A bell with a fundamental frequency of 880 Hz is moving toward an observer at 3.5 m/s. If the speed of sound is 343 m/s, what pitch would be heard by the observer
Doppler shift
f₀ = f/(1- v/u) =
=880/(1-3.5/343)=889.07 Hz
To determine the pitch heard by the observer, we need to consider the Doppler effect. The Doppler effect is the change in frequency of a wave (in this case, sound) due to the relative motion between the source of the wave (the bell) and the observer.
The formula to calculate the apparent frequency (heard by the observer) due to the Doppler effect is as follows:
f' = f * (v + v_o) / (v + v_s)
Where:
f' is the apparent frequency heard by the observer,
f is the actual frequency of the sound (fundamental frequency of the bell),
v is the speed of sound in the medium (343 m/s),
v_o is the velocity of the observer (positive if moving toward the bell, negative if moving away), and
v_s is the velocity of the source (positive if moving toward the observer, negative if moving away).
In this case, the actual frequency of the bell is 880 Hz, the speed of sound is 343 m/s, and the observer is moving toward the bell at 3.5 m/s. Therefore, we can substitute these values into the formula:
f' = 880 Hz * (343 m/s + 3.5 m/s) / (343 m/s + 0)
Simplifying the equation:
f' = 880 Hz * 346.5 m/s / 343 m/s
f' = 892.57 Hz
Therefore, the pitch heard by the observer would be approximately 892.57 Hz.
To determine the observed pitch, we need to consider the Doppler effect. This effect describes how the frequency of a wave changes when the source or observer is in relative motion.
The formula for the observed frequency, f', due to the Doppler effect is as follows:
f' = f * (v + v₀) / (v + vᵢ),
where:
f' is the observed frequency,
f is the original frequency (fundamental frequency of the bell),
v is the speed of sound,
v₀ is the velocity of the observer, and
vᵢ is the velocity of the source (bell).
Given values:
Original frequency, f = 880 Hz
Speed of sound, v = 343 m/s
Velocity of the observer, v₀ = 3.5 m/s
Velocity of the source, vᵢ = 0 m/s (since it is not mentioned whether the source is moving)
Let's substitute the values into the formula:
f' = 880 * (343 + 3.5) / (343 + 0)
Simplifying the equation:
f' = 880 * 346.5 / 343
f' ≈ 891.1 Hz
Therefore, the observed pitch, or frequency, heard by the observer would be approximately 891.1 Hz.