Overtones

When a musical instrument or a vibrating system (like a string or an air column) produces sound, it doesn’t just vibrate at a single frequency. Instead, it vibrates at a fundamental frequency and also at higher frequencies called overtones.

1. What is the Fundamental Frequency?

  • This is the lowest frequency at which an object vibrates.
  • It is also called the first harmonic.
  • Example: If a guitar string vibrates at 200 Hz, this is the fundamental frequency.

2. What are Overtones?

  • Overtones are the extra higher frequencies produced along with the fundamental frequency.
  • These are caused by different parts of the vibrating object moving in smaller sections.
  • Overtones add richness to the sound and are responsible for the different tones of musical instruments.

3. Overtones vs Harmonics

  • Harmonics are whole-number multiples of the fundamental frequency.
    • 1st harmonic = fundamental frequency
    • 2nd harmonic = 2 × fundamental frequency
    • 3rd harmonic = 3 × fundamental frequency (and so on…)
  • Overtones are any frequency higher than the fundamental frequency.
    • 1st overtone = 2nd harmonic
    • 2nd overtone = 3rd harmonic

📌 Key Difference: The first overtone is actually the second harmonic! This is because the fundamental frequency is not counted as an overtone.

4. Overtones in Different Systems

  1. Vibrating Strings (Guitar, Violin, Piano)
    • Both ends fixed.
    • Overtones follow whole-number harmonics.
  2. Open Air Columns (Flute, Pipe Organ)
    • Both ends open.
    • Overtones follow whole-number harmonics.
  3. Closed Air Columns (Clarinet, Some Organ Pipes)
    • One end closed, one end open.
    • Only odd-numbered harmonics are present (1st, 3rd, 5th…).

5. Real-Life Example

  • When you pluck a guitar string, it vibrates at its fundamental frequency.
  • However, if you lightly touch it at half its length, it will vibrate at twice the frequency (1st overtone / 2nd harmonic).
  • This is used in harmonics playing techniques in music.

Summary

✅ Fundamental Frequency = Lowest natural vibration
✅ Overtones = Higher vibrations above the fundamental
✅ Harmonics = Whole-number multiples of the fundamental
✅ Overtones and harmonics are related, but not the same

Here is a visual representation of overtones and harmonics in a vibrating system. The plot shows:

  • Fundamental frequency (1st Harmonic) – The simplest wave (solid line).
  • 1st Overtone (2nd Harmonic) – A wave with one additional node (dashed line).
  • 2nd Overtone (3rd Harmonic) – A wave with two additional nodes (dotted line).

Each overtone corresponds to a higher harmonic, and the frequency increases as more nodes appear.

Exercise A – Questions

Conceptual Questions

  1. Define a standing wave and explain how it is formed in a stretched string or an air column.
  2. Differentiate between nodes and antinodes in a standing wave. Where do they occur, and why?
  3. Explain the difference between harmonics and overtones in a vibrating system such as a string or an air column.
  4. A guitar string is vibrating in its third harmonic. How many nodes and antinodes are present?
  5. A closed pipe and an open pipe of the same length produce sound waves. How do their harmonic series differ? Explain your answer.

Calculation-Based Questions

  1. A stretched string of length 0.75 m is vibrating in its second harmonic. If the speed of the wave is 300 m/s, calculate the fundamental frequency.
  2. An open pipe produces a fundamental frequency of 250 Hz. Calculate the frequency of the second and third harmonics.
  3. A closed pipe (one end closed, one end open) has a length of 1.2 m. If the speed of sound in air is 340 m/s, calculate the frequency of the first and third harmonics.
  4. A standing wave is formed in a stretched string with five nodes. What is the harmonic number of this mode?
  5. A vibrating string of length 0.50 m produces a first overtone at 600 Hz. Calculate its fundamental frequency.

Exercise A – Questions with Solutions

A-Level Physics – Harmonics and Standing Waves
Conceptual Questions & Answers
1. Define a standing wave and explain how it is formed in a stretched string or an air column.

A standing wave is a wave pattern that results from the superposition of two waves traveling in opposite directions with the same frequency and amplitude. It is formed when a wave reflects back from a boundary and interferes with the incoming wave, creating nodes (points of no displacement) and antinodes (points of maximum displacement).

