Standing Wave Formation

Introduction to Standing Waves

• A standing wave (also known as a stationary wave) is a wave that appears to stay in one place – it doesn’t seem to travel through the medium.
• Standing waves are formed when two waves of the same frequency and amplitude, traveling in opposite directions, interfere with each other.
• These waves are typically a travelling wave and its reflection.

KEY TAKEAWAY: Standing waves result from the superposition of two identical waves moving in opposite directions.

Formation of Standing Waves

1. Travelling Wave: A wave propagates through a medium, carrying energy.
2. Reflection: When the travelling wave reaches a boundary (e.g., a fixed end of a string), it is reflected. The reflected wave is inverted if the boundary is fixed.
3. Superposition: The incident (original) travelling wave and the reflected wave superimpose (interfere) with each other.
   • The principle of superposition states that the resultant displacement at any point is the vector sum of the displacements of the individual waves.
4. Interference: The superposition results in interference patterns:
   • Constructive Interference: Occurs where the waves are in phase (crests align with crests, troughs with troughs). This results in points of maximum displacement called antinodes.
   • Destructive Interference: Occurs where the waves are out of phase (crest aligns with trough). This results in points of zero displacement called nodes.
5. Standing Wave Pattern: The interference creates a pattern of alternating nodes and antinodes. The wave appears to be standing still because the nodes and antinodes remain in fixed positions.

Visual Representation

Imagine a string fixed at both ends. When you pluck the string, a wave travels down the string and is reflected at the fixed end. The reflected wave interferes with the original wave, creating a standing wave pattern.

• Nodes: Points where the string remains stationary.
• Antinodes: Points where the string oscillates with maximum amplitude.

STUDY HINT: Draw diagrams of standing waves with different numbers of nodes and antinodes to visualize the relationship between wavelength and the length of the medium.

Key Properties of Standing Waves

• Nodes: Points of zero displacement.
• Antinodes: Points of maximum displacement.
• Wavelength (λ): The distance between two consecutive nodes or two consecutive antinodes is half a wavelength (λ/2). Therefore, the distance between a node and an adjacent antinode is a quarter of a wavelength (λ/4).
• Frequency (f): The frequency of the standing wave is determined by the frequency of the original travelling wave.
• Amplitude: The amplitude varies along the standing wave, being maximum at the antinodes and zero at the nodes.
• Energy: Energy is not transmitted along the string in a standing wave; it is trapped between the nodes.

Relationship between Wavelength, Length, and Frequency

For a string fixed at both ends, the possible wavelengths and frequencies of standing waves are quantized (they can only take on certain discrete values).

• Wavelength: The wavelength of the standing wave is related to the length (L) of the string by the equation:

$$λ = \frac{2L}{n}$$

where $n$ is an integer (1, 2, 3, …) representing the harmonic number.

• Frequency: The frequency of the standing wave is related to the length (L) of the string and the wave speed (v) by the equation:

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

where $n$ is an integer (1, 2, 3, …) representing the harmonic number.

Harmonics

• Fundamental Frequency (First Harmonic): The lowest possible frequency for a standing wave on a string. In this case, $n = 1$.

$$f_1 = \frac{v}{2L}$$

• Harmonics (Overtones): Frequencies that are integer multiples of the fundamental frequency. The nth harmonic has a frequency of $f_n = nf_1$.

VCAA FOCUS: VCAA often asks questions about the relationship between the length of a string, the wavelength of a standing wave, and the frequency of the wave. Understand how changing one parameter affects the others.

Examples of Standing Waves

• Musical Instruments: Stringed instruments (guitar, violin, piano) and wind instruments (flute, trumpet) rely on standing waves to produce sound.
• Tacoma Narrows Bridge: The collapse of the Tacoma Narrows Bridge in 1940 was caused by standing waves induced by wind.
• Microwave Ovens: Standing waves of microwaves can create hot and cold spots in a microwave oven.

REMEMBER: Nodes are No Displacement. Antinodes are Awesome Amplitude.

Analyzing Standing Waves (Nodes at Both Ends)

The most common scenario in VCE Physics involves standing waves formed on a string fixed at both ends. For this specific case:

• Boundary Conditions: The ends of the string must be nodes because they are fixed.
• Harmonic Series: The frequencies of the standing waves form a harmonic series, where each harmonic is an integer multiple of the fundamental frequency.

EXAM TIP: When solving problems involving standing waves, always start by drawing a diagram of the standing wave pattern. This will help you visualize the relationship between the wavelength and the length of the string.

Summary

Standing waves are formed by the superposition of a travelling wave and its reflection. They are characterized by fixed nodes and antinodes and play a crucial role in various physical phenomena, from musical instruments to structural vibrations. Understanding the relationship between wavelength, frequency, and the length of the medium is essential for solving problems related to standing waves.

COMMON MISTAKE: Forgetting that the distance between consecutive nodes (or antinodes) is half a wavelength, not a full wavelength.