Question

Ultrasonic sound emitters are used to repel rodents. They emit sound at a very high frequency that is unpleasant to rodents, but imperceptible to the human ear.

You are asked to design one such device. You must ensure that it is safe to be placed along highways where the speed limit is 98 km/hr. That is, drivers on the highway must not distracted on account of hearing the sound. The highest frequency perceived by the human ear is 20000 Hz. The speed of sound in air is 320 m/s.
A driver approaches the emitter, while driving at the speed limit. Sound reaching the driver must be perceived at or above the threshold frequency of 20000 Hz. What is the minimum frequency of the emitter to ensure this?
Another driver travels away from the emitter, while travelling at the speed limit. This driver must also perceive sound at or above the threshold frequency of 20000 Hz. What should be the minimum emitter frequency to ensure this?
What is the lowest possible frequency where the device remains safe for all drivers (driving in either direction, with speed below the speed limit)?

Answer 1

To ensure the sound reaching the driver is perceived at or above the threshold frequency of 20,000 Hz, we use the Doppler effect and calculate the minimum frequency of the emitter. When the driver approaches the emitter, the minimum frequency of the emitter should be 20,193 Hz. When the driver travels away from the emitter, the minimum frequency should be 19,807 Hz. The lowest possible frequency where the device remains safe for all drivers is in the range of 19,807 Hz to 20,193 Hz. let's find ;-

Now,

To ensure that the sound reaching the driver is perceived at or above the threshold frequency of 20,000 Hz, we need to consider the Doppler effect. The Doppler effect is the change in frequency of a wave perceived by an observer moving relative to the source of the wave.

When the driver approaches the emitter, the frequency of the sound waves will be shifted higher due to the Doppler effect. To calculate the minimum frequency of the emitter, we can use the equation:

f' = (v + vd) / (v + vs) * f,

where f' is the perceived frequency, f is the actual frequency, v is the speed of sound, vd is the speed of the driver, and vs is the speed of the source (emitter). Rearranging the equation, we get:

fs ≥ f * (v + vd) / (v + vs),

where fs is the minimum frequency of the emitter to ensure the sound is perceived at or above 20,000 Hz.

Using the given values, v = 320 m/s, vd = 98 km/hr = 27.22 m/s, and vs = 0 m/s, we can calculate:

fs ≥ 20,000 Hz * (320 + 27.22) / (320 + 0) = 20,193.18 Hz.

Therefore, the minimum frequency of the emitter should be 20,193 Hz to ensure the sound is perceived at or above the threshold frequency while the driver approaches the emitter at the speed limit.

Similarly, when the driver travels away from the emitter, the frequency of the sound waves will be shifted lower. Using the same equation and values, we can calculate that the minimum frequency of the emitter should be 19,807 Hz to ensure the sound is perceived at or above the threshold frequency while the driver travels away from the emitter at the speed limit.

To determine the lowest possible frequency where the device remains safe for all drivers in either direction, the emitter frequency should be between 19,807 Hz and 20,193 Hz. This range ensures that sound reaching the drivers will be perceived at or above the threshold frequency of 20,000 Hz, regardless of their direction of travel.