2. Differentiate between nodes and antinodes in a standing wave. Where do they occur, and why?

Nodes are points in a standing wave where there is no displacement due to complete destructive interference. Antinodes are points of maximum displacement due to constructive interference. Nodes occur at fixed points (such as ends of a string), while antinodes occur at points of maximum amplitude between nodes.

3. Explain the difference between harmonics and overtones in a vibrating system such as a string or an air column.

Harmonics are integer multiples of the fundamental frequency. The first harmonic is the fundamental frequency itself, the second harmonic is twice the fundamental frequency, and so on. Overtones, on the other hand, are any frequencies above the fundamental frequency, meaning the first overtone is the second harmonic, the second overtone is the third harmonic, etc.

4. A guitar string is vibrating in its third harmonic. How many nodes and antinodes are present?

The third harmonic has four nodes and three antinodes because the number of nodes is always equal to the harmonic number plus one, and the number of antinodes is equal to the harmonic number.

5. A closed pipe and an open pipe of the same length produce sound waves. How do their harmonic series differ? Explain your answer.

An open pipe supports all harmonics (1st, 2nd, 3rd, etc.), whereas a closed pipe only supports odd harmonics (1st, 3rd, 5th, etc.). This is because the closed end must always be a node and the open end must be an antinode, restricting the possible wave patterns.

Calculation-Based Questions & Solutions
6. A stretched string of length 0.75 m is vibrating in its second harmonic. If the speed of the wave is 300 m/s, calculate the fundamental frequency.

Formula: \( f_n = \frac{n v}{2L} \)

Given: \( n = 2, v = 300 \) m/s, \( L = 0.75 \) m

\( f_2 = \frac{2 \times 300}{2 \times 0.75} = 200 \) Hz

Fundamental frequency (first harmonic): \( f_1 = \frac{f_2}{2} = 100 \) Hz

7. An open pipe produces a fundamental frequency of 250 Hz. Calculate the frequency of the second and third harmonics.

Open pipes follow the harmonic series: \( f_n = n f_1 \)

Given: \( f_1 = 250 \) Hz

Second harmonic: \( f_2 = 2 \times 250 = 500 \) Hz

Third harmonic: \( f_3 = 3 \times 250 = 750 \) Hz

8. A closed pipe (one end closed, one end open) has a length of 1.2 m. If the speed of sound in air is 340 m/s, calculate the frequency of the first and third harmonics.

Closed pipes only support odd harmonics: \( f_n = \frac{n v}{4L} \)

Given: \( v = 340 \) m/s, \( L = 1.2 \) m

First harmonic: \( f_1 = \frac{1 \times 340}{4 \times 1.2} = 70.83 \) Hz

Third harmonic: \( f_3 = 3 \times 70.83 = 212.5 \) Hz

9. A standing wave is formed in a stretched string with five nodes. What is the harmonic number of this mode?

The number of nodes is always harmonic number + 1

Given: Nodes = 5

Harmonic number: \( n = 5 – 1 = 4 \) (Fourth harmonic)

10. A vibrating string of length 0.50 m produces a first overtone at 600 Hz. Calculate its fundamental frequency.

The first overtone is the second harmonic, so \( f_2 = 600 \) Hz

The fundamental frequency is \( f_1 = \frac{f_2}{2} = 300 \) Hz

This document provides a mix of conceptual and calculation-based questions to enhance understanding of harmonics, overtones, nodes, antinodes, and standing waves. Let me know if you need further explanations!

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Challenging A-Level Physics Questions

Exercise B


Questions with Final Answers
1. A string of length \(1.2 m\) is vibrating in its fifth harmonic. If the wave speed is \(340 m/s\), find its fundamental frequency.

Final Answer: \( f_1 = 56.67 Hz \)

2. A pipe closed at one end has a fundamental frequency of \(150 Hz\). Calculate the frequency of the fourth harmonic.

Final Answer: \( f_4 = 750 Hz \)

3. A stretched string supports a standing wave with three antinodes. If its length is \(0.9 m\), what is the wavelength?

Final Answer: \( \lambda = 0.6 m \)

4. An open organ pipe of length \(1.5 m\) produces a fundamental frequency. If the speed of sound is \(343 m/s\), find the frequency.

