Radiation of Elastic Waves

Acoustic waves emitted by sources of finite dimensions diverge, so that acoustic pressure and power density decrease with the distance from the transmitter. Understanding the radiation of sources is therefore essential to predict the acoustic field emitted by transducers used for medical diagnosis or in sonar systems, as well as in defect detectors in metallurgy or in other instruments and sensors.

The chapter is divided into three sections. In the first section, we study the acoustic radiation in a perfect fluid by a vibrating plane surface, then by a focused transducer. In the harmonic case and in a medium where the acoustic wave is defined by a scalar quantity such as the velocity potential or the acoustic pressure, the results are similar to those of the scalar theory of diffraction in optics. One specificity of acoustics is the use of short impulse waves or wave trains of varying durations. By drastically reducing the interferences between the waves emitted by the different points of the source, the conditions of the impulse diffraction, presented at the end of this section, are very different from those observed for time-harmonic waves. In the second section, we analyze the radiation of bulk elastic waves and Rayleigh waves in isotropic or anisotropic solids. With the exception of the explosive sources used in the field of oil exploration, these sources are most often located on the solid surface. The modeling of a seismic or a thermoelastic source is examined in the case of a linear distribution of impulsive forces. The last section is devoted to the radiation from an elementary spherical source embedded in an isotropic solid.

In a perfect fluid and assuming that the wave is a small perturbation of the reference state (pressure *p*_{0}, mass density *ρ*_{0}), the acoustic pressure *p _{a}* =

and to equation [1.322] from Volume 1:

resulting from the linearization of the continuity equation and the pressure–density relation of the fluid. * F* (

In the absence of sources in the volume (__ F__ =

The time-derivation of equation [1.1] shows that the potential *φ* satisfies the equation:

By introducing the potential *s*(* x, t*) of the volume sources, defined by:

equation [1.5] takes the form:

In the *harmonic case*, the velocity potential and volume source potential vary as *e*^{−iωt:}

The spectral components Φ(* x, ω*) and

where *k* = *ω/c* is the wave number.

The determination of the velocity potential^{1} Φ(* x* ) in a closed domain is not always easy. One way to do this is based on the integral formulation of the Helmholtz equation, also called Kirchhoff–Helmholtz equation.

Let us consider a volume *V* delimited by a surface *S* of unit normal __ n__, oriented toward outside. The potential Φ(

Let *G*(__ x__,

The Green’s function verifies the reciprocity principle, which is expressed by the relation:

Multiplying equation [1.9] by *G*(__ x__,

The integration of this equation over the volume *V* leads to the expression:

or again, after using the second Green’s identity:

where *∂*Φ(__ x__)

the scalar potential Φ(__ x__) is the sum of two integrals with different properties. The volume integral expresses the acoustic field generated by “primary” sources located inside the volume

The objective of this section is to determine the field radiated by a plane surface *S*_{0}, whose points vibrate with the normal velocity *v _{n}*(

which expresses the Kirchhoff–Helmholtz theorem.

Given the relation [1.3] between the particle velocity and the potential Φ, the continuity of the normal velocity imposes:

The velocity potential at point __ x__ takes the form:

In the absence of any information on the value of the velocity potential on the surface *S* and in order to remove the term Φ(*x*_{0}) in the integral equation, we must find a Green’s function satisfying a Neumann condition (*∂G/∂z*_{0} = 0) on this surface. The appropriate Green’s function is obtained using the image-source method, by the summation of the free field Green’s functions (Bruneau 2006):

corresponding to a source located at point *x*_{0} and to the source image, with respect to the plane *z* = 0, located at *x*^{′}0 (Figure 1.2):

By writing:

the derivative of the Green’s function with respect to the normal to the surface *z* = 0 is given by:

or again, considering expressions [1.21] for *R* and *R*^{′}:

At *z*_{0} = 0, the distances *R* and *R*^{′} are equal and therefore the normal derivative of the Green’s function is zero on the surface *z*_{0} = 0. The Green’s function *G*^{+} satisfies a Neumann condition on the surface *S*. Taking into account this condition and the expression [1.20] for a source point located at *z*_{0} = 0:

relation [1.18] can be written in the form of the *Rayleigh integral*:

The Rayleigh integral was established in the harmonic case under the hypothesis that all points on the emitting surface *S*_{0} vibrate in phase with a normal velocity *v _{n}*(

This distinction is independent of the type of excitation, that is, of the operating regime (harmonic, impulse, transient, etc.), as shown in Table 1.1.

REMARK.– The velocity potential Φ(* x, ω*) is naturally the Fourier transform of

There exists another operating mode specific to multi-element antennas (or arrays) used in sonars or ultrasound scanners. The source consists of a large number of piezoelectric components, usually rectangular, whose width is small compared to the wavelength. The electric voltage applied to each element is either delayed (in the impulse regime) or phase-shifted (in the harmonic case) according to a linear law, depending on the position of the elements (section 1.1.4.4). The electronic control of the delay or of the phase-shift law ensures the dynamic scanning of these ultrasound imaging devices.

If we choose the difference of the free field Green’s functions, the Green’s function obtained:

is zero at any point on the plane surface *S*, because *R* = *R*^{′} when *z*_{0} = 0. By changing the + sign to the – sign in the right-hand side of equation [1.23], the normal derivative to the surface *S* is expressed as:

Let us carry out two approximations, justified in optics, to calculate the field diffracted by an aperture:

- – the incident wave is little modified on the surface of the aperture, whose dimensions are large with respect to the wavelength
*λ*; - – the observation point M is located at a large distance, that is,
*kR >>*1.

Under these conditions, equation [1.18] takes the form:

where *θ* is the angle between the vector __x__ − __x___{0} and the *z* axis (cos *θ* = *z/R*). This *Rayleigh–Sommerfeld formula* provides the field *φ _{d}*(

We start by analyzing the radiation of time-harmonic acoustic waves by planar sources such as a circular disk, a rectangular element, or a multi-element linear antenna. These reference sources serve as a model for piezoelectric transducers used in various fields, such as non-destructive evaluation of materials, medical echography and underwater exploration by ultrasound.

