Question

One of the main things allowing humans to determine whether a sound is coming from the left or the right is the fact that the sound will reach one ear before the other. Given that the speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\) and that human ears are typically \(20.0 \mathrm{~cm}\) apart, what is the maximum time resolution for human hearing that allows sounds coming from the left to be distinguished from sounds coming from the right? Why is it impossible for a diver to be able to tell from which direction the sound of a motor boat is coming? The speed of sound in water is \(1.50 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\).

Short answer

Answer: The maximum time resolution for human hearing that allows sounds from the left to be distinguished from sounds coming from the right is approximately \(5.83 \times 10^{-4} \mathrm{s}\) in air. It is impossible for a diver to determine the direction of the sound of a motorboat because the time resolution for sounds in water is significantly smaller (approximately \(1.33 \times 10^{-4} \mathrm{s}\)), making it harder for the brain to process and distinguish the direction of the sound due to the tiny time differences.

Step-by-step solution

Find the time it takes for sound to travel the distance between human ears in air

The distance between human ears is given as \(20.0 \mathrm{~cm}\), which is equivalent to \(0.20\mathrm{~m}\). We are also given that speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\). To find the time it takes for sound to travel this distance, we can use the equation: \( \mathrm{time} (t) = \frac{\mathrm{distance} (d)} {\mathrm{speed} (v)} \) So we have: \(t_\mathrm{air} = \frac{0.20 \mathrm{~m}}{343 \mathrm{~m/s}} \approx 5.83 \times 10^{-4} \mathrm{s}\).

Find the time it takes for sound to travel the distance between human ears in water

The distance between human ears remains the same, \(20.0 \mathrm{~cm}\) or \(0.20\mathrm{~m}\). We are given that the speed of sound in water is \(1.50 \times 10^3 \mathrm{~m/s}\). Using the same equation: \( \mathrm{time} (t) = \frac{\mathrm{distance} (d)} {\mathrm{speed} (v)} \) We get: \(t_\mathrm{water} = \frac{0.20 \mathrm{~m}}{1.50 \times 10^3 \mathrm{~m/s}} \approx 1.33 \times 10^{-4} \mathrm{s}\).

Compare the time difference and explain the difference in perception between air and water

The maximum time resolution for human hearing that allows sounds coming from the left to be distinguished from sounds coming from the right is the time it takes for sound to travel between human ears in air, which we calculated as \(t_\mathrm{air} \approx 5.83 \times 10^{-4} \mathrm{s}\). Now, let's address why it's impossible for a diver to tell the direction of sound from a motorboat. The time resolution for sounds in the water is \(t_\mathrm{water} \approx 1.33 \times 10^{-4} \mathrm{s}\). Since this time difference (between the sound reaching one ear and the other) is significantly smaller than the time difference in air, it is much harder to distinguish direction. This is because the human brain has difficulty processing such tiny time differences.

Key concepts

Time Resolution of Human Hearing

Humans have the remarkable ability to locate sounds based on the tiny differences in time it takes for sound to reach each ear. This ability is called time resolution of hearing, which is crucial for our sense of directionality with sound. Consider a sound source directly to your left - the sound will hit your left ear slightly before your right. This time difference is what your brain processes to tell you the sound is coming from the left.

Using the distance between your ears, roughly 20 cm, and the speed of sound in air, we can calculate that it takes sound about 0.000583 seconds to travel this distance. This minuscule interval, discernible by the human brain, is what allows us to localize sounds in our environment. However, for our auditory system to compute this directional information, these interaural time differences need to be within a certain threshold. If the time difference is too small, such as underwater, our brain does not process the sound's direction as effectively.

Speed of Sound in Different Media

Sound travels at different speeds depending on the medium through which it's passing. In air, the speed of sound is approximately 343 meters per second. However, when sound travels through a denser medium such as water, the speed of sound increases because molecules are closer together, allowing the sound waves to be transmitted more quickly from one to the next.

Mathematically, the speed at which sound travels \(v\) in any medium is calculated using the formula: \[v = \frac{\text{distance}}{\text{time}}\]. For instance, while the speed of sound in air is 343 m/s, it's about 1500 m/s in water, signifying a profound difference. This variance in sound speed has interesting implications for how sound is perceived in different settings, which is especially notable when comparing experiences in air to underwater environments.

Directional Hearing Underwater

Underwater, the speed of sound increases significantly, affecting our ability to determine the direction of a sound source. The fast speed of sound in water means that the time differences between sounds reaching each ear are much smaller than in air. For example, a diver may find it challenging to pinpoint the direction of a motorboat overhead.

Even though the sound travels faster, the actual time difference for the sound to reach each ear underwater is only about 0.000133 seconds - a much tighter window for our auditory system to decipher the direction. This smaller time difference, below the threshold that humans can resolve, explains why directional hearing is compromised underwater. Our ears and brains are not adapted to such quick calculations in this dense medium, making sound localization a tough task for divers.