Problem 24 - University Physics with Modern Physics

Question

We define the index of refraction of a material for sound waves to be the ratio of the speed of sound in air to the speed of sound in the material. Snell's law then applies to the refraction of sound waves. The speed of a sound wave is 344 \(\mathrm{m} / \mathrm{s}\) in air and 1320 \(\mathrm{m} / \mathrm{s}\) in water. (a) Which medium has the higher index of refraction for sound? (b) What is the critical angle for a sound wave incident on the surface between air and water? (c) For total internal reflection to occur, must the sound wave be traveling in the air or in the water? (d) Use your results to explain why it is possible to hear people on the opposite shore of a river or small lake extremely clearly.

Step-by-Step Solution

Answer

(a) Air (b) 15.1° (c) Sound travels from air (d) Sound refraction and low water index enhances clarity.

Step 1: Determine Index of Refraction for Air

The index of refraction \( n \) for sound waves is given by the ratio of the speed of sound in air to the speed of sound in the medium. For air, since we are comparing air to air itself, the index of refraction \( n_{\text{air}} \) is 1.

Step 2: Calculate Index of Refraction for Water

Using the formula for the index of refraction \( n = \frac{v_{\text{air}}}{v_{\text{medium}}} \), we find the index for water: \[ n_{\text{water}} = \frac{344 \ \text{m/s}}{1320 \ \text{m/s}} \approx 0.26. \]

Step 3: Identify Medium with Higher Index of Refraction

Compare the indices of refraction: \( n_{\text{air}} = 1 \) and \( n_{\text{water}} = 0.26 \). The higher index is in the air.

Step 4: Compute Critical Angle for Sound

The critical angle \( \theta_c \) is found using Snell's Law: \( n_{\text{air}} \sin(\theta_{i}) = n_{\text{water}} \sin(\theta_{t}) \), setting \( \theta_{t} = 90^\circ \). \[ \sin(\theta_c) = \frac{n_{\text{water}}}{n_{\text{air}}} = 0.26. \] So, \( \theta_c = \arcsin(0.26) \approx 15.1^\circ. \)

Step 5: Determine Medium for Total Internal Reflection

Total internal reflection occurs only when sound is traveling from the medium with a higher index to a lower one. Since \( n_{\text{air}} > n_{\text{water}} \), sound must travel from air to water.

Step 6: Explain Sound Clarity Across Water

Even when traveling in air, sound waves can refract into water and travel long distances due to the low index of refraction. This phenomenon allows sound to carry over obstructions such as rivers or lakes, making voices heard across water.

Key Concepts

Index of Refraction

The index of refraction for sound waves is an important concept. It's defined as the ratio of the speed of sound in air to the speed in the medium.
This gives us a way to understand how sound behaves as it travels through different materials.

• The speed of sound in air is typically about 344 meters per second.
• In water, it is about 1320 meters per second.

Using the formula \[ n = \frac{v_{\text{air}}}{v_{\text{medium}}} \]we can find the index of refraction for each medium. For air, since it is compared to itself, the index is always 1.
For water, with the provided speeds, the index is calculated to be approximately 0.26.
This means air has a higher index of refraction for sound compared to water. A higher index means sound waves bend more when entering from a different medium.

Snell's Law

Snell’s Law is key to understanding how sound waves change direction when moving between different media.
This law states that the product of the index of refraction and the sine of the angle of incidence (the angle at which the wave enters the new medium) is constant for both media:\[ n_{1} \sin(\theta_{i}) = n_{2} \sin(\theta_{t}) \]Here,

• \( n_{1} \): Index of refraction of the first medium.
• \( \theta_{i} \): Angle of incidence.
• \( n_{2} \): Index of refraction of the second medium.
• \( \theta_{t} \): Angle of transmission.

For sound waves going from air to water, since air has a higher index, Snell's Law helps us predict that the angle of transmission is smaller in water.
This principle explains why sound waves change path, often being refracted toward the normal (an imaginary line perpendicular to the surface) when moving from air to water.

Critical Angle

The critical angle is the angle of incidence beyond which all light or sound is reflected back into the medium with a higher index of refraction.
When sound travels from air into water, we look for the critical angle to understand when refraction stops.
Using Snell’s Law, the critical angle \(\theta_{c}\) can be calculated as follows:For a transition from air (higher index) to water (lower index):\[ \sin(\theta_{c}) = \frac{n_{\text{water}}}{n_{\text{air}}} \]Plugging the values, \[ \sin(\theta_{c}) = 0.26 \]which gives:\[ \theta_{c} = \arcsin(0.26) \approx 15.1^\circ \]This means that any sound wave hitting the interface above this angle will not enter the water but will reflect back into the air, showcasing total internal reflection.

Total Internal Reflection

Total internal reflection (TIR) is a fascinating phenomenon where sound waves are completely reflected within a medium rather than passing through the boundary.
This occurs when the wave travels from a denser medium (i.e., higher index) to a less dense one (i.e., lower index), such as from air to water, at an angle greater than the critical angle.

• TIR ensures that no wave energy gets transmitted into the new medium.
• In the context of sound, this is why standing near water can enhance hearing over distances.
• The waves reflect between the air and water surfaces, allowing sound to travel further.

This behavior also results in sound clarity across water bodies, as waves reflect back into the air rather than being absorbed or refracted by the water.