The case of a plane baffled piston is a major problem in acoustics, as it allows us to model many acoustic sources, such as loudspeakers or piezoelectric transducers. The circular source of radius *a*, vibrating uniformly, is included in an infinite rigid screen (Figure 1.3). The calculation of the Rayleigh integral requires the use of polar and spherical coordinates. The position of a source point P on the surface of the disk is defined by the vector __x___{0} = (*r*_{0} cos *ϕ*_{0}*, r*_{0} sin *ϕ*_{0}, 0) and the observation point M in the domain *V* is marked by the vector * x*= (

The particle velocity, normal to the plane surface, is assumed to be uniform:

Since the problem is axisymmetric, calculations can be performed by taking *ϕ* = 0, without any loss of generality. The distance *R* = |__x__ – __x___{0}| between the source point and the observation point is equal to:

and the potential at the observation point M is given by:

This integral cannot be expressed analytically in the general case, but it is possible to evaluate it exactly along the piston axis and approximately in the far field.

In the harmonic case, the acoustic pressure *P _{a}*(

where *Z*_{0} = *ρ*_{0}*c* is the acoustic impedance of the fluid. On the piston axis (*θ* = 0), this pressure can be expressed as:

By performing the change of variable , the acoustic pressure takes the form:

Thus, the pressure field results from the interference of a traveling plane wave emitted by the whole piston, and a wave generated by diffraction on the edge of the piston.

The amplitude of the pressure field normalized by *Z*_{0}*v*_{0} is plotted in Figure 1.4, as a function of the ratio *z/a* for four different values of the product *ka*: (a) *ka* = 1, (b) *ka* = 10, (c) *ka* = 50 and (d) *ka* = 100. For *ka* = 1, that is, when the wavelength *λ* = 2*πa* is larger than the piston radius, the acoustic pressure on the piston axis decreases rapidly. If the wavelength is comparable to the radius (*ka* = 10), this pressure passes through a single maximum and then decreases as *z* increases. In situations where the radius *a* is greater than the wavelength *λ*, the amplitude of the normalized pressure oscillates between 0 and 2 and then subsequently decreases. The larger the product *ka* is, the greater the number of oscillations in the near field. The position of the last maximum marks the limit between the *Fresnel zone*, representing the near field, and the *Fraunhofer zone*, representing the far field.

The positions of the pressure maxima and zeros are easily deduced from its modulus. According to relation [1.34]:

the expression for the modulus of the pressure normalized by *Z*_{0}*v*_{0} is:

The pressure maxima result from *constructive interference*. They correspond to the extreme values of the sine function:

therefore, for the values of *z* that satisfy relation:

The maximum that is farthest from the source is located at the distance *z*_{1} (*m* = 1):

The pressure cancellations result from *destructive interference*, which takes place when the argument of the sine is equal to *nπ*, that is, for values of *z _{n}* that satisfy relation:

The position of the last pressure maximum is often used as the lower limit of the far field. If the piston radiates at high frequency, the product *ka* is large and the position of the last pressure maximum is:

This expression remains valid as long as the distance *z* between the far field input and the source is large compared to the source diameter and to the wavelength.

ORDER OF MAGNITUDE.– In immersion experiments, it may be important to place the sample to be probed in the far field of the emitter. For a transducer of diameter 2*a* = 13 mm, which generates longitudinal waves in water at a frequency *f* = 10 MHz, the far field corresponds to *z > a*^{2}*/λ* ≈ 28 cm.

In the far field, the distance *z* is large in front of the radius *a*. A limited development over the quantity shows that the pressure decreases as 1*/z*:

As previously mentioned, the Fraunhofer zone corresponds to the far field, that is, to distances *r* larger than the radius *a* of the piston. In this case, the distance |__x__ – *x*_{0}| appearing in the expression [1.31] of the velocity potential can be approximated by:

so that:

or again:

Knowing that the zero-order and first-order cylindrical Bessel functions satisfy the relations:

with *x* = *kr*_{0} sin *θ*, the velocity potential takes the form:

*Q* = *πa*^{2}*v*_{0} is the piston flow and *D*(*θ*) is the directivity factor, given by:

Since *J*_{1}(*x*) = *x/*2 for small values of *x* = *ka* sin *θ*, the function *D*(*θ*) is maximal and equal to unity for *θ* = 0.

In Figure 1.5, the directivity factor is plotted for two values of the product *ka*. If *ka* is small compared to unity, that is, at low frequency, the directivity factor is close to unity and depends little on *θ*. The radiation is omnidirectional and the source behaves like a point source (which is in agreement with the fact that its dimensions are small compared to the wavelength). When the product *ka* increases, the number of side lobes increases and the width of the main lobe shrinks: the piston is more directional at high frequency.

Two parameters are used to characterize the directivity of the emitted acoustic beam:

- –
*The divergence angle*defined by the angular difference 2*θ*_{0}between the zeros on either side of the main lobe. Knowing that the first zero of the Bessel function*J*_{1}(*x*) corresponds to the argument*x*= 3.83, the divergence angle of the beam is given by:

Since the maximum of the first side lobe is equal to only 0.133, that is –17 dB relative to that of the central lobe, the energy emitted by the disk is contained, for the most part, in the cone of apex angle 2*θ*_{0} ≈ 1.22*λ/a*, if *λ << a*.

- –
*The –3 dB angular aperture*is defined by the angular difference 2*θ*_{1/2}between the two directions (in a meridian plane) for which the acoustic power is divided by two. It is twice the angle*θ*_{1/2}for which the directivity factor is divided by that is,

The emission by the piston disk is more directional when its diameter 2*a* is large with respect to the wavelength *λ*.

