A sound wave has a frequency of 686 Hz in air and a wavelength of 0.50 m. What is the temperature of the air?

-Find wavelengths for 20 hz:
-Find wavelengths for 20,000hz:

To find the temperature of the air when given the frequency and wavelength of a sound wave, we can use the formula for the speed of sound in air:

Speed of sound = wavelength × frequency

The speed of sound in air at a specific temperature can be calculated using the following approximation:

Speed of sound (m/s) = 331.4 + 0.6 × temperature (in Celsius)

Now we can solve for the temperature:

Speed of sound = wavelength × frequency
331.4 + 0.6 × temperature = wavelength × frequency

For the first part of your question:
Given the frequency of 686 Hz and a wavelength of 0.50 m, we can substitute these values into the equation:

331.4 + 0.6 × temperature = 0.50 × 686

Solving for the temperature:

0.6 × temperature = (0.50 × 686) - 331.4
Temperature (in Celsius) = [(0.50 × 686) - 331.4] / 0.6

Now you can calculate the temperature using the given formula.

For the second part of your question:
To find the wavelengths for different frequencies, we can use the formula for wavelength:

Wavelength (m) = Speed of sound / Frequency

Given a frequency of 20 Hz, we can substitute this value into the equation:

Wavelength = Speed of sound / 20

Using the speed of sound formula mentioned earlier (331.4 + 0.6 × temperature), we can substitute this formula into the equation:

Wavelength (m) = (331.4 + 0.6 × temperature) / 20

Similarly, for the frequency of 20,000 Hz, substitute this value into the equation and solve:

Wavelength (m) = (331.4 + 0.6 × temperature) / 20,000

By solving these equations, you can find the wavelengths for both 20 Hz and 20,000 Hz at the given temperature.