A radio acoustic sounding system (RASS) instrument uses 404-MHz radio waves (wavelength = 0.743 m).

(a) What is the wavelength of the sound waves used to measure air temperature?
ë = ___ m

(b) If the Doppler shift of the radio wave reflected from a region several hundred meters above the ground indicates a sound speed of 330 m/s, what is the air temperature?
T = ___°K

To answer these questions, we need to use some formulas related to wavelength, sound speed, and temperature.

(a) The wavelength of the sound waves used to measure air temperature can be found using the formula:

wavelength = speed of sound / frequency

Since we are given the frequency for the radio waves (404 MHz) and the wavelength for the radio waves (0.743 m), we can rearrange the formula to solve for the wavelength of the sound waves:

wavelength_sound = speed of sound / frequency_sound

First, we need to convert the frequency of the radio waves to Hz:

frequency_sound = 404 MHz = 404 x 10^6 Hz

Then, we can substitute the values into the formula:

wavelength_sound = speed of sound / frequency_sound

Substituting the given radio wave frequency and speed of sound:

wavelength_sound = 330 m/s / (404 x 10^6 Hz)

Calculating the wavelength_sound:

wavelength_sound = 0.0008168 m

Therefore, the wavelength of the sound waves used to measure air temperature is approximately 0.0008168 meters.

(b) To find the air temperature, we can use the Doppler shift equation:

Doppler shift = (change in frequency) / original frequency
Doppler shift = (speed of sound - speed of observer) / speed of sound

In this case, the Doppler shift is related to the temperature difference between the region several hundred meters above the ground and the ground.

Given:
Doppler shift = change in frequency = speed of sound - speed of observer = speed of sound - 0 m/s = speed of sound
original frequency = speed of sound

Now we can rearrange the Doppler shift equation to solve for the speed of sound:

Doppler shift = (speed of sound - 0) / speed of sound
Doppler shift = 1 - 0 / 1 = 1

Therefore, the speed of sound is equal to the Doppler shift, which is 1.

To find the air temperature, we can use the formula:

speed of sound = √(γRT)
where γ is the heat capacity ratio (approximately 1.4 for air),
R is the gas constant (approximately 287 J/(kg·K)),
and T is the air temperature in Kelvin.

Using the given speed of sound (330 m/s):

330 = √(1.4 × 287 × T)

Now we can solve for T:

330^2 = 1.4 × 287 × T

T = (330^2) / (1.4 × 287)

Calculating T:

T ≈ 202.99 K

Therefore, the air temperature is approximately 202.99 Kelvin.

(a) To find the wavelength of the sound waves used to measure air temperature, we can use the formula:

wavelength = speed of sound / frequency of sound

The speed of sound can be approximated as 330 m/s, and the frequency can be calculated by using the formula:

frequency = speed of light / wavelength of radio waves

Given that the wavelength of the radio waves is 0.743 m, we can substitute these values into the formula:

frequency = (3.00 × 10^8 m/s) / 0.743 m

Simplifying, we get:

frequency ≈ 4.04 × 10^8 Hz

Now we can use this frequency to calculate the wavelength of the sound waves:

wavelength = speed of sound / frequency of sound
wavelength ≈ (330 m/s) / (4.04 × 10^8 Hz)

Using a calculator, we find that the wavelength of the sound waves is approximately 8.166 × 10^-7 m. Therefore:

λ ≈ 8.166 × 10^-7 m

(b) To calculate the air temperature using the Doppler shift, we need to use the formula:

Doppler shift = (change in frequency) / (original frequency) = (-speed of sound) / (speed of light)

Using this formula, we can solve for the change in frequency:

(change in frequency) = (Doppler shift) × (original frequency) = (-speed of sound) × (frequency)

Substituting the known values:

(change in frequency) = (-330 m/s) × (4.04 × 10^8 Hz)

Using a calculator, we find that the change in frequency is approximately -1.334 × 10^11 Hz.

Now, we can calculate the change in temperature using the formula:

(change in temperature) = (change in frequency) × (speed of sound) / (frequency of sound)

Substituting the values known so far:

(change in temperature) = (-1.334 × 10^11 Hz) × (330 m/s) / (4.04 × 10^8 Hz)

Using a calculator, we find that the change in temperature is approximately -109 °C.

Since the Doppler shift is negative, the temperature is colder than the standard temperature. To get the air temperature, we add the change in temperature to the standard temperature.

Assuming the standard temperature is 20 °C, we can calculate the air temperature:

T = (standard temperature) + (change in temperature)
T = 20 °C + (-109 °C)

Using a calculator, we find that the air temperature is approximately -89 °C.

Therefore:

T ≈ -89 °C or 184 K.