The antennas (transmitters and receivers) of underwater exploration and localization systems (sonar) or the arrays of medical ultrasound equipment are made up of a large number of rectangular piezeoelectric elements. The direction of the emitted acoustic beam is controlled by applying a delay law or a phase law to the different elements. The angular width of the swept sector is limited by the aperture of the beam emitted by an isolated element, whose width must be smaller than the wavelength *λ*. The angular resolution is all the better as the length *L* of the antenna is large compared to *λ*. Therefore, the number 2*N* of elements of the antenna must be large, typically 32 ≤ 2*N* ≤ 256.

Let us first examine the acoustic radiation in the far field of a rectangular element of width 2*d* (along *x*) and height 2*h* (along *y*), located at *z* = 0 and included in the surface *S* (Figure 1.6). A point on its surface *S*_{0} is defined by the vector __x___{0} = (*x*_{0}*, y*_{0}, 0), where *x*_{0} ∈ [*–d, d*] and *y*_{0} ∈ [–*h, h*]. An observation point M is marked in the domain *V* by the vector * x*= (

Assuming that *x*_{0} and *y*_{0} are small compared to *r*, the norm of vector |__x__ – __x___{0}| is expressed by:

In the harmonic case, the velocity potential results from the Rayleigh integral [1.25]; in the far field, its approximate expression becomes:

Since the vibratory velocity *v _{n}* is zero in the plane

where is the double spatial Fourier transform, defined by the relation:

This formulation is general: in the far field, the velocity potential emitted by a plane surface *S* vibrating in the piston mode (*v _{n}* =

Thus, the expression [1.45] for the acoustic field radiated by the uniform circular piston in Figure 1.3 can be found directly. In the *xz* plane (*ϕ* = 0), equivalent to any plane due to the symmetry of revolution: *k _{x}* =

where *D*(*θ*) is the directivity factor [1.48].

If the rectangular element vibrates in a uniform piston mode, the normal velocity is independent of the point:

The expression for the velocity potential is:

where *Q* = 4*dhv*_{0} is the piston flow and the directivity factor *D*(*θ, ϕ*) is a double cardinal sine function:

The radiation in the far field of a rectangular element of width 2*d* = 1 mm and height 2*h* = 3 mm vibrating at 10 MHz is represented in Figure 1.7. The field is calculated at a distance *r* = 50 mm. The color scale is saturated in order to visualize the side lobes.

Let us now consider the case of a distribution of 2*N* linear rectangular elements of width 2*d* and height 2*h* (Figure 1.8). The velocity potential is given by the relation:

with, in the far field:

In order for the plane P, inclined at an angle *β* with respect to the normal, to be equiphase, it is necessary to compensate the phase difference Δ*ψ* = 2*kd* sin *β* between waves emitted by two adjacent elements, 2*d* apart, by applying to the emission a phase law *ψ _{m}* =

the velocity potential in the *xz* plane (*ϕ* = 0) takes the form:

with *ψ* = *kd*(sin *θ* sin*β*). Given expression [1.59] for the directivity factor of a rectangular element:

and by recognizing the geometric series Σm *q ^{m}*, we obtain:

In Figure 1.9, the radiation pattern for a linear antenna composed of 2*N* = 64 rectangular elements, of width 2*d* = 4 mm and height 2*h* = 4 cm, is plotted in the *xz* plane (*ϕ* = 0) as a function of angle *θ* for three phase-shift angles: *β* = 0^{◦}, 30^{◦} and 60^{◦}. This example is representative of an equipment for shallow-water sonar (depth 5–500 m) operating at 100 kHz (*λ* = 15 mm). The width of each element, of the order of *λ/*4, is small enough to ensure an efficient scanning over a 100^{◦} sector. For *β >* 50^{◦}, the emitted amplitude decreases due to the elemental directivity factor *D*(*θ, ϕ*), and the angular aperture of the beam increases. This characteristic is defined by the angle Δ*θ* between two directions in the *xz* plane that are symmetric with respect to the beam axis and for which the emitted power is divided by two. For an antenna of length *L* = 4*Nd* and a small angle *θ*, the directivity function [1.65] takes the form:

Its maximum value (2*N*) is divided by when *α* = ±0.885*π/*2, that is:

i.e. Δ*θ* ≈ 3^{◦} for *L* = 256 mm and *λ* = 15 mm.

Figure 1.10 is an example of the measurement of the dispersion curves of a 1-mm thick duralumin plate using a linear array composed of 128 rectangular elements of dimensions 0.2 ×10 mm^{2} and spaced apart by 0.25 mm. A film of water ensures the coupling between the probe and the sample. A programmable electronic device transmits pulses to the first element, of center frequency 8 MHz and duration 4 *μ*s, whose spectrum ranges from 4 to 12 MHz. This electronic device records the 128 signals received by all the elements. The Fourier transforms with respect to the time and space of these signals, acquired over 25 *μ*s, are stored in a matrix. The *N* non-zero components are stored in a *N* × 3 (*k _{n}*,

In domains like medical ultrasound imaging or non-destructive testing of materials, transducers that generate acoustic waves are often excited by short pulses. The velocity potential at point M and angular frequency *ω* is obtained by replacing the wave number *k* by *ω/c* in the expression [1.25] for the Rayleigh integral, established in the harmonic case:

Φ(* x, ω*) is the spectral component, at angular frequency

is written as given below, by changing the integration order:

The integral in square brackets represents the normal component of the particle velocity at point P and time *t – R/c*, that is, *v _{n}*(

Let us consider the case where all points of the source vibrate with the same normal velocity *v*_{0}(*t*):

where *P _{n}*(

By substituting equations [1.72] and [1.73] into relation [1.71], we get:

with:

The velocity potential at point M is the convolution product of the velocity *v*_{0}(*t*) at the piston surface with the space-time function *h*(* x, t*), called the

The Fourier transform of a convolution product is the simple product of transforms:

the *diffraction frequency response H*(* x, ω*) being the Fourier transform of

To calculate the diffraction impulse response of a plane transducer vibrating uniformly: *P _{n}*(

The surface element is d*S* = *L*(*R*) d*ρ* where *L*(*R*) is the length of the circular arc. Since *δ*(*t – R/c*) = *cδ*(*R – ct*), the impulse response [1.75] takes the form Stepanishen (1971):

where *R*_{1} = *ct*_{1} and *R*_{2} = *ct*_{2} denote, respectively, the minimum and maximum distances from the points on the disk to the observation point M. The integration amounts to replace *R* with *ct*:

An even simpler expression is obtained by introducing the angle Ω(*R* = *ct*) which, from its center M_{0} (Figure 1.11), subtends the circular arc AB of length:

The impulse response [1.80] takes the equivalent form:

In the case of the *uniform disk* of radius *a*, the field only depends on the variables *r*, *z* and *t*. For points located on the piston axis (*r* = 0) and for time *t* such that:

the arcs that are equidistant from the observation point are complete circles, therefore Ω(*ct*) =2*π*:

Since it is zero at other instants, the diffraction impulse response is a rectangular function whose duration:

decreases as the distance *z* increases (Figure 1.12).

