Problem 48 - College Physics

Question

\(\bullet\) A sound wave has a speed of \(340 \mathrm{~m} / \mathrm{s}\) in air. If this wave produces a tone with a frequency of \(1000 \mathrm{~Hz}\), what is its wavelength?

Step-by-Step Solution

Answer

The wavelength is 0.34 meters.

Step 1: Identify Given Values

We are given the speed of sound in air as \( v = 340 \, \text{m/s} \) and the frequency of the sound wave as \( f = 1000 \, \text{Hz} \).

Step 2: Recall the Wave Speed Formula

The speed of a wave \( v \) is the product of its frequency \( f \) and wavelength \( \lambda \). The formula is \( v = f \times \lambda \).

Step 3: Solve for Wavelength

Rearrange the formula for wavelength \( \lambda \): \( \lambda = \frac{v}{f} \).

Step 4: Substitute Values into the Formula

Substitute the given values into the formula: \( \lambda = \frac{340 \, \text{m/s}}{1000 \, \text{Hz}} \).

Step 5: Calculate the Wavelength

Calculate \( \lambda \): \( \lambda = 0.34 \, \text{m} \).

Key Concepts

Wave Speed Formula

The wave speed formula is a fundamental concept when studying sound waves. It relates the speed of a wave to its frequency and the wavelength. Sound waves travel through the air at a certain speed, which is influenced by factors such as temperature and medium, but for many exercises, you will use a standard value. For sound in air, this speed is usually around 340 meters per second.

• Key formula: \( v = f \times \lambda \)

In this equation:- \( v \) represents the wave speed, the speed at which sound travels through a medium.- \( f \) is the frequency, or how many wave cycles occur in a second.- \( \lambda \) is the wavelength, the physical length of one cycle of the wave. When solving problems, rearrange the formula based on what information you have and what you need to find. Knowing how to apply this formula is essential for solving most sound-related physics problems.

Frequency and Wavelength

Understanding frequency and wavelength is essential for interpreting sound waves. Frequency refers to the number of complete wave cycles that pass a given point per second. It is measured in hertz (Hz), where one hertz equals one cycle per second. On the other hand, the wavelength is the distance between consecutive corresponding points of a wave, like from crest to crest or trough to trough. It's usually measured in meters (m).

• High frequency: Short wavelengths
• Low frequency: Long wavelengths

Inverse relationship: As the frequency of a wave increases, the wavelength decreases, provided the speed of the wave remains constant. This relationship is evident in the wave speed formula \( v = f \times \lambda \), which can be rearranged to \( \lambda = \frac{v}{f} \). This formula shows how frequency and wavelength are inversely related. For example, in the original exercise, a sound with a frequency of 1000 Hz has a wavelength of 0.34 meters when traveling at the speed of 340 m/s.

Acoustics

Acoustics is the science of sound, which encompasses the production, transmission, and effects of sound. It plays a crucial role in many real-world applications. Understanding acoustics can help in designing auditoriums, improving speaker systems, and enhancing sound clarity. In the context of our exercise, acoustics helps us understand how sound waves behave in air. Sound waves travel as mechanical waves, meaning they require a medium like air or water to move. These waves transfer energy from the source to the receiver through vibrations.

• Propagation: Refers to how sound waves move through a medium
• Reflection and Absorption: How surfaces interact with sound waves

The study of acoustics also involves understanding how sound waves interact with different environments and materials. This understanding helps in the precise control and manipulation of sound to suit particular needs. Whether it's improving concert hall acoustics or designing noise-canceling headphones, the principles of acoustics are applied to enhance sound quality and hearing experiences.