Sound

Learning goals

What students should be able to explain, calculate and investigate after completing Vibrations and Sound.

Goals marked Higher Level are required only for Higher Level students; the remaining goals are for all students.

• Explain why sound needs a medium to travel and describe how its speed varies between media.
• Explain how a stationary (standing) wave forms on a stretched string.
• Identify nodes, antinodes, harmonics and the fundamental frequency on a standing wave diagram.
• Higher Level: Use the sonometer to investigate the fundamental frequency of a stretched string.
• Describe ultrasound and its real-world applications.

Key definitions

Precise terms, symbols and SI units used in Vibrations and Sound.

Sound as a Mechanical Wave

Sound is produced by a vibrating source (e.g. a speaker cone, a guitar string, vocal cords) and travels as a longitudinal wave through a medium.

Sound needs a material medium because it is carried by oscillations and interactions between particles. Its speed depends mainly on the medium’s elastic stiffness and density. Solids are often much stiffer than liquids and gases, so sound usually travels faster in them, but “particles are closer together” is not a complete explanation.

As air is removed from a bell jar containing a ringing bell or buzzer, the sound becomes fainter although the source can still be seen vibrating. This shows that sound needs a material medium; it cannot travel through a vacuum.

Pitch, Loudness and Wave Characteristics

Pitch is mainly related to frequency; loudness is related to the amplitude and resulting intensity of a sound wave.

A higher frequency is heard as a higher pitch. A greater amplitude transfers more energy and is generally heard as a louder sound. Loudness is a human perception, so it also depends on frequency and the sensitivity of the ear; doubling amplitude does not mean a sound is perceived as twice as loud.

Observation task: use an oscilloscope or microphone display to compare waveforms while changing pitch and then loudness. Changing pitch changes the spacing of cycles; changing loudness mainly changes their height.

Standing Waves on a Stretched String

When a wave travelling along a stretched string (fixed at both ends) reflects off one end, the reflected wave travels back along the string in the opposite direction to the original wave.

The incident and reflected waves have the same frequency and travel in opposite directions. Their superposition forms fixed nodes and antinodes. For a string fixed at both ends, each end is a node and the allowed patterns fit an integer number of half-wavelengths into the string length.

Harmonics and the Fundamental Frequency

Because both ends of the string are fixed, they must always be nodes, which restricts the standing wave patterns that can form to a specific set of allowed frequencies (harmonics).

The simplest possible pattern (a single loop, one antinode in the middle) is the fundamental frequency (first harmonic); higher harmonics have progressively more nodes and antinodes, and frequencies that are whole-number multiples of the fundamental.

Ultrasound

Ultrasound consists of sound waves with frequencies too high for humans to hear (above about 20 kHz).

An ultrasound pulse is sent into the body or material. Part of the pulse reflects at a boundary and returns as an echo. The measured time is for the outward and return journeys, so the depth is d = vt/2. A higher frequency gives a shorter wavelength and can improve resolution, but it is usually attenuated more strongly.

• Ultrasound has frequency above 20 kHz.
• A pulse can be reflected at a boundary and its travel time used to determine depth.
• Medical imaging uses differences in reflection between tissues to build an image.

• Pulse sent → partial reflection at a boundary → echo returns.
• Use the echo time and sound speed to calculate depth.
• Divide by 2 because the pulse travels to the boundary and back.

Diagrams and formulae

Key diagrams and equations for reviewing Vibrations and Sound.

• Standing waves on a stretched string: the fundamental (1st harmonic) has one loop; higher harmonics have progressively more nodes and antinodes.
• Sonometer apparatus: a known tension is applied to a wire over movable bridges, allowing the vibrating length to be varied and the fundamental frequency investigated.

Standing waves on a stretched string: the fundamental (1st harmonic) has one loop; higher harmonics have progressively more nodes and antinodes.

Sonometer apparatus: a known tension is applied to a wire over movable bridges, allowing the vibrating length to be varied and the fundamental frequency investigated.

Worked Example

A step-by-step calculation applying the central relationships in Vibrations and Sound.

A sonometer wire of length 0.6 m vibrates at its fundamental frequency. Calculate the wavelength of the standing wave pattern.

• For the fundamental (1st harmonic), the string length equals half a wavelength: l = λ/2
• λ = 2l = 2 × 0.6 = 1.2 m

Practical / Investigation

Experimental planning, measurement and evaluation for Vibrations and Sound.

Experiment: Investigating the fundamental frequency of a stretched string using a sonometer

• Apparatus: sonometer, wire under known tension, movable bridges, tuning forks (or a signal generator with a small driver) of known frequency.
• Method: adjust the vibrating length of the wire (moving one bridge) until it resonates strongly with a tuning fork of known frequency held near it (or driven electrically); record the resonant length for several different tuning-fork frequencies.
• Independent variable: frequency of the tuning fork (or driving frequency). Dependent variable: resonant vibrating length. Controlled variables: tension in the wire, the wire used (mass per unit length).
• Data-analysis: since f = (1/2l)√(T/μ), plotting f against 1/l should give a straight line through the origin; the slope relates to √(T/μ).
• Sources of error: judging exact resonance by ear/observation is imprecise; the wire's own stiffness slightly affects the ideal formula's accuracy.
• Improvement: use a signal generator and a small detector (e.g. a paper rider that visibly jumps at resonance) instead of judging by ear, for a more objective resonance point.