The *acoustic pressure* radiated by a circular disk, animated by a velocity *v*_{0}(*t*), can be deduced from the time derivative of the velocity potential (equation [1.4]), that is, considering the convolution product:

On the piston axis, the derivative of the impulse response (Figure 1.12(b)) consists of a positive delta at time *t*_{1} and a negative delta at time *t*_{2}; we obtain:

Figure 1.13(a) shows that in the vicinity of the disk (*z << a*), the acoustic pressure is formed of a first term similar to the surface velocity (plane wave) and a second term, of opposite sign, whose arrival time corresponds to a path coming from the edge of the disk. This *edge wave* expresses the diffraction effects related to the finite radius *a* of the source. At a larger distance (*z* = 2*a*), the two waves begin to overlap (Figure 1.13(b)). The interference phenomena, which are responsible for the oscillations of the acoustic pressure observed on the piston axis in the sinusoidal case (Figure 1.4), disappear in the impulse regime, because the acoustic field received at one point and at a given time arises from a well-defined area on the surface of the transducer.

When the Fresnel approximation is valid, the duration Θ of the impulse response is inversely proportional to the distance *z*:

When *z*^{2} *>> a*^{2}, the diffraction frequency response *H*(* x, ω*), that is, the Fourier transform of

In the *far field* (Fraunhofer zone), the diffraction impulse response tends toward a Dirac pulse, that is, considering the unit area of the Dirac delta function:

The velocity potential is similar to the velocity *v*_{0}(*t*) of the points on the surface of the piston disk (formula [1.74]):

and the acoustic pressure:

is similar to the derivative of the surface velocity of the disk (Figure 1.13(d)).

Figure 1.14 shows that the acoustic pressure passes through a maximum at a distance such that the opposite contributions of plane wave and edge wave (equation [1.87]) are added, that is, when their path difference is equal to half a wavelength:

This “pseudo-focusing” distance corresponds to the position, *z*_{1}, of the last maximum (equation [1.41]), that is, the limit between the near field and the far field in the harmonic case.

REMARK.– The analysis of acoustic wave radiation by a transducer operating in the piston mode was conducted in two extreme cases, corresponding to the first two entries in Table 1.1:

- – in the harmonic case, that is, for a sinusoidally vibrating piston with a constant amplitude;
- – in the impulse regime, that is, for an infinitely short excitation (Dirac pulse).

We have shown that the radiated acoustic fields were very different (see, for example, Figures 1.4 and 1.14). In the ultrasonic domain, the need for a good axial resolution leads to the reduction of the time duration of the transmitted wave train. However, the center frequency and the bandwidth of the transducer impose a low limit on this duration (section 3.1.3). In practice, it varies from one period (very large bandwidth transducer) to a few tens of periods (narrow band transducer). The space-time evolution of the acoustic field, emitted in this transient regime, can then be calculated in two ways: either by summation of the contributions of the various spectral components (Fourier analysis) or, directly in the time domain, by convolution between the piston surface velocity and the diffraction impulse response.

In the far field, the acoustic beam emitted by a plane disk spreads due to diffraction: the acoustic intensity and lateral resolution decrease. A solution to improve these two parameters, essential to any imaging system, is to focus the acoustic beam either through a lens, like in optics, or by simply using a curved emitting surface.

The geometry of a focused transducer whose surface is a spherical cup is defined by its radius of curvature or focal length *F*, and by the radius *a* of its circular edge (Figure 1.15(a)). The wave radiated into the fluid by a concave source can be diffracted by its own surface, giving rise to a secondary radiation. Strictly speaking, the Rayleigh integral no longer applies. However, this secondary diffraction is negligible if the curvature of the surface is slight (*F >> a*). In this case, the angle *θ _{n}*, between the normal to the emitting surface and the

In the case of a uniform vibrating surface: *P _{n}*(

As for the plane disk, the surface element d*S* is equal to *L*(*R*) d*ζ*, where *L*(*R*) is the length of the intersection of the spherical cup of radius *F*, with the sphere of radius *R* = *ct* centered at the observation point M (Figure 1.15(b)), and d*ζ* is the width of the band intercepted on the spherical cup. The result obtained by Penttinen and Luukkala (1976) is:

where:

is the distance between the observation point M(*z*, *r*) and the focus F(*F*, 0). *t*_{1} and *t*_{2} are the extreme arrival times of the disturbance emitted, at time *t* = 0, by all the points on the transducer. Ω(*ct*) is the angle that subtends the arc formed by the points on the cup located at a distance *R* = *ct* from point M.

