Describe how changes in amplitude and frequency affect the loudness and pitch of sound waves

3.4 Sound

Objective

Explain how changes in amplitude and frequency affect the loudness and pitch of sound waves, and understand the related concepts required by the Cambridge IGCSE/A‑Level Physics syllabus.

1. Production of Sound

• A sound is produced when a source vibrates, causing the particles of a surrounding medium to oscillate about their equilibrium positions.
• The oscillating particles generate alternating regions of:

  • Compression – a region of higher pressure (particles pushed together).
  • Rarefaction – a region of lower pressure (particles spread apart).

• These compressions and rarefactions travel through the medium as a longitudinal wave.

2. Nature of Sound Waves

• Longitudinal wave: particle displacement is parallel to the direction of wave propagation.
• In solids a transverse (shear) component can also exist, but ordinary sound in gases and liquids is purely longitudinal.

Key Terms

3. Speed of Sound

The speed of sound depends on the medium’s elasticity and density. In general:

• Sound travels fastest in solids, slower in liquids, and slowest in gases.
• Typical values (20 °C):

  • Air: v ≈ 340 m s⁻¹ (range 330–350 m s⁻¹)
  • Water: v ≈ 1500 m s⁻¹
  • Steel: v ≈ 5000 m s⁻¹

The fundamental relationship is

\(v = f\lambda\)

Practical Determination of the Speed of Sound (AO3)

1. Place a loudspeaker and a microphone a known distance \(d\) apart (e.g., 5 m).
2. Emit a short pulse (or a burst of a known frequency) from the speaker.
3. Measure the time‑of‑flight \(t\) between emission and detection using an oscilloscope or digital timer.
4. Calculate the speed: \(\displaystyle v = \frac{d}{t}\).
5. Repeat at least three times, record the spread of results and estimate the uncertainty (AO2).

4. Amplitude, Intensity and Loudness

• The intensity \(I\) (power per unit area) of a sound wave is proportional to the square of its amplitude:

  \(I \propto A^{2}\)

  A more complete expression for a plane progressive wave in a fluid is

  \(I = \frac{1}{2}\,\rho v \omega^{2} A^{2}\)

  where \(\rho\) is the density of the medium and \(\omega = 2\pi f\) the angular frequency.

• Loudness is a logarithmic response to intensity. The sound‑level \(L\) in decibels (dB) is defined as

  \(L = 10\log{10}\!\left(\dfrac{I}{I{0}}\right)\)

  with \(I_{0}=1\times10^{-12}\,\text{W m}^{-2}\) (threshold of hearing).

• Consequences:

  • Doubling the amplitude → intensity increases by a factor of four → sound level rises by \(10\log_{10}4 \approx 6\) dB (perceptibly louder).
  • Halving the amplitude → intensity falls to one‑quarter → level drops by ≈ 6 dB (softer).

5. Frequency and Pitch

• Pitch is the perceptual correlate of frequency. For pure tones the relationship is approximately linear:

  \(\text{Pitch} \propto f\)

• Human audible range: 20 Hz – 20 kHz. Within this range the ear can discriminate frequency changes as small as 1 % for mid‑range tones.
• Examples:

  • Middle C on a piano ≈ 261 Hz (low pitch).
  • Whistle ≈ 3 kHz (high pitch).

6. Echo, Reverberation and Ultrasound

• Echo: a reflected sound wave that returns to the listener after travelling to a surface and back. The distance to the reflecting surface is obtained from the time delay \(\Delta t\):

  \(d = \dfrac{v\Delta t}{2}\)

• Reverberation: a rapid series of overlapping echoes that give a sense of spaciousness in a room.
• Ultrasound: sound with frequency > 20 kHz.

  • Medical imaging (2–15 MHz) – high‑frequency waves penetrate tissue and are reflected by internal structures.
  • Non‑destructive testing – detection of cracks or voids in metal.
  • Industrial cleaning and sonar navigation.

7. Combined Effects of Amplitude and Frequency

• Amplitude controls loudness; frequency controls pitch. They are independent, but many real sounds vary both simultaneously.
• Examples:

  • Piano – Striking a key harder increases the amplitude (louder) and, because the hammer strikes the string slightly closer to its centre, may raise the frequency a few cents (sharper pitch).
  • Loudspeaker volume control – Adjusts the amplitude of the electrical signal; pitch remains unchanged.
  • Human voice – Speaking loudly requires larger air‑pressure variations (greater amplitude). Changing the tension or length of the vocal cords changes frequency (pitch).

8. Practical Examples (Application of Concepts)

1. String instrument (e.g., guitar) – Tightening a string increases tension, raising the frequency (higher pitch). Plucking harder increases amplitude (louder note).
2. Loudspeaker – The volume knob varies the amplitude of the audio signal; the pitch of the music is unchanged.
3. Human voice – Loud speech → larger amplitude of air‑pressure variations. Pitch changes are produced by altering the length/tension of the vocal cords.
4. Ultrasound scanner – Generates high‑frequency pulses (≈2–15 MHz). Reflected echoes are processed to form an image of internal body structures.

9. Common Misconceptions

• Loudness ≠ Pitch – Loudness depends on amplitude/intensity; pitch depends on frequency.
• High pitch does not guarantee loudness – A high‑frequency tone can be very soft if its amplitude is low.
• Sound travels faster in denser media – It travels fastest in media that are both dense and highly elastic (e.g., steel ≫ water ≫ air).
• Amplitude and frequency are unrelated – While mathematically independent, many real sources (e.g., musical instruments) change both when the player varies effort.

10. Summary Table

11. Suggested Diagrams (for classroom use)

• Longitudinal wave diagram showing compressions, rarefactions, amplitude (vertical displacement of particles), wavelength, and direction of propagation.
• Time‑of‑flight set‑up for measuring the speed of sound, with labelled speaker, microphone, distance \(d\) and oscilloscope trace.
• Echo illustration: source → reflecting surface → listener, with the round‑trip distance indicated.
• Ultrasound transducer emitting high‑frequency pulses into a tissue phantom and receiving echoes.
• Graph of sound‑level (dB) versus amplitude, highlighting the 6 dB change for a doubling of amplitude.

Key Take‑away

In sound waves, amplitude determines loudness through intensity and the decibel scale, while frequency determines pitch. The speed of sound varies with the medium (solids > liquids > gases) and is linked to frequency and wavelength by \(v = f\lambda\). Understanding these relationships enables students to analyse musical tones, solve acoustic‑engineering problems, and appreciate applications such as medical ultrasound and echo ranging.