire $m/L = const$ (the condition of the problem). But with icreasing length, the tension force is decrease, thus, velocity is decrease.

 

[Fluxxx]

with icreasing length, the tension force is decrease

What equation gives this relation?

 

[sergiokapone]

$F = AY \frac{\delta l }{l_0}$, but initial length $l_0$ elongate due to temperature, thus F decreased.

 

[theodoros.mihos]

I think you can use variations like:
$$ v(F,L) \,:\,\to\, \frac{dv}{dT} = \frac{\partial{v}}{\partial{L}}\frac{dL}{dT} - \frac{\partial{v}}{\partial{F}}\frac{dF}{dL}$$
combine your equation and that has given above.

 

[Fluxxx]

I think you can use variations like:
$$ v(F,L) \,:\,\to\, \frac{dv}{dT} = \frac{\partial{v}}{\partial{L}}\frac{dL}{dT} - \frac{\partial{v}}{\partial{F}}\frac{dF}{dL}$$
combine your equation and that has given above.

I have no idea how you got to this multivariable equation from what I wrote. Where did this come from?

 

[summerphysics]

A wire is strung tightly between two immovable posts.
The speed of a transverse wave on the wire is given by Equation 16.2: wave v = sqrt(F /(m / L)) .
The wire will expand because of the increase in temperature. Since the length of the wire
increases slightly, it will sag, and the tension in the wire will decrease. From Equation 16.2,
we see that the speed of the wave is directly proportional to the square root of the tension in
the wire. If we ignore any change in the mass per unit length of the wire, then we can
conclude that, when the temperature is increased, the speed of waves on the wire will
decrease.