Final Answer: \( f = 114.33 Hz \)

5. A string fixed at both ends produces a second harmonic at \(400 Hz\). Find the fundamental frequency.

Final Answer: \( f_1 = 200 Hz \)

6. A closed pipe produces a first overtone at \(450 Hz\). Find the fundamental frequency.

Final Answer: \( f_1 = 150 Hz \)

7. A standing wave in a string has a total of 7 nodes. Find the harmonic number.

Final Answer: \( n = 6 \)

8. A violin string of length \(0.65 m\) is vibrating at its third harmonic. If the wave speed is \(300 m/s\), find the frequency.

Final Answer: \( f = 692.31 Hz \)

9. A closed tube resonates at its third harmonic at \(270 Hz\). Find the fundamental frequency.

Final Answer: \( f_1 = 90 Hz \)

10. A stretched wire vibrates at a frequency of \(500 Hz\) when in its fourth harmonic. Find the fundamental frequency.

Final Answer: \( f_1 = 125 Hz \)

Questions with Step-by-Step Solutions
1. A string of length \(1.2 m\) is vibrating in its fifth harmonic. If the wave speed is \(340 m/s\), find its fundamental frequency.

\( f_n = \frac{n v}{2L} \)

\( f_5 = \frac{5 \times 340}{2 \times 1.2} = 708.33 Hz \)

\( f_1 = \frac{f_5}{5} = 56.67 Hz \)

2. A pipe closed at one end has a fundamental frequency of \(150 Hz\). Calculate the frequency of the fourth harmonic.

Closed pipes only allow odd harmonics, so \( f_4 = f_7 = 750 Hz \).

3. A stretched string supports a standing wave with three antinodes. If its length is \(0.9 m\), what is the wavelength?

For three antinodes, \( L = \frac{3}{2} \lambda \).

\( \lambda = \frac{2}{3} L = 0.6 m \).

4. An open organ pipe of length \(1.5 m\) produces a fundamental frequency. If the speed of sound is \(343 m/s\), find the frequency.

\( f_1 = \frac{v}{2L} = \frac{343}{2 \times 1.5} = 114.33 Hz \).

5. A string fixed at both ends produces a second harmonic at \(400 Hz\). Find the fundamental frequency.

\( f_2 = 2 f_1 \), so \( f_1 = \frac{400}{2} = 200 Hz \).

6. A closed pipe produces a first overtone at \(450 Hz\). Find the fundamental frequency.

First overtone = third harmonic, so \( f_3 = 450 Hz \).

\( f_1 = \frac{450}{3} = 150 Hz \).

7. A standing wave in a string has a total of 7 nodes. Find the harmonic number.

\( n = N – 1 = 6 \).

8. A violin string of length \(0.65 m\) is vibrating at its third harmonic. If the wave speed is \(300 m/s\), find the frequency.

\( f_3 = \frac{3 \times 300}{2 \times 0.65} = 692.31 Hz \).

9. A closed tube resonates at its third harmonic at \(270 Hz\). Find the fundamental frequency.

\( f_3 = 3 f_1 \), so \( f_1 = \frac{270}{3} = 90 Hz \).

10. A stretched wire vibrates at a frequency of \(500 Hz\) when in its fourth harmonic. Find the fundamental frequency.

\( f_4 = 4 f_1 \), so \( f_1 = \frac{500}{4} = 125 Hz \).

Step-by-Step Solutions – Challenging A-Level Physics Questions


Exercise B – Step-by-Step Solutions
1. A string of length \(1.2 m\) is vibrating in its fifth harmonic. If the wave speed is \(340 m/s\), find its fundamental frequency.

Formula: \( f_n = \frac{n v}{2L} \)

Given: \( n = 5, v = 340 \) m/s, \( L = 1.2 \) m

\( f_5 = \frac{5 \times 340}{2 \times 1.2} = 708.33 Hz \)

\( f_1 = \frac{f_5}{5} = 56.67 Hz \)

2. A pipe closed at one end has a fundamental frequency of \(150 Hz\). Calculate the frequency of the fourth harmonic.

Closed pipes only allow odd harmonics: \( f_4 = f_7 = 750 Hz \).