At any point on the axis (*r* = 0), since Ω(*ct*) = 2*π* and *d*(*z,* 0) = *|F – z*|, the diffraction impulse response takes the form of a rectangular function:

Denoting the thickness of the spherical cup by:

the minimum and maximum transit times are

for disturbances emitted by the center and the edge of the cup, respectively. It must be noted that if *z < F*, then *t*_{1} *< t*_{2} and, conversely, if *z > F*, then *t*_{1} *> t*_{2}. Considering expression [1.4], the acoustic pressure created by a surface velocity *v*_{0}*δ*(*t*) derives from the potential *φ*(*t*) = *v*_{0}*h*(*t*):

that is, for its Fourier transform:

Given the value [1.98] of the limiting times *t*_{1} and *t*_{2} of the rectangle, the ratio of the acoustic intensity *I*(*z*) = |*P _{a}*(

Figure 1.16 represents the variations of the acoustic intensity *I*(*z*)*/I*_{0} normalized by the intensity *I*_{0} on the surface of a transducer of focal length *F* = 50 mm, radius *a* = 10 mm, that is, *e* = 1 mm, excited at a frequency *f* = 2.4 MHz, so that *ke* = 10. Before the focus, the oscillations resulting from the interference between waves emitted by the edge and the center of the cup are much smaller than in the case of a plane disk. The maximum of the relative intensity is located a little before the focus; beyond this point the acoustic intensity decreases rapidly with the distance.

When the observation point goes toward the focus (*z → F* ), the duration of the response tends to zero, while its amplitude tends to infinity. The diffraction impulse response converges to a Dirac function, whose intensity *A* is equal to the limit, when *z* approaches *F*, of the area of the rectangle of height *cF/*|*F* − *z*| and duration |*t*_{1} − *t*_{2}|:

As to the first order in (*z* − *F* ):

the area *A* of the rectangle is equal to the thickness *e* of the spherical cup, so that:

At the transducer focus, the velocity potential is a replica of the velocity *v*_{0}(*t*) of the points on the emitting surface: *φ*(*F,* 0*, t*) = *ev*_{0}(*t – F/c*). The velocity along *z* is proportional to the derivative of *v*_{0}(*t*) delayed by *F/c*:

Beyond the focus, the shape of the velocity no longer changes and only its amplitude decreases due to the divergence of the beam. The particle velocity (or acoustic pressure) undergoes a time-derivative as it passes through the focus of the transducer.

As mentioned at the beginning of this section, the advantage of using a focused transducer is to achieve better performances in acoustic detection or ultrasonic imaging. In the following, we evaluate these improvements in terms of sensitivity and spatial resolution.

The acoustic pressure is given by relation [1.4] and the diffraction frequency response *H*(*F,* 0*, ω*) is the Fourier transform of the impulse response [1.104]:

Since, in the frequency domain Φ(*F,* 0*, ω*) = *H*(*F,* 0*, ω*)*V*_{0}(*ω*), the modulus of the acoustic pressure at the transducer focus is equal to:

On the surface of the spherical cup, *p _{a}*(0, 0

Assuming that *a*^{2} *<< F*^{2}, the thickness of the spherical cup can be written as *e* ≈ *a*^{2}/2*F* and the gain in intensity:

is larger than unity if the dimensionless parameter *λF/a*^{2} is less than *π*, that is, if the focal length *F* is smaller than the Rayleigh distance (Kino 1987), defined by:

Energy concentration is only possible in the near field, beyond this limit the natural divergence of the acoustic beam prevails over the focusing effect due to the curvature of the spherical transducer.

In the piston mode, the mean power emitted by the entire surface of the transducer is *πa*^{2}*I*_{0}. By defining the diameter of the acoustic beam at the focus by the value 2*w*, such that half the emitted power passes through the interior of a circle of radius *w*, we obtain:

The width of the acoustic beam at –3 dB is thus written as:

where *D* and *F/D* are, respectively, the diameter and the *F*-number of the transducer.

In order to examine the radiation of bulk and surface elastic waves by sources distributed on the surface of a solid, we use Fourier analysis and the formalism of the mixed matrix, discussed in Appendix 2. The approach is as follows:

- – expression of stresses in the harmonic case as a function of the amplitudes of the rising and falling waves;
- – matrix inversion to calculate, using the emission matrix, the amplitudes of the emitted waves as functions of the forces applied on the solid surface;
- – calculation of the inverse Fourier transform to obtain the spatiotemporal response. In the near field, the inversion is carried out numerically; in the far field, approximate analytical expressions for the mechanical displacements at the arrival time of the bulk wave fronts and the Rayleigh wave fronts are obtained using the stationary phase method.

The method is applied to the case of an infinitely long and thin line source located along the *x*_{3} axis, on the surface of a solid occupying the half-space *x*_{2} *>* 0; the impulsive forces are either normal or parallel to the surface (Figure 1.17).

The mixed matrix *M* relates the amplitude * a* of the upward bulk waves and the amplitude

To arrive at a two-dimensional problem, the material, *a priori* anisotropic, is assumed to have an orthotropic symmetry with a binary axis parallel to *x*_{3}. The propagation of **QL** and **QT** waves, polarized in the *x*_{1}*x*_{2} plane, is then decoupled from the propagation of the TH wave, polarized along *x*_{3} (Volume 1, section 2.2.2.1). The slowness component *s*_{3} is equal to zero and the propagation direction in the sagittal plane is defined by the ratio *m* = *s*_{2}*/s*_{1} of the two other components. By taking *s*_{1} as variable, the Christoffel equation is a fourth-degree polynomial equation in *s*_{2}, whose coefficients are real. This admits four real or complex roots. Each real root corresponds to a propagative bulk wave transporting energy. The direction of the power flux is given by the normal to the slowness surface. The complex roots appear in conjugated pairs and only the two solutions decreasing with *x*_{2} must be retained as components of the Rayleigh wave. These two acceptable roots (*s*_{2J} with *J* = 1,2) and all the elements of the mixed matrix are expressed as a function of a single variable: the slowness component *s*_{1}.

