define the specific acoustic impedance of a medium as Z = ρc, where c is the speed of sound in the medium

Topic 24 – Waves in Medical Physics

Learning Objectives

• Define the specific acoustic impedance of a medium and calculate it from density and sound speed.
• Explain how ultrasound is generated and detected using piezo‑electric, magnetostrictive and CMUT transducers.
• Describe diagnostic, therapeutic and industrial uses of ultrasound, including Doppler flow measurement.
• Apply the intensity‑reflection coefficient, transmission coefficient and attenuation law to predict how ultrasound behaves at material boundaries.
• Design a simple pulse‑echo experiment to measure the speed of sound in a liquid (AO3 – practical skills).
• Understand the production and use of X‑rays, the minimum‑wavelength equation and X‑ray attenuation.

24.1 Production and Use of Ultrasound

1. What Is Ultrasound?

• Sound waves with frequencies > 20 kHz (above the upper limit of human hearing).
• Medical and most industrial applications use the range 1 MHz – 20 MHz.
• At these frequencies the wavelength in soft tissue is a few mm to a few cm, comparable to the size of anatomical structures.
• Why this range?

  • Higher frequency → shorter wavelength → better spatial resolution.
  • Higher frequency → stronger attenuation → reduced penetration depth.
  • 1–20 MHz is a compromise that gives sufficient resolution while still reaching the required depth.

2. Production of Ultrasound

2.1 Piezo‑electric transducers

• Alternating voltage → electric field → crystal expands/ contracts → longitudinal sound wave (generation).
• Incoming wave compresses crystal → strain → voltage generated (reverse effect, reception).
• Typical resonant frequencies: 2 – 15 MHz for medical imaging probes.

2.2 Magnetostrictive transducers

• Ferromagnetic rod changes length when subjected to a varying magnetic field.
• Alternating magnetic field → vibration → ultrasound emission; reverse process for reception.

2.3 Capacitive Micromachined Ultrasonic Transducers (CMUTs)

• Thin silicon membrane suspended over a cavity; electrostatic force pulls membrane toward substrate when voltage is applied.
• Membrane vibration radiates ultrasound; incoming waves move the membrane, changing capacitance → electrical signal.
• Typical resonant frequency: 5 – 10 MHz for linear imaging arrays.

3. Uses of Ultrasound

1. Diagnostic imaging (sonography) – visualising soft tissue, fetal monitoring, Doppler flow measurement.
2. Therapeutic ultrasound – physiotherapy, lithotripsy, targeted drug delivery.
3. Industrial non‑destructive testing (NDT) – crack detection, thickness measurement, material characterisation.
4. Cleaning and sonochemistry – cavitation‑driven cleaning of delicate parts.
5. Doppler ultrasound – measures blood‑flow velocity using the frequency shift

   \[

   fD = \frac{2\,v\,f0\cos\theta}{c}

   \]

   where \(v\) is the flow speed, \(f_0\) the transmitted frequency, \(\theta\) the angle between beam and flow, and \(c\) the speed of sound in tissue.

4. Specific Acoustic Impedance

The specific acoustic impedance \(Z\) quantifies the resistance a medium offers to the passage of a sound wave.

4.1 Formula and Units

\[

Z = \rho\,c

\]

• \(\rho\) – density of the medium (kg m⁻³).
• \(c\) – speed of sound in the medium (m s⁻¹).
• Unit: rayl (1 rayl = 1 kg m⁻² s⁻¹). Values are usually expressed in kilorayl (kRayl = 10³ rayl).

4.2 Typical Acoustic Impedances

5. Reflection, Transmission and Matching

5.1 Intensity‑reflection and transmission coefficients

\[

R = \left(\frac{Z2 - Z1}{Z2 + Z1}\right)^{2}, \qquad

T = 1 - R

\]

• Large impedance mismatch (e.g., air ↔ tissue) → \(R\) ≈ 1 (almost total reflection).
• Coupling gel with impedance close to skin reduces mismatch, maximising transmitted energy.

