Suppose you are stopped for a traffic light, and an ambulance approaches you from behind with a speed of 19.0 m/s. The siren on the ambulance produces sound with a frequency of 960 Hz as measured when you are at rest relative to the ambulance. The speed of sound in air is 343 m/s. What is the wavelength of the sound reaching your ears?
To find the wavelength of the sound reaching your ears, we can use the formula:
wavelength = speed of sound / frequency
First, we need to determine the speed of sound relative to you. Since you are not moving, the speed of sound relative to you is still 343 m/s.
Next, we need to determine the frequency of the sound relative to you. This can be calculated using the Doppler effect formula:
frequency observed = frequency emitted * (speed of sound + speed of observer) / (speed of sound + speed of source)
In this case, you are at rest relative to the ambulance, so your speed is 0 m/s. The frequency emitted by the siren is 960 Hz.
Plugging these values into the formula, we get:
frequency observed = 960 Hz * (343 m/s + 0 m/s) / (343 m/s + 19 m/s)
Simplifying the expression, we have:
frequency observed = 960 Hz * (343 m/s) / (362 m/s)
The frequency observed is equal to 913.26 Hz.
Now, we can calculate the wavelength by dividing the speed of sound by the frequency observed:
wavelength = 343 m/s / 913.26 Hz
Calculating this, we find:
wavelength ≈ 0.375 m
Therefore, the wavelength of the sound reaching your ears is approximately 0.375 meters.