We first examine the generation of bulk waves polarized in the sagittal plane and then the generation of Rayleigh waves. In the harmonic case, an integral expression gives the mechanical displacement created in an elastic solid by a distribution of mechanical traction *T _{k}*(

As the mechanical traction *T _{k}* =

By introducing, for the mode *J*, the components *s*_{1} = *k*_{1}*/ω* and *s*_{2J} = *k*_{2J} */ω* of the slowness vector, the total amplitude emitted by the line source is given by the integral:

We begin by developing the method in the general case of a medium with orthotropic symmetry, before applying it to an isotropic solid, and then to a material of cubic symmetry. The propagation equations for the QL and QT plane waves polarized in the *x*_{1}*x*_{2} plane of an orthotropic solid were developed in sections 2.5.5 and 3.1.2.2 of Volume 1. With a mechanical displacement:

they are written in matrix form (Volume 1, system [3.43] with *m* = *k*_{2}*/k*_{1}):

Dividing by *ω*^{2} reveals the slownesses *s*_{1} = *k*_{1}*/ω* and *s*_{2} = *k*_{2}*/ω*. The characteristic equation:

is a biquadratic equation in

whose coefficients *α*_{0}, *α*_{2} and *α*_{4} are given by relations [2.172] of Volume 1. The two acceptable roots (*J* = 1, 2) corresponding to the quasi-longitudinal wave (*s*_{21}) and quasi-transverse wave (*s*_{22}) depend on the slowness *s*_{1} through the intermediary of coefficients *α*_{2} and *α*_{0}:

The polarization vector components for each wave, *q*_{1J} and *q*_{2J}, are related by:

where *s*_{2J} (*J* = 1, 2) are the two positive roots of equations [1.121]. The stresses *T _{i}* =

The expressions for these coefficients, identical to those for coefficients *A*_{iJ}, are given as functions of the parameter *m*_{J} = *s*_{2J} */s*_{1} by relations [2.150] in Volume 1, that is:

The inversion of the linear system [1.123], that is, of matrix * B*, provides the four components

Since the amplitudes of the partial waves, that is, the components of vector * b*, are obtained in the Fourier domain, the components of the displacement field in the real space are calculated by numerical inverse Fourier transforms. However, the presence of singularities associated with the poles or the branch points makes the inversion difficult. The introduction of a small imaginary part in the angular frequency allows us to move away from these singularities while keeping their significant influence (Weaver

The generation of elastic waves by a line source in a duralumin half-space is illustrated in Figure 1.18. The source, located at (*x*_{1}*, x*_{2}) = (0, 0), is a very short impulse. Four types of waves are visible: the first two are longitudinal (L) and transverse (T) bulk waves, which propagate with a cylindrical wave front, diverging from the line source. Two Rayleigh (R) waves propagate on the free surface on both sides of the source. Finally, two head waves (H) can also be observed on either side of the source. These head waves are transverse waves resulting from the conversion of the grazing longitudinal wave beyond the critical angle *θ _{c}* = arcsin(

The generation of elastic waves by an impulse line source in a copper half-space is illustrated in Figure 1.19. The comparison of this map at time *t* = 40 *μ*s with the wave surfaces in Figure 1.13(b) of Volume 1 clearly shows quasi-longitudinal and quasi-transverse wave fronts.

In this section, the integral [1.116] is evaluated by the stationary phase method. This approximation is valid in the far field and in the vicinity of the wave fronts of bulk waves, when they propagate independently. The directivity functions are plotted for an isotropic solid. In the case of an anisotropic material, the focusing effect due to the variations in curvature of the slowness surface is highlighted. Then, we calculate the contribution of the Rayleigh wave to the mechanical displacement of the solid surface.

When *k*_{1}*x*_{1} is very large compared to unity, the phase variations:

cause very rapid oscillations (as functions of *s*_{1}) of the exponential *e*^{iϕ(s1)}, which contribute little to the integral [1.116]. The major part of its value comes from the interval around the slowness where the phase *ϕ*(*s*_{1}) is stationary, that is, for which the derivative d*ϕ/*d*s*_{1} cancels. The second-order development is a good approximation of the phase in the vicinity of

Let us carry over this development into [1.116] and by assuming that the variations in factor *E*_{Jk}(*s*_{1}) are slow compared to the variations in the exponential, we get:

By changing the variable:

and since we obtain:

As:

the amplitude of the wave *J* emitted by the source is given by equation [1.116]:

As *s*_{1} = *s _{J}* sin

vanishes for the angle *θ*_{0}, so that tan *θ*_{0} = *x*_{1}*/x*_{2} and the second derivative is negative:

By introducing the wave number *k _{J}* =

With *ε* = −1, equation [1.132] gives the amplitude of the longitudinal (*J* = 1) and transverse (*J* = 2) displacements generated by a linear distribution of tangential (*k* = 1) or normal (*k* = 2) forces:

This amplitude decreases in as expected for cylindrical waves emitted by a line source. The last term of the right member is, in the two-dimensional case, the far field approximation of the spectral Green’s function of the Helmholtz equation:

Taking into account its asymptotic behavior, the zero-order Hankel function of first kind is the appropriate solution for a divergent wave and a temporal factor in *e*^{−iωt}:

Given the expression for the time Green’s function, that is, the Fourier transform of *G*(*r, ω*):

where *H*(*t*) is the unit step (Heaviside) function, the instantaneous displacement is given by:

*Normal force (k = 2)*

The coefficients *E*_{12}(*s*_{1}) and *E*_{22}(*s*_{1}) are calculated in Appendix 2:

where Δ(*s*_{1}) is the Rayleigh determinant:

- – In the case of a
*longitudinal wave*:*s*_{1}=*s*sin_{L}*θ*and*s*_{2L}=*s*cos_{L}*θ*, the Rayleigh determinant is expressed by:

with *κ* = *s _{L}*

and the displacement in the far field [1.140] is equal to:

*D*_{2L}(*θ*) is the dimensionless directivity factor for a normal force:

- – In the case of a
*transverse wave s*_{1}=*s*sin_{T}*θ*and*s*_{2T}=*s*_{T}cos*θ*, the Rayleigh determinant is expressed by:

By replacing *s*_{1} by *s*_{T} sin *θ* in the numerator of *E*_{22}, we obtain:

The transverse displacement in the far field is given by expression [1.145], by replacing *D*_{2L}(*θ*) with:

The directivity patterns *D*_{2L}(*θ*) and *D*_{2T} (*θ*) are plotted in Figure 1.20 in the case of duralumin (*κ* = *V _{T}*

ORDER OF MAGNITUDE.– The amplitude of the mechanical displacement, proportional to the ratio of the force density *F*_{2} (N/m) to the shear modulus *μ* (N/m^{2}), is expressed in meters. In the ultrasound domain, the displacements are of the order of a few nanometers. For example, with constants close to those of duralumin: *μ* = 25 GPa, *κ* = 0.5 and *F*_{2} =1 N/cm, we obtain *D*_{2L}(0) = *κ*^{2} = 0.25 and *u _{L}* =1 nm.