5.2 Matching‑layer design (optional for higher‑level study)

For a single matching layer the optimum impedance is the geometric mean:

\[

Zm = \sqrt{Z{\text{transducer}}\,Z_{\text{tissue}}}

\]

Example: quartz transducer \(Zt≈1.5×10^{7}\) Rayl, soft tissue \(Zs≈1.6×10^{6}\) Rayl → \(Z_m≈4.9×10^{6}\) Rayl. A thin polymer layer with this impedance improves transmission.

6. Attenuation of Ultrasound

Intensity decreases exponentially with distance due to absorption and scattering:

\[

I = I_0\,e^{-\mu x}

\]

• \(I_0\) – initial intensity.
• \(\mu\) – attenuation coefficient (Np m⁻¹ or dB cm⁻¹).
• \(x\) – propagation distance.

Higher frequency → larger \(\mu\) → reduced penetration depth.

7. Example Calculations

7.1 Impedance of Soft Tissue

\[

Z = (1050\;\text{kg m}^{-3})(1540\;\text{m s}^{-1}) = 1.62\times10^{6}\;\text{Rayl}

\]

7.2 Reflection at a Tissue–Bone Interface

\[

R = \left(\frac{7.75\times10^{6} - 1.62\times10^{6}}{7.75\times10^{6} + 1.62\times10^{6}}\right)^{2}

\approx 0.46

\]

≈ 46 % of the incident intensity is reflected; 54 % is transmitted.

7.3 Doppler Shift for Blood Flow

Assume a 5 MHz probe, blood flowing at 30 cm s⁻¹, beam‑to‑flow angle \(\theta = 60^{\circ}\), and \(c = 1540\) m s⁻¹:

\[

f_D = \frac{2(0.30\;\text{m s}^{-1})(5\times10^{6}\;\text{Hz})\cos60^{\circ}}{1540\;\text{m s}^{-1}}

\approx 975\;\text{Hz}

\]

7.4 Attenuation Over 5 cm in Muscle

Typical \(\mu_{\text{muscle}} \approx 0.5\;\text{dB cm}^{-1}\) at 5 MHz. Converting to Nepers (\(1\;\text{dB}=0.115\;\text{Np}\)):

\[

\mu = 0.5\times0.115 = 0.0575\;\text{Np cm}^{-1}

\]

\[

I = I0 e^{-\mu x}= I0 e^{-0.0575\times5}= I0 e^{-0.2875}\approx 0.75\,I0

\]

≈ 25 % loss of intensity after 5 cm.

8. Practical Activity – Pulse‑Echo Measurement of the Speed of Sound in Water

1. Aim: Determine the speed of sound in water using a pulse‑echo ultrasonic transducer.
2. Apparatus: Piezo‑electric transducer (5 MHz), function generator, oscilloscope, water tank, ruler or Vernier caliper, coupling gel.
3. Procedure:

   1. Mount the transducer vertically at the bottom of the tank, apply a thin layer of gel.
   2. Generate a short voltage pulse; the transducer emits an ultrasonic burst that travels to the water surface, reflects, and returns.
   3. Measure the time interval \(\Delta t\) between the transmitted pulse and the received echo on the oscilloscope (use the cursors for accuracy).
   4. Measure the distance \(d\) from the transducer face to the water surface (to the nearest 0.1 mm).
   5. Calculate the speed of sound using \(c = 2d/\Delta t\).

4. Data table (example)

   Trial	Distance \(d\) (mm)	Time \(\Delta t\) (µs)	Calculated \(c\) (m s⁻¹)
   1	45.0	58.6	—
   2	50.0	65.0	—
   3	55.0	71.5	—

5. Sample calculation (Trial 1):

   \[

   c = \frac{2\times0.045\;\text{m}}{58.6\times10^{-6}\;\text{s}} \approx 1535\;\text{m s}^{-1}

   \]

6. Evaluation (AO3):

   • Sources of error – timing resolution, temperature dependence of \(c\), mis‑reading of distance, imperfect coupling.
   • Improvements – use a temperature probe to correct for temperature, average multiple readings, employ a digital time‑interval counter.

