Lesson 7.5: Stationary Waves

Introduction

Welcome to Lesson 7.5 of Foundation Physics, where we’ll dive into the fascinating world of stationary waves! 🌊 By the end of this lesson, you will be able to:

• Understand how stationary waves are formed by two counter-propagating waves.
• Identify nodes and antinodes and relate them to the fringe pattern and wavelength.
• Describe harmonics in both stretched strings and air columns (open and closed pipes).
• Measure the speed of sound or wave speed using stationary waves.
• Explain the process of stationary wave formation.

Get ready to experience how waves behave in different mediums and understand their importance in the field of physics!

Understanding Stationary Waves

Stationary waves, also known as standing waves, occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. 🧑‍🔬

This can be demonstrated using a simple example related to a guitar string:

1. When you pluck a guitar string, it vibrates back and forth, producing a wave.
2. If you could observe the vibrations closely, you would see that the wave appears to “stand still,” rather than traveling along the string.

Formation of Stationary Waves

The mechanism behind stationary wave formation involves constructive and destructive interference. Here's how it works:

• When two waves traveling in opposite directions are in phase (i.e., their crests and troughs coincide), they undergo constructive interference, enhancing the wave amplitude.
• When they are out of phase (the crest of one wave coincides with the trough of the other), they undergo destructive interference, canceling each other out.

In a mathematical representation, let's consider two waves described by:

$$ y_1 = A \sin(kx - \omega t) $$

$$ y_2 = A \sin(kx + \omega t) $$

Where:

• $A$ is the amplitude,
• $k$ is the wave number,
• $\omega$ is the angular frequency, and
• $t$ is time.

The superposition of these two waves gives:

$$ y = y_1 + y_2 = A \sin(kx - \omega t) + A \sin(kx + \omega t) $$

Using the trigonometric identity:

$$ \sin(x) + \sin(y) = 2 \sin\left( \frac{x + y}{2}

$ight) \cos\left( \frac{x - y}{2} $

ight) $$

We simplify this to find the resulting stationary wave:

$$ y = 2A \sin(kx) \cos(\omega t) $$

Nodes and Antinodes

In the equation derived above, there are specific points along the string or medium that remain at rest. These points are called nodes. The maximum amplitude points, where the amplitude is largest, are called antinodes.

• Nodes are represented by the points where $y = 0$, occurring at:

$$ x_n = \frac{n\lambda}{2} $$

for $n = 0, 1, 2, \ldots$ where $\lambda$ is the wavelength.

• Antinodes occur at points where:

$$ x_{a} = \frac{(2n+1)\lambda}{4} $$

for $n = 0, 1, 2, \ldots$

Here's an easy way to remember: Nodes are where there’s no movement, and antinodes are the peaks of movement! 🎸

Harmonics and Pipe Sounds

When we talk about harmonics, we refer to the various frequencies at which stationary waves can form in different systems, like strings and air columns.

• On a stretched string, the harmonics can be defined as:

1. Fundamental Frequency (1st Harmonic): The simplest oscillation, with one antinode at the center.
2. 2nd Harmonic: The string oscillates in two segments with two antinodes.
3. 3rd Harmonic: The string vibrates in three segments, producing three antinodes.

The frequencies for a string of length $L$ fixed at both ends can be calculated as:

$$ f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}} $$

Where:

• $f_n$ is the frequency of the $n^{th}$ harmonic,
• $T$ is the tension in the string, and
• $\mu$ is the linear mass density of the string.

• In air columns, there are differences depending on whether the ends of the pipe are open or closed. For an open pipe, both ends are antinodes, while in a closed pipe, one end is a node and one end is an antinode.
• The fundamental frequency for an open pipe is:

$$ f = \frac{v}{2L} $$

• For a closed pipe, it’s:

$$ f = \frac{v}{4L} $$

Where $v$ is the speed of sound in air, and $L$ is the length of the pipe.

Measuring Wave Speed

Stationary waves provide a way to measure wave speed, particularly in contexts like sound waves. By using the relationships derived from harmonics, students can measure the frequencies produced by stationary waves and relate that back to speed.

If you can measure the distance between nodes, you can find the wavelength:

$$ \lambda = \frac{2d}{n} $$

Where $d$ is the distance between $n$ nodes. The wave speed can then be calculated as:

$$ v = f \lambda $$

Where:

• $v$ is the wave speed,
• $f$ is the frequency, and
• $\lambda$ is the wavelength.

This is useful not just for waves in strings or tubes, but also for sound waves in the air! 🎶

Conclusion

In this lesson, we explored the key concepts related to stationary waves, including their formation, the characteristics of nodes and antinodes, and how harmonics work in different mediums. We also learned how to measure wave speed using stationary waves. Remember that stationary waves are key to understanding much of the music we hear and the way waves behave in many physical contexts!

Study Notes

• Stationary waves are formed from the interference of two waves moving in opposite directions.
• Nodes are points of zero amplitude; antinodes are points of maximum amplitude.
• Fundamental frequency and harmonics depend on the medium (strings vs pipes).
• Wave speed can be calculated using frequency and wavelength relationships.
• Applications of stationary waves include musical instruments and sound measurement.