• Method: sound the tuning fork above the tube and change the air-column length slowly until the first strong resonance is heard. Measure L from the water surface to the open end and repeat the resonance position.
• Calculation: for the first resonance, the effective length is L + 0.3d, so λ ≈ 4(L + 0.3d) and c = fλ. Use metres throughout.
• Precautions: keep the fork close to but not touching the tube, avoid striking it on a hard surface, read the water level at eye level and repeat from both directions.

• Keep vibrating length l and mass per unit length μ fixed. Use several known driving frequencies and adjust the tension until the paper rider falls at resonance.
• Record f and T, then plot f against √T. A straight line through the origin supports f = (1/2l)√(T/μ).
• Check that the tension is stable before each reading, measure the vibrating length between bridge contact points, and repeat each resonance setting.

• For the f–l relationship, use the same wire and constant tension, find resonant length l for several known frequencies, and plot f against 1/l.
• Locate resonance using maximum amplitude or a paper rider; repeat the bridge position from both directions and average.
• Measure vibrating length between the two bridge contact points and keep the hanging load stationary and secure.

• Analyse supplied values of fundamental frequency while one of length, tension or mass per unit length is varied.
• For a length investigation, plot f against 1/l. A straight line supports f ∝ 1/l when tension and mass per unit length are constant.
• State the controlled variables, use a best-fit line and discuss whether the data support the string model within uncertainty.

Quick-Check Questions

Short retrieval questions covering the essential ideas in Vibrations and Sound.

1. Why can sound not travel through a vacuum?

2. Distinguish a node from an antinode.

3. What must always be true at the fixed ends of a vibrating string?

4. Define ultrasound.

5. Give one real-world application of ultrasound.

Common mistakes and exam tips

Common misconceptions and exam-technique checks for Vibrations and Sound.

• Avoid: Saying sound can travel through a vacuum. Correct idea: Sound is a mechanical wave and requires a material medium.
• Avoid: Describing sound transmission only as particles colliding. Correct idea: Describe vibrations as being passed between neighbouring particles through the medium.
• Avoid: Confusing nodes and antinodes. Correct idea: A node remains at zero displacement; an antinode has maximum displacement amplitude.
• Avoid: Treating a standing wave as one travelling wave. Correct idea: It is formed by superposition of waves of the same frequency travelling in opposite directions.

• Label nodes and antinodes on standing-wave diagrams.
• For a string fixed at both ends, use nodes at both ends to relate L to λ.
• In echo or ultrasound ranging, include the outward and return journey.

Exam-style questions

Five exam-style questions with worked answers on Vibrations and Sound.

Higher Level — Required Content

Required Higher Level content for this topic.

• At constant tension and mass per unit length, f ∝ 1/l.
• At constant length and mass per unit length, f² ∝ T.
• Test the relationships using suitable straight-line graphs and discuss uncertainty in the measurements.

End-of-unit revision

Formulae, definitions, experiments and mixed revision questions for Vibrations and Sound.

• f = (1/2l)√(T/μ) (fundamental frequency of a stretched string)
• v = fλ (recap from Waves and Wave Motion)

• Node, antinode
• Fundamental frequency, harmonics
• Ultrasound

• Standing wave patterns for the fundamental and higher harmonics, with nodes/antinodes labelled
• Sonometer apparatus

• Investigating the fundamental frequency of a stretched string using a sonometer

• Thinking sound travels through a vacuum
• Confusing nodes and antinodes
• Forgetting fixed ends must be nodes
• Thinking a standing wave transfers no energy

1. A sonometer wire of length 0.5 m vibrates at its fundamental frequency of 200 Hz. Calculate the speed of the wave on the wire.

2. Explain why the speed of sound is generally faster in solids than in gases.

3. A string fixed at both ends is 1.2 m long. Calculate the wavelength of its 4th harmonic.

4. Explain, using reflection of waves, how an ultrasound scanner can build an image of internal body structures.

1. λ = 2l = 1.0 m (fundamental). v = fλ = 200 × 1.0 = 200 m s⁻¹.

2. Sound speed depends mainly on elastic stiffness and density. Solids are usually much stiffer than gases, so vibrations generally travel through them more quickly; particle spacing alone is not a sufficient explanation.

3. In the 4th harmonic, the string length equals 4 half-wavelengths: l = 4×(λ/2) → λ = 2l/4 = 2×1.2/4 = 0.6 m.

4. Ultrasound waves partially reflect whenever they cross a boundary between different types of tissue (which have different densities); by emitting short ultrasound pulses and precisely timing and analysing the reflected echoes, a scanner can build up a detailed image of the different structures and boundaries inside the body.

The table below shows which equations from this unit appear in the official State Examinations Commission Formulae and Tables book, and where.