9. Summary

• The specific acoustic impedance \(Z = \rho c\) governs how ultrasound is reflected, transmitted and absorbed at material boundaries.
• Piezo‑electric, magnetostrictive and CMUT transducers convert electrical energy to ultrasound and back again.
• Choosing the appropriate frequency balances resolution against attenuation; coupling media minimise impedance mismatch.
• Reflection (\(R\)), transmission (\(T\)) and attenuation (\(I = I_0 e^{-\mu x}\)) equations allow quantitative predictions for imaging and therapeutic applications.
• Doppler ultrasound provides a non‑invasive method of measuring flow speed.
• A simple pulse‑echo experiment demonstrates the practical measurement of sound speed and reinforces experimental skills (AO3).

24.2 Production and Use of X‑rays

1. Production of X‑rays

• Bremsstrahlung (braking radiation): Decelerating high‑energy electrons in the target nucleus field produces a continuous spectrum.
• Characteristic X‑rays: Electron transitions between inner atomic shells of the target material give discrete lines (e.g., K\(\alpha\), K\(\beta\)).

1.1 Minimum‑wavelength equation

\[

\lambda_{\min} = \frac{hc}{eV}

\]

• \(h = 6.626\times10^{-34}\) J s (Planck constant).
• \(c = 3.00\times10^{8}\) m s⁻¹.
• \(e = 1.60\times10^{-19}\) C.
• \(V\) – accelerating voltage (V). The higher the voltage, the shorter the minimum wavelength (higher photon energy).

2. Uses of X‑rays

1. Diagnostic radiography – projection images of bone and chest.
2. Computed tomography (CT) – series of thin slices reconstructed into cross‑sectional images.
3. Therapeutic radiotherapy – high‑energy beams to destroy malignant tissue.
4. Industrial inspection – weld inspection, material density measurements.

3. Attenuation of X‑rays

Intensity follows an exponential law analogous to ultrasound:

\[

I = I_0\,e^{-\mu x}

\]

• \(\mu\) – linear attenuation coefficient (depends on photon energy and atomic number of the material).
• Higher‑Z materials (e.g., bone, metal) have larger \(\mu\) → appear white on radiographs.

4. Safety and Protection

• Use of lead shielding, collimation, and minimum‑necessary exposure time (ALARA principle).
• Personal dosimeters for staff, warning signs, and interlock systems on X‑ray rooms.

5. Practical Example – Calculating Required Tube Voltage

To produce X‑rays capable of penetrating a 10 cm thick aluminium plate (required \(\lambda_{\min} \le 0.02\) nm):

\[

\lambda_{\min}=0.02\times10^{-9}\;\text{m}

\quad\Rightarrow\quad

V = \frac{hc}{e\lambda_{\min}} = \frac{(6.626\times10^{-34})(3.00\times10^{8})}{(1.60\times10^{-19})(0.02\times10^{-9})}

\approx 62\;\text{kV}

\]

6. Summary of X‑ray Section

• Bremsstrahlung gives a continuous spectrum; characteristic lines arise from atomic transitions.
• The minimum‑wavelength equation links tube voltage to the highest photon energy.
• Exponential attenuation explains image contrast and the need for high‑Z contrast agents.
• Strict safety measures are essential because X‑rays are ionising.

Suggested Diagram (for both sections)

Cross‑section showing a piezo‑electric transducer, a thin coupling‑gel layer, layered tissues (skin → muscle → bone) with labelled impedances, incident, reflected and transmitted ultrasound beams; adjacent inset illustrating an X‑ray tube, target, emitted spectrum (continuous + characteristic lines), and a patient’s radiograph highlighting high‑Z contrast.