Problem 5 - University Physics with Modern Physics

Question

The speed of light waves in air is greater than the speed of sound in air by about a factor of a million. Given a sound wave and a light wave of the same wavelength, both traveling through air, which statement about their frequencies is true? a) The frequency of the sound wave will be about a million times greater than that of the light wave. b) The frequency of the sound wave will be about a thousand times greater than that of the light wave. c) The frequency of the light wave will be about a thousand times greater than that of the sound wave. d) The frequency of the light wave will be about a million times greater than that of the sound wave. e) There is insufficient information to determine the relationship between the two frequencies.

Step-by-Step Solution

Answer

a) The frequency of the light wave will be about the same as that of the sound wave. b) The frequency of the light wave will be about a hundred times greater than that of the sound wave. c) The frequency of the light wave will be about a thousand times greater than that of the sound wave. d) The frequency of the light wave will be about a million times greater than that of the sound wave. Answer: d) The frequency of the light wave will be about a million times greater than that of the sound wave.

Step 1: Recall the wave speed formula.

The wave speed formula relates the speed(v), frequency(f), and wavelength(λ) of a wave: v = fλ

Step 2: Write the given information

We are given that the speed of light waves in air is greater than the speed of sound in air by a factor of a million, meaning: Speed of light (v_l) = 1,000,000 × Speed of sound (v_s) And, we are also given that both light and sound waves have the same wavelength: λ_l = λ_s

Step 3: Use the wave speed formula for both waves

Use the wave speed formula to express the frequency of light waves (f_l) and sound waves (f_s) in terms of their speeds and wavelengths: For light waves: v_l = f_l λ_l For sound waves: v_s = f_s λ_s

Step 4: Relate the two equations and solve for the frequency ratio of light and sound waves

As λ_l = λ_s, we can divide the light wave equation by the sound wave equation: ( v_l = f_l λ_l ) / ( v_s = f_s λ_s ) Which simplifies to: (v_l / v_s) = (f_l / f_s) We know that (v_l / v_s) = 1,000,000, so: 1,000,000 = (f_l / f_s)

Step 5: Interpret the result and find the correct answer.

This result shows that the frequency of the light wave is a million times greater than the frequency of the sound wave. Therefore, the correct answer is: d) The frequency of the light wave will be about a million times greater than that of the sound wave.

Key Concepts

Frequency of Waves

The frequency of a wave is a fundamental characteristic that describes how often the wave oscillates per unit of time. It is measured in Hertz (Hz), which equates to the number of wave cycles occurring per second. When considering waves traveling through a medium, the formula \( v = f\lambda \) is key. It connects three parameters: speed (v), frequency (f), and wavelength (\( \lambda \)). The frequency remains constant when the wave changes media, but the speed and wavelength adjust correspondingly.
This change directly impacts the perceived sound or light, as frequency determines the pitch of sound and the color of light. For example, higher frequencies in sound produce higher pitches, while in light they yield colors towards the violet end of the spectrum. Understanding frequency helps predict how waves behave across different environments.

Speed of Light

The speed of light in a vacuum is a universal constant, approximately \( 299,792,458 \) meters per second. However, when light travels through different media, such as air or glass, its speed reduces slightly. In air, light travels nearly at its maximum speed due to minimal resistance and refraction. This speed is vast and countless times greater than that of most other waves like sound.
In practical applications, the speed of light serves as a benchmark for developing technologies like GPS, telecommunications, and high-speed Internet. It's also used in calculations involving astronomical distances, reflecting the wonders and scales of our universe. Light's speed is why we perceive events almost instantaneously over distances on Earth, providing our understanding of the "speed of information."

Speed of Sound

Sound travels through air by creating pressure waves that move molecules. The speed of sound is around 343 meters per second at room temperature; however, this speed can vary depending on the medium. Sound travels faster in denser media, like water or steel, due to better particle connectivity allowing quicker vibration transfer.
The speed of sound is crucial in many areas, such as acoustics, sonar, and engineering. It influences how music is composed for different environments or how engineers design auditorium acoustics to ensure high-quality sound delivery. In the atmosphere, sound's speed affects phenomena like sonic booms produced when objects break the sound barrier.

• Denser media: faster sound travel
• Temperature and humidity affect speed

Understanding the speed of sound aids in applications ranging from music production to aviation.

Wavelength

Wavelength is a spatial characteristic of waves, representing the distance between successive crests or identical points in a wave. It is denoted by the Greek letter lambda (\( \lambda \)) and usually measured in meters. Wavelength is inversely related to frequency, meaning as wavelength increases, frequency decreases, given a constant wave speed. This relationship is essential when applying the formula \( v = f\lambda \).
Wavelength determines numerous properties of waves. In light, it defines color; in sound, it relates to pitch. Longer wavelengths in sound are associated with lower pitches and bass tones, while shorter wavelengths correspond to higher pitches. In light, short wavelengths appear as violets or blues, whereas longer wavelengths appear red.
Understanding wavelength helps in sectors like astronomy, where it aids in identifying light from distant galaxies, or telecommunications, designing networks based on signal clarity and distance.