You are traveling in a car toward a hill at a speed of 42.1 mph. The car's horn emits sound waves of frequency 251 Hz, which move with a speed of 343 m/s.
a) Determine the frequency with which the waves strike the hill.
b) What is the frequency of the reflected sound waves you hear?
c) What is the beat frequency produced by the direct and the reflected sounds at your ears?
To answer these questions, we need to use the concept of the Doppler effect, which describes the change in frequency of a wave as the source and/or observer moves relative to each other. The formula to calculate the apparent frequency (the frequency observed by the observer) is as follows:
Apparent Frequency = (Actual Frequency * Speed of Sound) / (Speed of Sound ± Speed of Observer/Source)
Now let's apply this formula to each part of the question:
a) Determining the frequency with which the waves strike the hill:
In this case, the observer is the hill, not moving relative to the source (the car). Therefore, the speed of the observer is 0.
Apparent Frequency = (Actual Frequency * Speed of Sound) / (Speed of Sound ± 0)
Since the observer's speed is 0, the divisor will simply be the speed of sound.
Apparent Frequency = (Actual Frequency * Speed of Sound) / Speed of Sound
The speed of sound is given as 343 m/s, and the actual frequency is 251 Hz.
Apparent Frequency = (251 Hz * 343 m/s) / 343 m/s
Therefore, the frequency with which the waves strike the hill is 251 Hz.
b) Determining the frequency of the reflected sound waves you hear:
In this case, the observer (you) is stationary and the source (the horn in the car) is moving towards you at a speed of 42.1 mph.
Apparent Frequency = (Actual Frequency * Speed of Sound) / (Speed of Sound ± Speed of Observer/Source)
Applying the values, we have:
Apparent Frequency = (251 Hz * 343 m/s) / (343 m/s + (42.1 mph * 0.44704 m/s/mph))
Converting the speed of the car from mph to m/s gives:
Apparent Frequency = (251 Hz * 343 m/s) / (343 m/s + (42.1 * 0.44704 m/s))
Simplifying the expression:
Apparent Frequency = (251 Hz * 343 m/s) / (343 m/s + 18.85744 m/s)
= (251 Hz * 343 m/s) / 361.85744 m/s
Calculating the result:
Apparent Frequency ≈ 238 Hz
Therefore, the frequency of the reflected sound waves you hear is approximately 238 Hz.
c) Determining the beat frequency produced by the direct and reflected sounds at your ears:
The beat frequency is the difference between the frequencies of two interfering waves. In this case, the direct sound and the reflected sound waves are interfering.
Beat Frequency = |Frequency of Direct Sound - Frequency of Reflected Sound|
Using the calculated values:
Beat Frequency = |251 Hz - 238 Hz|
Therefore, the beat frequency produced by the direct and reflected sounds at your ears is 13 Hz.