A plane sound wave in air at 20^∘ C, with wavelength 589 mm, is incident on a smooth surface of

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A plane sound wave in air at 20^∘ C, with wavelength 589 mm,...

A plane sound wave in air at $20^{\circ} \mathrm{C},$ with wavelength $589 \mathrm{mm},$ is incident on a smooth surface of water at $25^{\circ} \mathrm{C}$ at an angle of incidence of $13.0^{\circ} .$ Determine (a) the angle of refraction for the sound wave and (b) the wavelength of the sound in water. A narrow beam of sodium yellow light, with wavelength 589 $\mathrm{nm}$ in vacuum, is incident from air onto a smooth water surface at an angle of incidence of $13.0^{\circ} .$ Determine (c) the angle of refraction and (d) the wavelength of the light in water. (e) Compare and contrast the behavior of the sound and light waves in this problem.

Key Concepts

Snell's Law

Snell's Law is the fundamental principle that governs the refraction of waves when they pass from one medium into another. It relates the sine of the angle of incidence to the sine of the angle of refraction proportionally to the speeds of the waves in the respective media or, equivalently, the indices of refraction. This law applies generally to all types of waves, including mechanical waves like sound and electromagnetic waves like light.

Wave Speed in a Medium

The speed of a wave is highly dependent on the properties of the medium through which it travels. For sound waves, factors such as temperature, density, and compressibility of the medium play significant roles, while for light waves in a medium, the index of refraction determines the speed. The differences in wave speeds across media are critical in understanding how the direction and wavelength change upon refraction.

Wavelength-Frequency Relationship

The wavelength of a wave in any medium is determined by the relationship wavelength = wave speed/frequency, with the frequency remaining constant when transitioning between media. This concept is essential because, despite changes in wave speed during refraction, the frequency does not change, resulting in a corresponding adjustment in wavelength.

Refractive Index

The refractive index is a dimensionless number that quantifies how much a wave, particularly light, slows down in a given medium compared to its speed in a vacuum. Although primarily used for electromagnetic waves, an analogous concept applies to mechanical waves like sound, where the change in speed is influenced by the medium's properties. It is a key factor in applying Snell's Law to calculate angles of refraction and changes in wavelength.

Comparison of Mechanical and Electromagnetic Wave Behavior

While both sound (mechanical) waves and light (electromagnetic) waves obey the same fundamental refraction principles, their natures cause them to interact with media in different ways. Sound, being a mechanical vibration that requires a medium, depends on the medium’s thermodynamic properties, whereas light, having both wave and particle characteristics, travels according to electromagnetic principles and is affected by the optical density of the medium. This leads to different behaviors in terms of speed adjustment and wavelength change despite a common underlying refraction law.

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