*Tangential force (k = 1)*

The elements *E*_{11}(*s*_{1}) and *E*_{21}(*s*_{1}) of the emission matrix [A2.24] are:

and the Rayleigh determinant is still expressed by:

Replacing *s*_{1} by *s _{L}* sin

where the directivity factor *D*_{1L}(*θ*) has the same denominator as *D*_{2L}(*θ*):

Similarly, by replacing *s*_{1} by *s _{T}* sin

The denominator of *D*_{1T}(θ) is identical to that of *D*_{2T}(θ).

Figure 1.21(a) shows that there is no radiation in the form of a longitudinal wave along the normal to the surface. The emission is maximum in a direction depending on the ratio *κ* = *V _{T}*

Since the ratio *κ*^{−2} = (*V _{L}*

The energy velocity vector * V ^{e}* is at all points normal to the slowness surface:

vanishes if tan *θ ^{e}* =

The associated point L on the slowness curve of mode *J* defines the propagation angle *θ* through:

Setting *OM* = *r*: *x*_{1} = *r* sin *θ ^{e}*,

where *ψ* = *θ ^{e}* −

By using the relation d*s/*d*θ* = –*s* tan *ψ* (Figure 1.22) to evaluate the derivative of *s*_{1} = *s*(*θ*) sin *θ*, we obtain:

By introducing the wave number corresponding to the energy velocity we get:

The amplitude of the wave *J* transmitted in the direction *θ* is given by [1.132]:

The phase is proportional to the time taken by the energy to reach the observation point distant from *r*. This result agrees with the fact that in a point source and point receiver configuration, the mechanical disturbance propagates at the energy velocity in the source–receiver direction.

In addition to the product *s _{J}* (

Therefore, the exact calculation of the dynamic response of an anisotropic solid half-space to a local and transient solicitation is complex. An analytical solution consists of determining the Green’s function *G*(*s*_{1}*, x*_{2}*, p*) in the space of the Laplace transform for time (complex variable *p*) and in the space of the Fourier transform according to *x*_{1} (complex variable *k*_{1} = –*ips*_{1}). The inversion of these transforms in order to calculate the impulse response *g*(*x*_{1}*, x*_{2}*, t*), for example, by using the Cagniard-De Hoop method, requires rigorous mathematical developments (Poncelet and Deschamps 2009).

On the surface of an elastic solid, the disturbance arising from a line source is composed of a part corresponding to the head wave and of a part corresponding to the Rayleigh wave (Figure 1.18). Unlike the head wave, the amplitude of the Rayleigh wave does not decrease during propagation. Then, far from the source, the Rayleigh wave predominates.

Relation [1.113], namely *u _{i}* =

where is the Fourier transform of the spatial distribution *T _{j}*(

is the spectral Green’s function. The integral only depends on a single variable: the product *ωx*_{1}. For an isotropic solid, the *Y _{ij}* components take the form (Appendix 2):

where Δ(*s*_{1}) is the Rayleigh determinant. The normal slownesses *s*_{2L} and *s*_{2T} are imaginary:

so that the displacements of the Rayleigh wave
decrease with depth *x*_{2} *>* 0 in the material. The Rayleigh determinant [1.142] is written as:

Each solution *s*_{1} = ±*s _{R}* with

The poles ±*s _{R}* are located on the integration contour because we have neglected the attenuation. To take this effect into account, we must associate a positive imaginary part with the pole +

that is:

Using the property which states that ( dΔ/ d*s*_{1}) is an odd function *s*_{1}, the contribution of the negative pole (integrating clockwise around C_{−}) is:

The components of the matrix *N* (formulae [A2.27] in Appendix 2) are equal to:

and:

The sign of the displacement depends on the parity of the numerator. Since *N*_{22} (*N*_{21}) is even (odd), a normal (tangential) force creates an equal (opposite) normal displacement moving off the source. The calculation (which is quite long) of the derivative of the Rayleigh determinant Δ(*s*_{1}) for *s*_{1} = *s _{R}* leads to an expression that depends on the parameters

Substituting the expressions for *N*_{12} and *N*_{22} in equation [1.168] provides the surface displacements (*u*_{1} and *u*_{2}) of the Rayleigh wave generated by a line source of normal force density *F*_{2} (Lamb 1904; Achenbach 1987). Similarly, substituting *N*_{11} and *N*_{21} provides the Rayleigh wave displacements (*u*_{1} and *u*_{2}) generated by a line source of tangential force density *F*_{1}; for example, for *x*_{1} *>* 0:

Considering [1.172], the normal displacement is given by:

REMARK.–

- – For
*x*_{1}*>*0, the displacement*u*_{2}is negative, because a positive tangential force*F*_{1}compresses the material; in the (*x*_{1}*<*0) region, the same force creates a depression at the level of the free surface and*u*_{2}is positive. - – The ratio of amplitudes
*b*_{21}*/b*_{11}=*N*_{21}(*s*)_{R}*/N*_{11}(*s*) is equal to that of transverse and longitudinal components of the Rayleigh wave (relation [3.23] in Volume 1): the disturbance propagates on the surface without deforming it._{R} - –
*N*_{21}is real,*N*_{11}is imaginary: the two displacement components are in phase quadrature.

ORDER OF MAGNITUDE.– For duralumin, *ζ* = 1.145 and *κ* = 0.486, hence *R* = 0.384; with *μ* ≈ 25 GPa, the amplitude of the normal displacement of the Rayleigh wave created by a linear tangential force, of density *F*_{1} = 1 N/cm, is *b*_{21}*F*_{1} = 0.38 nm.

