THE NATURE OF ULTRASONIC WAVES

Ultrasonic waves, like sound waves are mechanical vibrations having frequencies above the audible range. The audible range of frequencies is usually taken from 20 Hz to 20 kHz. Sound waves with frequencies higher than 20 kHz are known as ultrasonic waves. In general ultrasonic waves of frequency range 0.5 MHz to 20 MHz are used for the testing of materials. They can propagate in solid, liquid and gas but not in vacuum. Sound can travel in the form of beam similar to that of light and follows many of the physical rules of light. Ultrasonic beam can be reflected, refracted, scattered or diffracted.

Ultrasound is a form of mechanical vibration. To understand how ultrasonic motion occurs in a medium it is necessary to understand the mechanism which transfers the energy between two points in a medium. This can be understood by studying the vibration of a weight attached to a spring Figure 2.1a.

FIG. 2.1. (a) Weight attached to a spring, (b) Plot of displacement of W with time with respect to position A.

The two forces acting on W, while it is at rest, are force of gravity G and tension T in the spring.

Now if W is moved from its equilibrium position A to position B, tension T increases. If it is now released at position B, W would accelerate towards position A under the influence of this increase in tension. At A the gravity G and tension T will again be equal, but as now W is moving with a certain velocity, it will overshoot A. As it moves towards position C, tension T decreases and the relative increase in gravity G tends to decelerate W until it has used up all its kinetic energy and stops at C. At C, G is greater than T and so W falls towards A again. At A it possesses kinetic energy and once more overshoots. As W travels between A and B, T gradually increases and slows down W until it comes to rest at B. At B, T is greater than G, and the whole process again.

The sequence of displacements of W from position A to B, B to A, A to C and C to A, is termed a cycle. The number of such cycles per second is defined as the frequency of vibration. The time taken to complete one cycle is known as the time period T of the vibration, where T = 1/f.

The maximum displacement of W from A to B or A to C is called the amplitude of vibration.

All these concepts are illustrated in Figure 2.1b.

All materials are made of atoms (or molecules) which are connected to each other by interatomic forces. These atomic forces are elastic, i.e. the atoms can be considered to be connected to each other as if by means of springs. A simplified model of such a material is shown in Figure 2.2.

FIG.2.2. Model of an elastic material.

Now if an atom of the material is displaced from its original position by an applied stress, it would start to vibrate like the weight W of Figure 2.1a. Because of the interatomic coupling, vibration of this atom will also cause the adjacent atoms to vibrate. When the adjacent atoms have started to vibrate, the vibratory movement is transmitted to their neighbouring atoms and so forth. If all the atoms were interconnected rigidly, they would all start their movement simultaneously and remain constantly in the same state of motion, i.e. in the same phase. But since the atoms of a material are connected to each other by elastic forces instead, the vibration requires a certain time to be transmitted and the atoms reached later lag in phase behind those first excited.

When a mechanical wave traverses a medium, the displacement of a particle of the medium from its equilibrium position at any time ‘𝑡’ is given by:

(2.1) 𝑎 = 𝑎 sin 2𝜋 𝑓𝑡

Where,

a = displacement of the particle at time ‘𝑡’

a = amplitude of vibration of the particle and 𝑓 = frequency of vibration of the particle.

A graphical representation of Equation 2.1 is given in Figure 2.3.

FIG. 2.3. Graphical representation showing variation of particle displacement with time.

Equation 2.2 is the equation of motion of a mechanical wave through a medium. It gives the state of the particles (i.e. the phase) at various distances from the particle first excited at a certain time ‘t’.

(2.2) 𝑎 = 𝑎 sin 2𝜋 𝑓( )

Where,

𝑎 = displacement (at a time‘𝑡’ and distance ‘𝑥’ from the first excited particle) of a particle of the medium in which mechanical wave is travelling

𝑎 = amplitude of the wave which is the same as that of the amplitude of vibration of the particles of the medium

v = velocity of propagation of the wave f = frequency of the wave

x = distance of the particle from the first excited particle Figure 2.4 gives the graphical representation of Equation 2.2.

FIG. 2.4. Graphical representation of Equation 2.2.

Since in the time period T, a mechanical wave of velocity ‘𝑣’ travels a distance ‘’ in a medium, therefore we have:

𝑣 =/𝑇 (2.3)

But the time period ‘T’ is related to the frequency ‘𝑓 ’ by:

𝑓 = 1 𝑇 (2.4)

Combining Equations 2.3 and 2.4 we have the fundamental equation of all wave motion, i.e.

𝑣 =𝑓 (2.5)

In Equation 2.5 if ‘𝑓’ is in Hz, ‘’ in mm then ‘𝑣’ is in mm/s. Alternatively if ‘𝑓’ is in MHz,

‘’ in mm then ‘𝑣’ is in km/s.

