How can I use the speed of sound to calculate the air temperature?

[Sabres151]

Hey guys, need some help here with a problem. I've gotten half way thru and don't know where to go next. Here's the problem:

The wavelength of a 40,000-Hz ultrasound wave is measured to be 0.868cm. Find the air temperature.

Using v = frequency(wavelength)
40,000-Hz(0.868cm) = v
v = 347 m/s

I'm not sure where to go from here? ...Thanks for any advice!

 

[Andrew Mason]

Hey guys, need some help here with a problem. I've gotten half way thru and don't know where to go next. Here's the problem:

The wavelength of a 40,000-Hz ultrasound wave is measured to be 0.868cm. Find the air temperature.

Using v = frequency(wavelength)
40,000-Hz(0.868cm) = v
v = 347 m/s

I'm not sure where to go from here? ...Thanks for any advice!

You will need to know the relationship between the speed of sound in air and temperature.

$$v = \sqrt{\gamma RT/M}$$ so

$$T = Mv^2/R$$

where M is the molar weight of the gas (28.95 grams/mole for air) and $\gamma$ is Cp/Cv = 7/5 for a diatomic gas such as air.

See this link to the http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe3.html#c2"

AM

 

[kyle8921]

Hello,

It seems like you are on the right track with your calculations. To find the air temperature, you will need to use the formula v = √(γRT), where v is the speed of sound, γ is the adiabatic index for air (approximately 1.4), R is the gas constant for air (approximately 287 J/kg*K), and T is the temperature in Kelvin.

Since you have already found the speed of sound (v = 347 m/s), you can rearrange the formula to solve for T. It would look like this:

T = (v^2)/(γR)

Plugging in the values, you will get:

T = (347^2)/(1.4*287) = 195.8 K

To convert to Celsius, you can subtract 273.15 from the Kelvin temperature, giving you an air temperature of -77.35 °C.

I hope this helps! Let me know if you have any further questions. Good luck with your problem!