Harmonics

Why harmonics matter in anaesthesia

Ultrasound imaging uses harmonics to improve image quality (tissue harmonic imaging) and to reduce artefact.
- Harmonics are generated by non-linear propagation in tissue; the machine receives at a harmonic frequency (often 2nd harmonic).
Airway/respiratory acoustics: harmonic content affects sound quality and analysis (e.g., wheeze/stridor spectral content).
Electrical power systems: mains hum and harmonics can affect monitoring and filtering (50 Hz fundamental in the UK).
- Non-linear loads can generate higher-frequency harmonics that may leak into sensitive circuits if filtering/earthing is poor.

Core definitions

A harmonic is a sinusoidal component whose frequency is an integer multiple of a fundamental frequency.
Fundamental (1st harmonic): f1. 2nd harmonic: 2f1. 3rd harmonic: 3f1, etc.
Overtones: frequencies above the fundamental; the 1st overtone is usually the 2nd harmonic (but terminology can vary).
Harmonic series describes frequencies; relative amplitudes and phases determine waveform shape (Fourier series concept).

Standing waves, resonance and harmonics in strings/air columns

Standing waves form when incident and reflected waves of same frequency superimpose, creating nodes (zero displacement) and antinodes (max displacement).
Boundary conditions determine allowed harmonics (normal modes).
String fixed at both ends (or open-open air column): all integer harmonics are present.
- Allowed wavelengths: λn = 2L/n
- Frequencies: fn = n·v/(2L)
Air column closed at one end (closed-open): only odd harmonics are present (1st, 3rd, 5th…).
- Allowed wavelengths: λn = 4L/n where n is odd (1,3,5…).
- Frequencies: fn = n·v/(4L) where n is odd.
Resonance occurs when driving frequency matches a natural frequency; amplitude rises (limited by damping).

Harmonics and waveform shape (Fourier perspective)

Any periodic waveform can be expressed as a sum of sinusoids at the fundamental and its harmonics (Fourier series).
Square wave: contains odd harmonics with amplitudes falling as 1/n (idealised).
Sawtooth wave: contains all harmonics with amplitudes falling as 1/n (idealised).
Triangle wave: contains odd harmonics with amplitudes falling as 1/n^2 (idealised).
Phase relationships between harmonics affect waveform symmetry and peak shape; changing phase can change the waveform without changing the spectrum magnitude.

Non-linear propagation and harmonic generation (key for ultrasound)

In a linear medium, wave speed is independent of pressure amplitude; no new frequencies are generated.
In tissue, propagation is weakly non-linear: compressions travel slightly faster than rarefactions → waveform distortion → harmonic generation (especially 2nd harmonic).
As the wave propagates, energy transfers from the fundamental to higher harmonics; higher harmonics are more attenuated (frequency-dependent attenuation).
- Practical consequence: harmonic signal tends to originate from deeper tissue than the transducer face but not too deep (trade-off with attenuation).
Harmonic frequency has shorter wavelength: λ = c/f, so doubling f halves λ (improves axial resolution if bandwidth supports it).

Tissue Harmonic Imaging (THI): principles, benefits, limitations

Transmit at fundamental frequency; receive and process echoes at harmonic frequency (commonly 2nd harmonic).
Benefits: reduced near-field artefact and reverberation; narrower effective beam; improved lateral resolution and contrast; reduced clutter.
- Near-field improvement because harmonics are generated progressively with depth (little harmonic content right at the transducer face).
Limitations: reduced penetration (higher frequency attenuates more); lower signal in very deep targets; may reduce frame rate depending on implementation.
Pulse inversion harmonic imaging: transmit two pulses 180° out of phase; linear (fundamental) components cancel on summation, leaving non-linear (harmonic) components.

Harmonics, artefacts and safety in ultrasound (links to exam themes)

Reverberation artefact is mainly a fundamental-frequency phenomenon; THI reduces it because harmonic echoes are not generated strongly by superficial reverberations.
Side lobes/grating lobes contribute to clutter; THI can reduce their impact because harmonic generation is strongest in the main beam where intensity is highest.
Safety: harmonic imaging does not inherently make ultrasound 'more dangerous', but settings that increase output (MI/TI) can increase heating/cavitation risk; always follow ALARA.
- Mechanical Index (MI) relates to cavitation likelihood; Thermal Index (TI) relates to heating potential.

Key equations and quick facts

Wave speed: v = fλ. In soft tissue, c ≈ 1540 m·s⁻1 (assumed by ultrasound machines).
String/open-open resonator: fn = n·v/(2L). Closed-open resonator: fn = n·v/(4L), n odd.
If frequency doubles (2nd harmonic), wavelength halves (λ2 = λ1/2) in same medium.
Attenuation in tissue increases approximately linearly with frequency (rule of thumb often ~0.5 dB·cm⁻1·MHz⁻1, tissue dependent).

