CHAPTER 1
The Basic Physics of Ultrasound

SOUND WAVES

A sound wave is a fluctuating variation in pressure within a medium such as air, water, or solid material. Our ears are sensitive to such pressure changes in air, and we hear sounds all the time. The faster the changes in pressure take place, the higher the pitch or frequency of the sound we hear. Frequency is measured in hertz (Hz) and, for a young person, their hearing goes from 20?Hz to 20?kHz. Middle C on a piano is 261?Hz. A sound above 20?kHz is called ultrasound ('beyond sound'), in other words, we cannot hear it. Dogs can hear sounds of higher frequency than we can, and bats use ultrasound, up to 200?kHz, for echo location in the dark. The ultrasound we use for medical imaging is in the range of megahertz (MHz), far above anything we can hear. As we will see, the reason for going to such high frequencies is that we can then make narrow beams of ultrasound that we can point in a particular direction and which can produce high-resolution images showing fine details.

We can think of a medium like air or water as a collection of molecules with mass connected by springs that represent the forces between the molecules (Figure 1.1). If you push on one molecule, it will move closer to the adjacent molecule and exert a force so that it too begins to move. Adjacent molecules having been squashed together will then repel one another and recover to their resting position. They will keep moving beyond their resting position due to their momentum. The force holding them together then becomes an attractive force that pulls the molecules back towards their resting position again as they are then stretched apart. That is, each molecule will move forwards then backwards in the direction of the applied force. This oscillating molecular motion within a material is the basis of a sound wave.

FIGURE 1.1 Ball and spring model of a sound wave travelling in some medium.

DEFINITION

A sound wave is a longitudinal pressure wave travelling at the speed of sound through a medium (e.g. air, water, and soft tissue).

The pressure is the force exerted from the excess density (number of molecules per unit volume) of molecules above or below the average density of the medium as the molecules get alternately squashed and pulled apart from one another. In other words, it is the excess pressure about the mean resting pressure.

A sound wave may be generated by a piston moving forwards and backwards. This is what a normal loudspeaker does (Figure 1.2). It has a diaphragm that moves forwards and backwards driven by an electric signal. This will alternately push, then move away, from the molecules in front of it. These molecules will then alternately push on to those in front of them and then pull on them, and so on, as described above. This will create a series of compressions and rarefactions in the molecules of the medium that move away from the source of movement (the piston), i.e. a sound wave as shown in Figure 1.3.

FIGURE 1.2 A loudspeaker driven by an electric signal acts as a piston on the air in front of it.

FIGURE 1.3 Oscillating piston producing a longitudinal pressure wave, a sound wave consisting of compression (+) followed by rarefaction (-). c is the speed of sound, and ? is the wavelength.

The speed at which they move away from the source is called the speed of sound (c). The speed of sound can be considered constant for each medium, so the speed of sound in air is 330?m·s-1, and if a plane goes faster than this it breaks the sound barrier. The speed of sound in water varies with temperature and at 20?°C is 1480?m·s-1. The average speed of sound in soft tissue is 1540?m·s-1, and, as will be seen, this is a key number for ultrasound imaging.

For ultrasound imaging, the ultrasound transducer acts exactly like the loudspeaker pushing and pulling the molecules of the medium in front of it.

NOTE

The individual molecules oscillate backwards and forwards about a mean position, but the pressure disturbance (p) propagates forward at the speed of sound (c). It is the moving disturbance that is the sound wave.

A typical sine wave plot of a sound wave is seen if we plot the change in pressure (p) at a given point against time, or if we plot the change in pressure versus distance away from the piston.

Looking at one position in space versus time (t), we see the pressure increasing and decreasing as the sound wave passes by (Figure 1.4a).

The frequency (f) is the number of cycles (peaks) passing a given point in one?second.

The period (T) is the time taken to complete 1?cycle. The relationship between period and frequency is

Looking at one instant of time versus distance (x) away from the sound source (Figure 1.4b), we see the pressure increases and decreases as we move away from the transducer.

FIGURE 1.4 Illustration of the change in pressure with time at one point in space (a) and the change in pressure with distance x from the sound source (b).

The wavelength (?) of the sound is then defined as the distance in space between two successive peaks on the wave.

THE SPEED OF SOUND EQUATION

The relationship between speed of sound (c), frequency (?f?), and wavelength (?) is

Relationship Between Pressure, Particle Velocity and Particle Displacement

We can also plot the change in particle velocity (v) versus time and the particle displacement (s), from its resting position, versus time, to give similar graphs (Figure 1.5).

FIGURE 1.5 The sine waves produced by plotting the change in pressure, particle velocity, and displacement associated with a sound wave.

Comparing these waves, we see that the pressure is greatest when the particle velocity is greatest and the particle displacement is greatest when the rising pressure passes through its mean zero level.

When we simply talk about a 'sound wave,' we usually mean the (excess) pressure wave - also known as acoustic pressure. It is acoustic pressure that our ultrasound transducers are sensitive to and detect.

NOTE

Do not confuse particle velocity with the speed of sound. Particle velocity is movement at a molecular level, whereas the speed of sound is the speed at which the sound wave propagates through the medium.

A transducer is anything that converts one form of energy into another form. The loudspeaker and the ultrasound transducer both convert electrical energy into sound energy and so are transducers.

NOTES

• As a sound wave propagates its frequency remains constant at the same frequency as the transducer 'piston' oscillates.
• For a given medium, the speed of sound is constant (it may vary with temperature). This means that if the transmitted frequency increases, the wavelength must get shorter to balance the speed of the sound equation.
• Key Concept: 'High frequencies give short wavelengths'.

DESCRIBING WAVES

The amplitude of a wave is the difference between the peak value and the mean zero value. We can also define the peak-to-peak amplitude A+ to A- as shown in Figure 1.6.

FIGURE 1.6 Definition of wave amplitude A.

FIGURE 1.7 Illustration of a spinning bicycle wheel over a moving roll of paper.

FIGURE 1.8 Definition of the phase angle of a sound wave.

The phase of a wave is a point along the course of one period of the wave expressed as an angle.

Think of a spinning bicycle wheel mounted above a moving roll of paper as shown in Figure 1.7. On the paper, we mark the vertical distance of the valve away from the hub at each moment as the wheel spins round. What results is a sine wave drawn on the paper. By the time, the wheel has gone round once, you would have drawn one period of the sine wave and the valve would have travelled round 360°. So, by measuring the angle of the valve as it goes round, we can mark the phase angle along the sine wave as shown in Figure 1.8. One cycle is equal to 360° (hence, frequency is equal to 'cycles' per second).

This gives us a very useful way to compare two sine waves. If one wave has a phase angle of 45° at the same time another sine wave has a phase angle of 0°, we know where the peak of one wave is compared to the other (Figure 1.9). If we know the frequency, amplitude, and phase of a wave, we know everything about it. The wavelength will depend on the speed of sound of the medium the sound wave...