3. A stretched string supports a standing wave with three antinodes. If its length is \(0.9 m\), what is the wavelength?

For three antinodes, \( L = \frac{3}{2} \lambda \).

\( \lambda = \frac{2}{3} L = 0.6 m \).

4. An open organ pipe of length \(1.5 m\) produces a fundamental frequency. If the speed of sound is \(343 m/s\), find the frequency.

\( f_1 = \frac{v}{2L} = \frac{343}{2 \times 1.5} = 114.33 Hz \).

5. A string fixed at both ends produces a second harmonic at \(400 Hz\). Find the fundamental frequency.

\( f_2 = 2 f_1 \), so \( f_1 = \frac{400}{2} = 200 Hz \).

6. A closed pipe produces a first overtone at \(450 Hz\). Find the fundamental frequency.

First overtone = third harmonic, so \( f_3 = 450 Hz \).

\( f_1 = \frac{450}{3} = 150 Hz \).

7. A standing wave in a string has a total of 7 nodes. Find the harmonic number.

\( n = N – 1 = 6 \).

8. A violin string of length \(0.65 m\) is vibrating at its third harmonic. If the wave speed is \(300 m/s\), find the frequency.

\( f_3 = \frac{3 \times 300}{2 \times 0.65} = 692.31 Hz \).

9. A closed tube resonates at its third harmonic at \(270 Hz\). Find the fundamental frequency.

\( f_3 = 3 f_1 \), so \( f_1 = \frac{270}{3} = 90 Hz \).

10. A stretched wire vibrates at a frequency of \(500 Hz\) when in its fourth harmonic. Find the fundamental frequency.

\( f_4 = 4 f_1 \), so \( f_1 = \frac{500}{4} = 125 Hz \).

Standing Waves

There are two main types of standing waves, based on the medium in which they form:

1. Standing Waves on a String (Fixed at Both Ends)
  • Found in stretched strings (e.g., guitar, violin, piano).
  • Both ends are nodes (zero displacement).
  • The waveforms follow harmonic patterns (1st harmonic, 2nd harmonic, etc.).
  • Example: Vibrating guitar string.
2. Standing Waves in Air Columns (Sound Waves)
  • Found in musical instruments like flutes, organ pipes, and clarinets.
  • Two subtypes:
    1. Open Pipe (Both Ends Open)
      • Both ends are antinodes (maximum displacement).
      • Supports all harmonics (1st, 2nd, 3rd, …).
      • Example: Flute, open organ pipe.
    2. Closed Pipe (One End Open, One End Closed)
      • Closed end is a node, open end is an antinode.
      • Supports only odd harmonics (1st, 3rd, 5th, …).
      • Example: Clarinet, closed organ pipe.
Relation Between String Length and Wave
Relation Between String Length and Standing Waves
Mathematical Relation:

For a string of length \( L \), the wavelength \( \lambda_n \) of the standing wave in the \( n \)-th harmonic is given by:

\[ \lambda_n = \frac{2L}{n} \]

where:

  • \( L \) = length of the string
  • \( n \) = harmonic number (\( n = 1,2,3,… \))
  • \( \lambda_n \) = wavelength of the standing wave

The frequency \( f_n \) of the wave is related to the wave speed \( v \) by:

\[ f_n = \frac{n v}{2L} \]

where:

  • \( f_n \) = frequency of the \( n \)-th harmonic
  • \( v \) = wave speed on the string
Examples of Harmonics in a String:
1. First Harmonic (Fundamental Mode, \( n = 1 \))

\[ \lambda_1 = 2L \]

\[ f_1 = \frac{v}{2L} \]

2. Second Harmonic (\( n = 2 \))

\[ \lambda_2 = L \]

\[ f_2 = \frac{2v}{2L} = 2f_1 \]

3. Third Harmonic (\( n = 3 \))

\[ \lambda_3 = \frac{2L}{3} \]

\[ f_3 = \frac{3v}{2L} = 3f_1 \]

Key Observations:
  • Higher harmonics have shorter wavelengths and higher frequencies.
  • The fundamental frequency \( f_1 \) is the lowest possible frequency at which the string vibrates.
  • Each harmonic is a whole number multiple of the fundamental frequency.