The time displacement, that is, the Fourier transform of *U*_{2}(*x*_{1}*, ω*):

is a Dirac function:

The mechanical displacement generated by a distribution of impulsive forces *F*_{1q}(*t*) is obtained through the convolution with *q*(*t*):

It has the same time-dependence *q*(*t*) as the force that created it, while the temporal form of the tangential component is the Hilbert transform of *q*(*t*) (Royer 2001).

In the case of a normal force of density *F*_{2}, since *N*_{12}(*s _{R}*) = −

Then, it is the normal component which is the Hilbert transform of *q*(*t*).

In practice, the propagation medium has finite dimensions. Let us study the generation in a plate, of thickness *d*, by a line source of normal forces. The observation point is located on the same side of the plate at a distance *ℓ* from the source. Assuming a Gaussian spatial distribution, uniform along *x*_{3}, of width *a* along *x*_{1}, the force density is of the form:

The function *q*(*t*) is normalized to unity and its duration at half-height is Δ*t* ≈ 2.4*τ*. The width at −3 dB of its frequency spectrum *Q*(*ω*) is Δ*f* ≈ 0.1*/τ* (Figure 1.24):

Compared to the case of a semi-infinite medium (section 1.2.1), the model takes into account all the upward waves (of amplitude *a _{I}*) and downward waves (of amplitude

The normal components of the wave vector (*k*_{2I} *, k*_{2J} ) and the polarizations (*p _{lI}*

Figure 1.25 shows the mechanical displacement generated in a stainless steel plate of thickness *d* = 4 mm, for *τ* = 10 ns and a force *F* = 10^{−2} N, that is, *F/a* = 100 N/m, when *a* = 0.1 mm. The tangential component *u*_{1} and normal component *u*_{2} were calculated at a distance *ℓ* = 6 mm on the same face as the source. For both components, the first signal H corresponds to the head wave that propagates on the surface at a velocity *V _{L}* = 5 634 m/s. The largest displacement R is that of the Rayleigh wave that propagates at the velocity

With a period *τ* = 10 ns, that is, Δ*f* = 10 MHz, and for a frequency *f* = 5 MHz, the wavelength *λ _{R}* =

The other echoes are due to the successive reflections with the possible conversion of bulk waves on each free surface (section 2.4.1, Volume 1). For example, the echo 1T1L corresponds to a transverse wave on the forward path and a longitudinal wave on the return path converted during the reflection on the opposite face (Figure 1.26). Their arrival times are predicted by a ray model, using the relation:

where integers *n* and *m*, whose sum is necessarily even, correspond to the number of single paths from one face to another for L and T waves, between the line source and the receiver. For equal times of flight (for example, when *n* = *m*: *t _{nT}*

The angles of incidence *θ _{L}* and

Since angles *θ _{L}* and

The echo H2T is different: it arises from the radiation of the head wave into a transverse wave at the critical angle *θ ^{L}* = arcsin(

Due to the energy loss through radiation toward the core of the plate, the head wave H and the echo H2T are rapidly attenuated when the distance *ℓ* increases.

In the case of a thin plate, whose thickness *d* is no longer large with respect to the wavelengths *λ _{R}*,

In this section, we calculate the mechanical displacement radiated in an isotropic medium by a moving sphere. This sphere can expand and contract along its radius, or oscillate in a rotational or translational motion.

Let *R _{s}* be the reference frame with spherical coordinates (

with:

where *φ _{L}*,

By assuming that the motions are independent of the angle *ϕ*, the components of the displacement field are expressed in spherical coordinates as:

Let us consider a sphere of radius *R* animated by a purely radial displacement (pulsating sphere). When the three potentials *φ _{L}*,

This is the solution to the equation:

hence:

The first term represents a divergent wave, the second one a convergent wave, with amplitudes *a _{L}* and

In the following, two cases are examined. If the sphere is animated by a radial displacement of amplitude *U*, the continuity of the displacement *u _{r}*(

or again, if

If the sphere is a cavity with a pressure *p*_{0} that is assumed to be uniform on its surface, the continuity of the radial stress *σ _{rr}*(

or again, if

Let there be a sphere animated by a rotational motion around the *z* axis, of angular amplitude Ω:

Since the movement of the sphere is, on the one hand, independent of the angle *ϕ* and, on the other hand, only polarized along the __e___{φ} axis, the wave excited by the rotation of the sphere derives only from the potential *ϕ _{S}*:

The displacement of the sphere imposes the following form for the potential *ϕ _{S}*:

where the function *f _{S}* satisfies the Helmholtz equation [A1.22]:

The solution is a linear combination of first-order spherical Hankel functions of the first and second kinds, that is:

These functions are associated with divergent and convergent waves. Only the divergent wave is generated by the motion of the sphere. With *b _{S}* = 0, the displacement is given by:

The continuity of the displacement *u*_{φ} on the surface of the sphere [1.197] imposes:

At low frequency and in the far field (*k _{T}*

Let there be a sphere animated by a translational motion of amplitude *U* along the *z* axis. In the reference frame with spherical coordinates, the displacement components at each point on the sphere are expressed by:

Since the motion of the sphere is independent of the angle *φ* and is contained in the (__e___{r}*, e*

The functions *f _{M}* (

and the non-zero components of the displacement are given by:

Considering expression [A1.39], the application of the boundary conditions at *r* = *R* leads to the following relations, by writing

The resolution of these equations leads to the expressions:

that is, when the longitudinal and transverse wavelengths are large compared to the radius *R* of the sphere (_{M} *<<* 1):

Given the limit of the function for large arguments (formula [A1.40]), the approximate expressions for the displacement components in the far field are:

The mechanical displacement is calculated in the far field for a bead buried in an epoxy matrix. The components *u _{r}* and

- 1 In the following and to simplify the notations, the reference to the angular frequency
*ω*is removed for all quantities. - 2 On the surface
*S*, the outgoing unit normal is −*e*_{z}, imposing a change of sign between equations [1.15] and [1.16].

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