Define a harmonic and an overtone. How do they relate to the fundamental frequency?

Be precise with terminology and acknowledge variable use of 'overtone'.

A harmonic is a sinusoidal component with frequency equal to an integer multiple of the fundamental: fn = n·f1.
The fundamental is the 1st harmonic (n=1). The 2nd harmonic is 2f1, etc.
An overtone is any frequency above the fundamental; commonly the 1st overtone corresponds to the 2nd harmonic (but definitions vary).

A string is fixed at both ends. Derive the expression for the allowed wavelengths and harmonic frequencies.

Use boundary conditions: displacement must be zero at both ends.

Standing wave condition: L contains an integer number of half-wavelengths: L = n(λ/2).
Therefore λn = 2L/n.
Using v = fλ gives fn = v/λn = n·v/(2L).

An air column is closed at one end and open at the other. Which harmonics are present and why?

State node/antinode conditions at each end.

Closed end: displacement node. Open end: displacement antinode.
Fundamental corresponds to a quarter wavelength in the tube: L = λ/4.
Next allowed modes add half-wavelengths: L = 3λ/4, 5λ/4 … so only odd harmonics occur (n = 1,3,5…).
Frequencies: fn = n·v/(4L) for n odd.

Explain how harmonics arise during ultrasound propagation through tissue.

This is about non-linear propagation and waveform distortion.

Tissue is weakly non-linear: local sound speed increases slightly with pressure, so compressions travel faster than rarefactions.
This distorts the initially sinusoidal wave (steepening), which mathematically introduces higher-frequency components at integer multiples of the fundamental.
The 2nd harmonic is typically the strongest clinically used harmonic; higher harmonics exist but are more attenuated.

Describe tissue harmonic imaging (THI). What are the advantages and disadvantages for regional anaesthesia ultrasound?

Link mechanism to image quality changes relevant to nerve blocks.

Transmit at fundamental frequency; receive/process at harmonic (often 2nd harmonic) generated within tissue.
Advantages: reduced near-field clutter and reverberation; improved contrast; improved lateral resolution due to narrower effective beam; cleaner needle/nerve interface in some cases.
Disadvantages: reduced penetration (higher frequency attenuates more); may be less useful for deep targets (e.g., deep blocks in obese patients).

Explain pulse inversion harmonic imaging and why it suppresses the fundamental component.

This is a common physics viva topic in ultrasound.

Two pulses are transmitted along the same line, the second inverted by 180°.
In a linear system, received echoes would be exact inverses; summing them cancels the linear (fundamental) component.
Non-linear propagation generates harmonics that do not invert perfectly; summation preserves/enhances harmonic content.

A transducer transmits at 3 MHz in soft tissue (c = 1540 m/s). What are the wavelengths of the fundamental and 2nd harmonic?

Use λ = c/f.

Fundamental: λ1 = 1540 / (3×10^6) ≈ 0.513 mm.
2nd harmonic frequency = 6 MHz → λ2 = 1540 / (6×10^6) ≈ 0.257 mm (half of λ1).

Why does harmonic imaging often reduce reverberation artefact and improve near-field image quality?

Tie artefact origin to where harmonics are generated.

Reverberation arises from repeated reflections between strong reflectors, typically dominated by the transmitted fundamental frequency.
Harmonics are generated within tissue as the wave propagates; there is relatively little harmonic signal in the immediate near field.
By receiving at the harmonic, the system rejects much of the superficial clutter/reverberation that lacks harmonic content.

Give an FRCA-style explanation of why higher harmonics are more attenuated in tissue.

State frequency dependence and its practical consequence.

Soft tissue attenuation increases approximately with frequency (often approximated as linear with MHz).
Therefore, harmonic frequencies (e.g., 2f) experience greater attenuation per unit distance than the fundamental.
This limits penetration of harmonic imaging and biases harmonic signal toward shallower depths compared with what would be expected if attenuation were frequency-independent.

A tube is open at both ends and has length 0.85 m. Taking speed of sound in air as 340 m/s, calculate the fundamental frequency and the next two harmonics.

Open-open: fn = n·v/(2L).

f1 = v/(2L) = 340 / (2×0.85) = 340 / 1.70 = 200 Hz.
2nd harmonic: f2 = 2f1 = 400 Hz.
3rd harmonic: f3 = 3f1 = 600 Hz.

How do harmonics relate to Fourier analysis and why is this relevant to monitoring/filters in anaesthesia?

Connect waveform distortion to spectral content and filtering.

Fourier analysis represents a periodic waveform as a sum of the fundamental and harmonics with specific amplitudes and phases.
Non-sinusoidal interference (e.g., from non-linear electrical loads) contains harmonics of mains frequency; filters designed only for 50 Hz may not remove higher harmonics.
Understanding harmonic content helps interpret artefact and design/choose appropriate filtering without distorting physiological signals.