/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 The frequency \((f)\) of a wave ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 3

The frequency \((f)\) of a wave is defined as the number of wavelengths per second which travel past a given point. a) For a wave traveling at a given speed, \(c\), how does the frequency depend on the wavelength, if at all? b) Provide a mathematical expression showing the relationship between \(f, \lambda\) and \(c\) for a wave. (Hint: consider how you determined answers to CTQ 2ab).

Short Answer

Expert verified
a) The frequency depends on the wavelength inversely, given the speed of the wave is constant. b) The mathematical relationship is given by \(f = \frac{c}{\lambda}\), where \(c\) is the wave speed, \(f\) is the frequency, and \(\lambda\) is the wavelength.

Step by step solution

01

Define the Terms

First, understand what the terms mean. The frequency (\(f\)) of a wave is the number of waves passing a point per second. A wavelength (\(\lambda\)) is the distance over which the wave's shape repeats. The speed (\(c\)) is the distance traveled by the wave per unit of time. In this case, we are told that the speed of the wave is constant.
02

Relate Frequency and Wavelength

Cause of wave speed being constant, if the wavelength increases, then fewer waves can pass a point per second. This implies that the frequency decreases. Similarly, if the wavelength decreases, more waves can pass a point per second, indicating an increase in frequency. Hence, frequency is inversely proportional to wavelength.
03

Formulate the Relationship Mathematically

The relationship derived in the previous step can be expressed mathematically as \(f = \frac{c}{\lambda}\). This equation shows how frequency (f), wavelength (\(\lambda\)) and speed (c) are related. The wave speed equals the product of its frequency and wavelength. This relationship holds true for all types of waves.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Speed
Wave speed is a fundamental concept when discussing wave motion. It describes how quickly a wave propagates through a medium. In other words, it's the rate at which wave crests (or troughs) move forward. The speed of a wave is generally determined by the properties of the medium it's traveling through. For example, sound waves travel faster in water than in air.

One important thing to remember is that wave speed, often denoted by the letter \(c\) or \(v\), reflects a constant quantity when conditions remain the same. This means that if you know the wave speed and the frequency or wavelength, you can figure out the other values using the relationship \(c = f \lambda\).
  • **Constant nature**: Wave speed remains constant in a given medium.
  • **Calculated by**: The product of frequency \(f\) and wavelength \(\lambda\).
  • **Unit**: Measured in meters per second (m/s).
Wavelength
Wavelength is a crucial characteristic of waves that measures the distance between two successive identical points on the wave, such as crest to crest or trough to trough. This distance determines the wave's nature and its interactions with the environment. Longer wavelengths are characteristic of lower-frequency waves, while shorter wavelengths are associated with higher frequencies.

Understanding wavelength can be vital in applications like radio broadcasting, optics, and even understanding the colors of light we see. Shorter wavelengths can penetrate objects more easily than longer ones, influencing how waves are used in communication and technology.
  • **Definition**: The length of one complete wave cycle.
  • **Relation to frequency**: Inversely proportional in wave motion.
  • **Impact**: Influences wave's ability to travel through mediums.
Inverse Proportionality
Inverse proportionality is a key relationship in any wave context, especially between wave frequency and wavelength. When something is inversely proportional, as one value increases, the other decreases proportionally. This is precisely how frequency and wavelength interact with one another for waves.

For constant wave speeds, if the wavelength (\(\lambda\)) of a wave increases, the frequency (\(f\)) must decrease to maintain the constant speed (\(c\)). Mathematically, we express this relationship as \(f = \frac{c}{\lambda}\). This inverse proportionality is a guiding principle in understanding wave behaviors.
  • **Definition**: One quantity increases while the other decreases.
  • **Wave Formula**: Expressed as \(f = \frac{c}{\lambda}\).
  • **Applications**: Useful in calculations in physics and various technologies.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Chemistry Textbooks

Making Measurements

Read Explanation

The Earths Atmosphere

Read Explanation

Chemical Reactions

Read Explanation

Chemistry Branches

Read Explanation

Nuclear Chemistry

Read Explanation

Physical Chemistry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.