14.1 Physics of ultrasonic waves

Ultrasonic waves are mechanical vibrations with frequencies above the human hearing range (greater than 20 kHz). Understanding their physics is essential for applications like medical imaging, industrial testing, and nondestructive evaluation. The behavior of these waves depends on their frequency, wavelength, propagation speed, and the properties of the medium they travel through.

Generating and detecting ultrasonic waves typically relies on piezoelectric transducers, which convert electrical energy into mechanical vibrations and back again. Other methods exist too, including magnetostrictive transducers, electromagnetic acoustic transducers (EMATs), and laser-generated ultrasound.

Fundamentals of Ultrasonic Waves

Characteristics of ultrasonic waves

Ultrasonic waves are mechanical waves, which means they need a medium to travel through (air, water, solids). They can't propagate in a vacuum. What sets them apart from audible sound is simply their frequency: anything above 20 kHz counts as ultrasonic, and practical applications use frequencies ranging from tens of kilohertz up to several gigahertz.

Three core properties define any ultrasonic wave:

• Frequency measures wave cycles per second in Hertz. Higher frequencies give better resolution in imaging but get absorbed more quickly by the medium.
• Wavelength is the distance between consecutive wave crests, inversely proportional to frequency:

$\lambda = \frac{v}{f}$

where $\lambda$ is wavelength, $v$ is wave speed, and $f$ is frequency. A higher frequency means a shorter wavelength, which is why high-frequency ultrasound can detect smaller features.

• Propagation speed is how fast the wave moves through a given medium. It depends on the material's density and elasticity, not on the wave's frequency (in most cases). For a fluid, it's calculated as:

$v = \sqrt{\frac{B}{\rho}}$

where $B$ is the bulk modulus (a measure of the material's resistance to compression) and $\rho$ is its density. Stiffer materials with lower density transmit ultrasound faster. For example, ultrasound travels at roughly 1,480 m/s in water but around 5,960 m/s in steel.

Generation and detection methods

Piezoelectric transducers are by far the most common way to generate and detect ultrasound. They rely on the piezoelectric effect, where certain materials (quartz crystals, lead zirconate titanate ceramics) deform when voltage is applied, and generate voltage when deformed.

How generation works:

1. An alternating electrical signal is applied to the piezoelectric crystal.
2. The crystal vibrates at the frequency of the applied signal.
3. These mechanical vibrations propagate into the surrounding medium as ultrasonic waves.

How detection works:

1. Incoming ultrasonic waves strike the crystal and cause it to deform.
2. The deformation generates a small electrical signal.
3. That signal is amplified and processed to extract information about the wave.

The same transducer often serves as both transmitter and receiver, switching between roles rapidly.

Alternative generation methods include:

• Magnetostrictive transducers exploit the tendency of ferromagnetic materials to change shape in a magnetic field. They're commonly used at lower ultrasonic frequencies.
• Electromagnetic acoustic transducers (EMATs) generate ultrasound directly in a conductive material using the Lorentz force, without needing physical contact with the surface.
• Laser-generated ultrasound uses a pulsed laser to create rapid, localized thermal expansion on a surface, launching an ultrasonic wave. This is useful for remote or non-contact inspection.

Ultrasonic Wave Behavior and Propagation

Interaction with matter

When ultrasonic waves encounter boundaries or travel through materials, three main phenomena occur:

• Reflection happens at boundaries between different media. The angle of incidence equals the angle of reflection, just like with light. How much energy reflects depends on the acoustic impedance mismatch between the two materials. A large mismatch (like a water-air interface) reflects almost all the energy, which is why gel is used in medical ultrasound to eliminate air gaps between the transducer and skin.
• Refraction bends the wave's path when it passes from one medium into another at an angle. This follows Snell's law:

$\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2}$

where $\theta_1$ and $\theta_2$ are the angles in each medium, and $v_1$ and $v_2$ are the respective wave speeds. If the wave enters a faster medium, it bends away from the normal.

• Attenuation is the gradual loss of wave energy as it travels, caused by absorption (energy converted to heat) and scattering (energy redirected in other directions). It follows an exponential decay:

$A(x) = A_0 e^{-\alpha x}$

where $A_0$ is the initial amplitude, $x$ is the distance traveled, and $\alpha$ is the attenuation coefficient (which depends on the material and frequency). Higher frequencies attenuate faster, creating a trade-off between resolution and penetration depth.

Factors in wave propagation

Acoustic impedance is one of the most important concepts for understanding how ultrasound behaves at interfaces. It's defined as:

$Z = \rho v$

where $\rho$ is density and $v$ is wave speed. When two materials have very different $Z$ values, most of the wave energy reflects at their boundary. When their $Z$ values are similar, most energy transmits through. This is the principle behind tissue imaging: different tissue types have slightly different impedances, producing reflections that form the image.

Medium properties that affect wave behavior include:

• Density influences both speed and acoustic impedance.
• Elasticity (bulk modulus, Young's modulus) directly impacts propagation speed.
• Temperature alters both density and elasticity, changing wave speed. In water, for instance, ultrasound speed increases by about 2.5 m/s per degree Celsius near room temperature.

Several additional effects shape how ultrasonic waves propagate:

• Dispersion occurs when wave speed varies with frequency. This can distort short ultrasonic pulses as they travel, since different frequency components arrive at different times.
• Scattering redirects wave energy when the wave encounters particles or inhomogeneities smaller than or comparable to the wavelength. In medical ultrasound, scattering from small structures in tissue contributes to the speckle pattern seen in images.
• Nonlinear effects become significant at high wave amplitudes. The wave shape distorts as it propagates, generating harmonics (multiples of the original frequency). This is exploited in techniques like harmonic imaging and even acoustic levitation.
• Mode conversion can occur at boundaries, where a wave changes type (for example, a longitudinal wave partially converting into a shear wave upon reflection from a solid surface). This is particularly relevant in ultrasonic welding and angle-beam inspection of solids.