What is the minimum speed at which a source must travel toward you for you to be able to hear that its frequency is Doppler-shifted? That is, what speed produces a shift of 0.300 percent on a day when the speed of sound is 331 meters per second?
We’re told in this statement that the speed of sound is 331 meters per second, which we’ll refer to as 𝑣. We want to know the minimum speed at which a Doppler shift is detectable to our ear, a speed we’ll call 𝑣 sub min.
Let’s start by drawing a sketch of this scenario. Imagine, to start, that there’s a stationary sound source near a stationary observer. In this case, there would be no Doppler shift because nothing is moving, but then imagine that we set the source of the sound in motion toward the observer with the speed 𝑣 sub 𝑠. If we did that, the waves behind the moving source would be stretched out; there would be more distance between peaks and troughs, while the waves in front of the source would be compressed.
If we say the human ear is able to detect a percent difference in frequency, Δ percent 𝑓, of 0.300 percent, then we want to solve for the source speed 𝑣 sub 𝑠 that leads to this change. We can figure that out using the Doppler equation, where the observed frequency 𝑓 sub 𝑜 is equal to the source frequency 𝑓 sub 𝑠 multiplied by the speed of sound 𝑣 plus the speed of the observer 𝑣 sub 𝑜 divided by the speed of sound 𝑣 minus the speed of the source 𝑣 sub 𝑠.
When we apply this relationship to our scenario, we can first cancel out 𝑣 sub zero because that term is zero; our observer is stationary. Next, we can use the fact that Δ percent 𝑓 is equal to 0.300 percent; that’s the change in the frequency from the source to the observer due to the motion of the source.
The observed frequency is equal to the source frequency 𝑓 sub 𝑠 times one plus Δ percent 𝑓 divided by 100, or one plus 0.003 times 𝑓 sub 𝑠. Now we can replace the 𝑓 sub 𝑜 in our equation with this revised expression. And now that we have 𝑓 sub 𝑠 on both sides of the equation, we can cancel that term out. Our result is independent of the source frequency.
Now we want to rearrange this equation to solve for 𝑣 sub min. When we do that, we find it’s equal to the speed of sound 𝑣 times 0.003 divided by 1.003. When we plug in 331 meters per second for the speed of sound and calculate this fraction, we find that 𝑣 min is equal to 0.990 meters per second. That’s the minimum speed the sound source needs to take on towards us in order for our ear to detect a Doppler shift in frequency.