The Physics of Sound
Sound is a vibration that propagates as an acoustic wave through a medium by means of compression and rarefaction of particles in the medium.
Speed of Sound
The wavelength, frequency, and speed of a sonic wave is governed by the relationship:
- c is the speed of sound in the medium defined as the distance traveled by the sonic wave in a medium per unit time
- λ is the wavelength which is defined as the distance traveled during one full oscillatory cycle
- f is the frequency of oscillation which is defined as the number of cycles completed each second
Importantly, the speed of sound in a medium varies widely depending on the properties of the medium. Speed of sound for a given medium is determined by the bulk modulus, B, which is a measure of a material’s resistance to compression and the density of the medium, ρ.
- B is the bulk modulus of the medium
- ρ is the density of the medium
|Material||Density (kg/m3)||c (m/s)||Acoustic Impedance (Rayls)
Key Point: Speed of sound is greatest for materials which are stiff (do not compress easily) and materials that have a low density.
Acoustic Pressure and Intensity
Acoustic rarefaction and compression cause changes in the local pressure of the medium. The pressure amplitude is defined as the difference between the maximum or minimum pressure of the wave and the average pressure of the medium in absence of the wave.
The SI unit of pressure is the Pascal (Pa) where 1Pa = 1 kg/m. In ultrasound, pressure amplitude is typically around 1 MPa which is approximately 10 times atmospheric pressure.
Acoustic intensity, loudness, is a measure of power per unit area and, in the ultrasound setting, is usually represented with units of mW/cm2. Intensity is proportional to the square of pressure (I ∝ P2).
Key Point: In ultrasound applications, the compression pressure amplitude is significantly greater than the rarefaction pressure amplitude.
Relative Intensity (Decibel (dB) Scale)
The decibel (dB) is a measure of relative intensity on a base-10 logarithmic scale.
Key Point: The loudness humans perceive is based on a logarithmic scale. The logarithmic definition of the decibel scale means that it is approximately proportional to human perception of loudness.
Acoustic Wave Interactions
Modern ultrasound equipment uses multiple sound emitters that create sound beams independently. These sound waves will interact with each other based on their phase (position in the periodic waveform), amplitude, and frequency.
As ultrasound propagates through a medium, it interacts with it. These interactions include reflection, refraction, and attenuation. The type of interaction is governed by differences in the acoustic impedance of different types of media.
Acoustic impedance (Z) is similar to the stiffness of a spring and is defined as the product of material density (p) times the speed of sound in the material (c). Acoustic impedance has the SI unit of Rayl, where 1 Rayl = 1 kg/m2s.
Acoustic impedance is a useful quantity because differences in acoustic impedance govern the flow of acoustic energy between connected media. When two connected materials have similar acoustic impedance, energy will flow from one material to another without much loss at the transition. However, when the materials have very different acoustic impedance values (e.g. soft tissue/lung interface) a great deal of acoustic energy will be reflected at the interface. This reflected acoustic energy is used to produce images in ultrasound imaging.
Reflection is the redirection of acoustic energy propagation which occurs at the interface of materials with different acoustic impedance. The angle of reflection relative to normal incidence is equal to the angle of incidence relative to normal incidence.
The fraction of acoustic energy reflected depends on the magnitude of the difference in acoustic impedance.
Refraction is the change in direction of the transmitted portion of an acoustic wave incident upon an interface. The angle of transmitted energy (θt) depends on both the speed of sound in each material (c1, c2) and on the angle of incidence (θi).
Attenuation is a loss of acoustic wave intensity due to interactions between the acoustic wave and the medium. Attenuation is measured by a change in decibel levels and is caused by the scattering and absorption of acoustic energy.
Scattering occurs either because of small non-uniformities within a medium, or because a reflective interface has a non-specular (rough) surface.
Absorption and Attenuation
Some acoustic energy is lost via absorption; in which the material converts acoustic energy to heat energy.
Key Point: Attenuation rate in soft tissue = 0.5 dB/cm/MHz.
Doppler Frequency Shift
A Doppler frequency shift is the change in frequency which occurs when an ultrasound wave travels through and is reflected by objects moving axially relative to the transducer. A Doppler shift in ultrasound is the same phenomenon that occurs when the sound of a car appears to change from a higher to a lower pitch as the car approaches, passes, and moves away from an observer. In the case of ultrasound, it is the usually the medium (blood) that is moving while the source and receiver (the transducer) is stationary.
The observed frequency can be found as follows:
- fr is the frequency at the receiver
- fs is the frequency at the source
- vr is the velocity of the medium relative to the receiver along the direction of wave propagation
- vs is the velocity of the medium relative to the source along the direction of wave propagation
Key Point: Doppler shifts only occur for motion along the direction of the wave propagation.
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