Question

At what temperature, speed of sound in air will be doubled than its speed in air at STP?

Answer 1

Hint: Speed of a longitudinal wave when propagating in air is directly proportional to the temperature of air at the time of wave propagation. Sound is also a longitudinal wave, and it's on-book speed which is $343m{s^{ - 1}}$, is recorded at STP $(0^\circ C,1atm)$ .

Complete step by step answer:
Sound waves are longitudinal waves. Their speed of propagation is related to the absolute temperature of the atmosphere by the following relation.
$v \propto \sqrt T $
Where, $v$ is speed of sound
$T$ is absolute temperature ( measured in Kelvin scale)
Now let us look at the problem.
Let us assume that at standard temperature and pressure or at temperature ${T_1}$, speed of sound is ${v_1}$
Let ${v_2}$ is the doubled spee$ \Rightarrow {T_2} = 1092K$d of sound at temperature ${T_2}$. $({v_2} = 2{v_1})$
As we know, speed of sound is directly proportional to the square root of temperature, we can write
${v_1} \propto \sqrt {{T_1}} $ ……….. (1)
And ${v_2} \propto \sqrt {{T_2}} $ ………. (2)
Now, $\dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{{{T_1}}}{{{T_2}}}} $
Substituting the value of ${v_2}$ in the above equation, we get,
$ \Rightarrow \dfrac{1}{2} = \sqrt {\dfrac{{{T_1}}}{{{T_2}}}} $
At STP, ${T_1}$ is $273K$
So, $ \Rightarrow \dfrac{{273}}{{{T_2}}} = \dfrac{1}{4}$
${T_2} = 1092K$


So, the answer is, at $1092K$ or $819^\circ C$, the speed of sound will be double the value of its speed at STP.

Additional information:
There are two distinct classes of wave motion. Transverse and longitudinal.
In a transverse wave motion, the particles of the medium oscillate about their mean and equilibrium position at right angles to the direction of wave propagation. Transverse waves are associated with crests, troughs and wavelengths.
A longitudinal wave is different from a transverse wave in terms of angles of oscillations of media particles from their mean and equilibrium position. The particles oscillate along the direction of wave propagation. The longitudinal waves consist of compressions and rarefactions.
Examples of transverse waves are light waves, heat waves and examples of longitudinal waves are sound waves.

Notes: Sound travels at a speed of $340m{s^{ - 1}}$ at STP. Speed of any longitudinal wave depends on its media of propagation, if the media allows fast propagation, the wave will do so. By increasing the temperature of the medium, one can achieve a speed increase of sound waves. It is directly proportional to square root of the absolute temperature of the media. At $819^\circ C$ we can achieve the doubled speed of sound, but practically it is not possible in day to day life.