Physics | Standing waves and resonance

Welcome to Standing Waves and Resonance!

Hello future physicists! This chapter, C.4, is absolutely crucial because it explains the physics behind musical instruments, acoustics, and even radio tuning. You've already learned about travelling waves (like ripples moving across water); now we look at what happens when waves get trapped and appear to stand still.

Don't worry if this seems tricky at first—we'll break down the concepts of nodes, antinodes, and harmonics step-by-step. By the end, you’ll understand how objects use resonance to amplify sound and energy!

Section 1: The Magic of Standing Waves

1.1 How Standing Waves Form

A standing wave (or stationary wave) is fundamentally different from a travelling wave. It isn't actually going anywhere! It occurs when two identical waves—meaning they have the same speed, frequency, and amplitude—travel through the same medium in opposite directions and superpose.

In most practical examples (like a string instrument), this happens when a wave generated at one end hits a fixed boundary and is reflected back, interfering with the incoming wave.

Key Characteristics of a Standing Wave:

• Energy Transfer: Unlike a travelling wave, a standing wave does not transport energy continuously through the medium. Energy is localized and trapped between the nodes.
• Oscillation Pattern: The pattern itself (the envelope) remains fixed in space, but the particles in the medium oscillate perpendicular to the axis (for transverse waves) or parallel (for longitudinal waves).

1.2 Essential Terminology: Nodes and Antinodes

The most important feature of a standing wave is the existence of fixed points where the superposition always results in destructive interference, and points where it always results in constructive interference.

Nodes (N)

Nodes are points along the standing wave where the particles of the medium have zero displacement at all times.

• These are the points where the two waves interfering always cancel each other out (perfect destructive interference).
• Nodes appear stationary.

Antinodes (A)

Antinodes are points along the standing wave where the particles of the medium oscillate with maximum amplitude.

• These are the points where the two waves always reinforce each other (perfect constructive interference).
• Antinodes represent the points of maximum energy and maximum movement.

Memory Aid: Think of a Node as "No motion" (N = N). Think of an Antinode as "Amplitude maximum" (A = A).

Section 2: Resonance in Strings and Pipes (Harmonics)

2.1 Understanding Harmonics

When a medium (like a string or a column of air) is fixed in size, it can only support specific standing wave patterns. These specific patterns are called harmonics (or modes of vibration).

The simplest possible standing wave pattern is the fundamental frequency (or first harmonic, \(f_1\)). All other possible frequencies are integer multiples of this fundamental frequency.

2.2 Standing Waves in Strings (Fixed at Both Ends)

For a string instrument (like a guitar or violin), both ends are fixed. A fixed end cannot move, so the boundary condition for a string is always a Node (N) at both ends.

Mathematical Model for Strings:

If \(L\) is the length of the string, the condition for a standing wave is that the length \(L\) must contain an integer number \(n\) of half-wavelengths:

$$L = n \frac{\lambda_n}{2}$$

Where \(n = 1, 2, 3, \ldots\) (the harmonic number).

The frequency of the \(n\)th harmonic (\(f_n\)) is found using the wave speed formula \(v = f\lambda\). Since \(v\) (the speed of the wave in the string) is constant:

$$f_n = n \left( \frac{v}{2L} \right) = n f_1$$

Harmonic Examples for Strings:

• \(n=1\) (Fundamental / First Harmonic): The string vibrates in one segment. \(L = \lambda_1/2\). It has 2 Nodes (at the ends) and 1 Antinode (in the middle).
• \(n=2\) (Second Harmonic / First Overtone): The string vibrates in two segments. \(L = \lambda_2\). It has 3 Nodes and 2 Antinodes. \(f_2 = 2 f_1\).
• \(n=3\) (Third Harmonic / Second Overtone): The string vibrates in three segments. \(L = 3\lambda_3/2\). \(f_3 = 3 f_1\).

2.3 Standing Waves in Air Columns (Pipes)

Sound waves are longitudinal, but the concept of nodes and antinodes still applies:

• Closed End: Air particles cannot move, resulting in a Node (N) of displacement.
• Open End: Air particles are free to move and oscillate with maximum amplitude, resulting in an Antinode (A) of displacement.

Case A: Pipe Open at Both Ends (Open Pipe)

An open pipe must have an Antinode (A) at both ends. This means the mathematics are exactly the same as for a fixed string!

$$L = n \frac{\lambda_n}{2} \quad \text{and} \quad f_n = n f_1 \quad (n = 1, 2, 3, \ldots)$$

Example: Flutes or whistles often behave like open pipes.

Case B: Pipe Closed at One End (Closed Pipe)

A closed pipe has a Node (N) at the closed end and an Antinode (A) at the open end. Since the shortest distance from N to A is \(\lambda/4\), the patterns are restricted.

Critical difference: Only odd harmonics are possible in a closed pipe.

The length \(L\) must contain an odd number of quarter wavelengths:

$$L = m \frac{\lambda_m}{4}$$

Where \(m = 1, 3, 5, \ldots\) (only odd integers).

The frequency relationships are:

$$f_m = m f_1 \quad \text{where} \quad f_1 = \frac{v}{4L}$$

Harmonic Examples for Closed Pipes:

• \(m=1\) (Fundamental / First Harmonic): \(L = \lambda_1/4\). It has 1 N and 1 A.
• \(m=3\) (Third Harmonic): \(L = 3\lambda_3/4\). It has 2 Ns and 2 As. \(f_3 = 3 f_1\).
• \(m=5\) (Fifth Harmonic): \(L = 5\lambda_5/4\). \(f_5 = 5 f_1\).

Section 3: What is Resonance?

3.1 Definition of Resonance

Resonance is a phenomenon that occurs when an external driving force (vibrating source) acts on an oscillating system at a frequency equal to the system’s natural frequency (or one of its harmonic frequencies).

When resonance occurs, there is maximum energy transfer from the driving force to the system, causing the amplitude of oscillation in the system to dramatically increase.

If you push a swing at exactly the right time (its natural frequency), the swing goes very high. If you push randomly, nothing much happens. Resonance is pushing at the "right time."

Resonant Frequencies

The frequencies at which a system naturally vibrates are called its natural frequencies or resonant frequencies. These are the same frequencies we calculated in Section 2 (the fundamental and all its harmonics).

3.2 Practical Applications of Resonance

• Musical Instruments: When you pluck a guitar string, it vibrates at its fundamental frequency (\(f_1\)). But that vibration alone is quiet. The sound box (or body) of the guitar, which has its own air cavity, is designed to resonate at that same frequency, amplifying the sound tremendously.
• Tuning a Radio: When you tune a radio, you are adjusting the natural electrical frequency of the receiver circuit until it matches the frequency of the radio waves being broadcast (the driving frequency). When the frequencies match, resonance occurs, and the signal strength increases dramatically, allowing you to hear the station clearly.
• Acoustics: Opera singers can sometimes shatter a glass if they sing a note that perfectly matches the natural frequency of the glass, causing the glass's amplitude of vibration to increase until the material fails.

Did you know?

The famous collapse of the Tacoma Narrows Bridge in 1940 is often mistakenly cited as a pure resonance failure. While wind excited the bridge at its natural frequency initially, the catastrophic failure was actually due to a more complex aeroelastic flutter phenomenon. Nonetheless, it remains a dramatic (and terrifying) illustration of how large structures can absorb energy when driven near a natural frequency.

Key Takeaways and Summary

Understanding standing waves is vital not just for exam success, but for appreciating the physics that shapes the world around you, from the notes you hear to the structures you inhabit. Keep practising those boundary condition calculations—you've got this!