2.2. CHARACTERISTICS OF WAVE PROPAGATION 2.2.1. Frequency

The frequency of a wave is the same as that of the vibration or oscillation of the atoms of the medium in which the wave is travelling. It is usually denoted by the letter ‘𝑓’ and is expressed as the number of cycles per second. The international term for a cycle per second is named after the physicist H. Hertz and is abbreviated as Hz.

1 Hz = 1 cycle per second

1 kHz = 1,000 Hz = 1,000 cycles per second 1 MHz = 1,000,000 Hz = 1,000,000 cycles per second 1 GHz = 1,000,000,000 Hz = 1,000,000,000 cycles per second Frequency plays an important role in the detection and evaluation of defects.

2.2.2. Amplitude

The displacement of the weight from its position of rest in Figure 2.1 and that of the particles of a medium in Figures 2.3 and 2.4 is called the amplitude. In Equation 2.2 ‘𝑎’ is the amplitude at any time ‘𝑡’ while ‘𝑎 ’ is the maximum amplitude.

2.2.3. Velocity

The speed with which energy is transported between two points in a medium by the motion of waves is known as the velocity of the waves. It is usually denoted by the letter ‘𝑣’. SI unit of velocity is meter per second (m/s).

2.2.4. Wavelength

During the time period of vibration T, a wave travels a certain distance in the medium. This distance is defined as the wavelength of the wave and is denoted by the Greek letter . Atoms in a medium, separated by distance ‘’ will be in the same sate of motion (i.e. in the same phase) when a wave passes through the medium.

The relationship between ‘’ ‘𝑓’ and ‘𝑣’ is given in Equation 2.5 which shows that in a particular medium the wavelength is the reciprocal of frequency. Therefore, higher the frequency shorter the wavelength and vice versa. In practical testing usually flaws of the order of /2 or /3 can be detected. Therefore smaller the wavelength, smaller are the detectable defects. Thus smaller wavelength or higher frequency ultrasound waves provide a better flaw sensitivity. This is further elaborated by the following example.

Example : Compare the flaw sensitivities for probes of frequencies 1 MHz and 6 MHz in steel.

Let us assume that flaw sensitivity is of the order of /3. Then for a 1 MHz frequency we have Flaw sensitivity = /3 = 1.98 mm

= 𝑣/𝑓

= (5940 (for steel) × 1000)/(1 × 1000000) mm = 5.94 mm

For the 6 MHz frequency we have

 = (5940 × 1000) / (6 × 1000000) mm = 0.99 mm Flaw sensitivity = /3 = 0.33 mm

2.2.5. Acoustic impedance

The resistance offered to the propagation of an ultrasonic wave by a material is known as the acoustic impedance. It is denoted by the letter ‘Z’ and is determined by multiplying the density of the material by the velocity ‘v’ of the ultrasonic wave in the material, i.e.

Z = 𝜌 𝑣 (2.6)

The value of the acoustic impedance for a given material depends only on its physical properties and thus to be independent of the wave characteristics and the frequency. Values of acoustic impedances for a number of familiar materials are given in Table 2-I.

2.2.6. Acoustic Pressure

Acoustic pressure is a term most often used to denote the amplitude of alternating stresses on a material by propagating ultrasonic wave. Acoustic pressure ‘P’ is related to the acoustic impedance ‘Z’ and the amplitude of particle vibration ‘a’ as:

P = Z 𝑎 (2.7)

2.2.7. Acoustic Intensity

The transmission of mechanical energy by ultrasonic waves through a unit cross section area, which is perpendicular to the direction of propagation of the waves, is called the intensity of the ultrasonic waves. Intensity of the ultrasonic waves is commonly denoted by the letter ‘I’.

Intensity ‘I’ of ultrasonic waves is related to the acoustic pressure P, acoustic impedance ‘Z’

and the amplitude of vibration of the particles ‘𝑎’ as:

𝐼 = 𝑃 /2𝑍 (2.8)

and

𝐼 = 𝑃 𝑎/2 (2.9)

TABLE 2.1. DENSITIES, SOUND VELOCITIES AND ACOUSTIC IMPEDANCES OF SOME COMMON

Polyvinylchloride (pvc hard) 1400 1060 2395 3353

quartz 2650 - 5760 15264

quartz glass 2600 3515 5570 14482

rubber vulcanized 1200 - 2300 2800

silver 10500 1590 3600 37800

steel (low alloy) 7850 3250 5940 46620

steel (calibration block) 7850 3250 5920 46472

steel (stainless) 7800 3130 5740 44800