# Speed of wave with temperature

1. Jun 6, 2015

### Fluxxx

1. The problem statement, all variables and given/known data
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.

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

3. 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 text book is "Decreases". Why?

2. Jun 6, 2015

### 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.

3. Jun 6, 2015

### Fluxxx

What equation gives this relation?

4. Jun 6, 2015

### sergiokapone

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

Last edited: Jun 6, 2015
5. Jun 6, 2015

### 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.

6. Jun 6, 2015

### Fluxxx

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