Properties of Ultrasonic Signals in Common Materials PH380 Alasdair Moles November 30, 2021 Lab Dates: October 27 th to November 24 th 2021 Abstract Di ↵ erent materials exhibit unique characteristics influencing the ve- locity and attenuation of ultrasonic signals propagating a given medium sample. Common modes of propagation for ultrasonic signals are longi- tudinal compressive waves, transverse shear waves, and Rayleigh waves. This study is to analyse the properties that can be observed experimen- tally and analytically for said materials and wave propagation methods, and draw conclusions about the relationship between the mechanical prop- erties of a medium and the allowed transmission of wave-types generated via piezoelectric transducers. Figure 1: A water drop impact causing surface waves 1 1 Introduction and Underlying Concepts 1.1 Ultrasonic Waves An ultrasonic wave is defined as any sound wave with a frequency in the ’Ultra- sound’ region - above 20,000 Hertz. Sound waves in this category do not di ↵ er in terms of physical properties from conventional waves, and they are only defined as being ’Ultrasound’ in that they cannot be heard by human ears (1) Such waves are a powerful tool for imaging materials and spaces. For example, a bat will emit an ultrasonic ’pulse’ that will ricochet o ↵ of surfaces before returning to the bat, telling it roughly how far the surface is from its position by taking into consideration the time taken for the ’pulse’ to return. This principle of measuring response time is a common application of ultrasonics - and is used for the common medical scan known as a diagnostic medical sonography. Figure 2: A photo of a foetus captured through an ultrasound scan An ultrasound scan is performed by measuring the response time of waves emit- ted by an ultrasonic probe (also known as a transducer) and calculating the distance between the probe and the object that deflected it back to the trans- ducer (2) . These values for distance are then displayed in a light/dark palette, with longer response times being darker (as shown in fig 1). The equations used to analyse the distance between an ultrasonic source and a surface are as follows: v = d/t [1.1.1] v = f � [1.1.2] d = v/ 2 t [1.1.3] It is important to recognise that the time elapsed for the signal to return will correspond to travelling twice the distance, as it is a roundtrip. 2 1.2 Transducers Ultrasound waves can be produced by converting electric signals into ultrasonic signals via a device called a piezoelectric transducer. When the transducer receives an AC voltage from the supply, the size of the piezoelectric ceramic (certain crystals are also used) will fluctuate at the same frequency of the AC supply, producing a sound wave (this also works in reverse, if the ceramic is com- pressed or stretched - it will generate an electric charge). The ceramic changes in size due to the ’Piezoelectric E ↵ ect’ (3) , which arises from the electromechan- ical interaction between charges in the lattice of the material and a compressive mechanical force. The excess charges introduced by the incident voltage alter the polarisation of the material, introducing small poling forces. This deforms the piezoelectric substance by applying a mechanical stress as a result of the internal poling. Figure 3: A diagram of a piezoelectric transducer The resulting oscillation creates waves in essentially the same fashion as a di- aphragm component of a loudspeaker. The variation in pressure produced by the oscillating ceramic forms wavefronts that can propagate through a medium. Importantly, the piezoelectric substance is clad in a conducting metal, and a gel is used to remove any air pockets between the transducer and medium, allowing for greatened accuracy. The transducers used for the experiments performed in this study were transceivers, meaning they can both produce ultrasonic signals from an electric signal, and also read ultrasonic signals to produce an electric signal. There were five types of transducer used for the experiments in this study. Transducer Properties Transducer Type Model Type Angle Damping Type A, B Single Compression 0 o Low Type C Single Compression 0 o High Type D Angled Beam 37 5 o Medium Type E Angle Beam 65 o Medium 3 1.3 Longitudinal Compressive Waves A longitudinal wave (L-wave) is defined as a wave where the direction of propa- gation is identical to the direction of the periodic ’vibration’ in a medium. The frequency and wavelength of L-waves can be expressed by the following formula; where y is the displacement of the point on the wave, y 0 is the amplitude of the wave, ! is the angular frequency, t is the elapsed time, x is the displacement from the point to the source, and v is the speed of the wave (4) y ( x, t ) = y 0 cos ( ! ( t � x/v ))[1 3 1] Figure 4: A longitudinal wave on a spring As shown in figure 4, the peak of the waves represents the maximum ( y 0 ) of a cluster of vibrational density in the medium (5) , which travels along the axis of propagation. This is pertinent to what will be observed on the analysis of di ↵ erent materials speed of propagation. L-waves can propagate through most solids, liquids, and gases. If the thickness of a medium is known, and transceiver transducer is used measure the response time of its outgoing signal, the velocity of L-waves can be calculated through equation [1.1.1] taking into account that it is a roundtrip. 1.4 Transverse Shear Waves A transverse wave (T-wave) is defined as a wave where the oscillations of the wave are perpendicular to the direction of propagation (6) The vertical dis- placement of any point on a sinusoidal transverse wave can be expressed by the following equation; where p is the horizontal displacement of any particle in the medium of propagation, t is the time, A is the amplitude of the wave, T is the period of the wave, v is the speed of the waves propagation in the medium, and � is the phase of the wave. S ( p, t ) = A u sin (( t � ( p � b ) · d /v ) /T + � )[1 4 1] The variable d is vector describing the direction of travel, and the variable u represents the direction of the oscillations of the wave. The variable b represents a reference point within the medium of propagation that defines the amplitude, period, speed and phase. Unlike L-waves, T-waves can only propagate through solids, as gases and liquids do not possess the ability to produce forward-driving rigidity - which is essential as the medium must resist changes in its shape (7) In an elastic medium, any deformation caused by the propagation of the wave will return to its original position once the exciting energy of the wave is no longer present. In general, T-waves travel at around half the speed of L-waves. 4 1.5 Rayleigh Wave A Rayleigh wave (SAW - surface acoustic wave) is defined as a wave that travels at the surface of solids. These waves contain characteristics of both transverse and longitudinal waves. At the surface of the solid, the waves propagate in an elliptical motion in planes at normals to the surface of the medium - this causes the ’sea-like’ appearance of the surface. Deeper into the medium, the waves amplitude attenuates to zero, and the wave does not propagate at this depth (8) . In general, Rayleigh waves propagate slightly slower than transverse shear waves - and much slower (1 / 2 ⇥ ) than longitudinal compressive waves. Figure 5: A Rayleigh wave propagating on the surface of a medium Rayleigh waves are commonly produced in the earths ground by strong impact forces, and are one of the types of seismic waves produced by an earthquake. 1.6 Young’s Modulus Young’s Modulus ( E , also known as the modulus of elasticity) is an aspect of solids that defines the sti ↵ ness (resistance to deformities) of the material. This is intrinsically linked to the speed at which waves propagate a medium, as a more rigid material will propagate waves at a greater speed. The formula for deriving Young’s Modulus is as follows; where � is the tensile/compressive stress, and ✏ is the strain (10) E = � / ✏ [1.6.1] Stress essentially measures the internal forces that adjacent particle elements of a cohesive material exert on each other under a deforming force, whereas strain represents the amount of deformation that takes place in the material as a fraction of the total length of the material. The following equations describe these relationships; where F is the deforming force applied to the material, A is the cross-sectional area of the material, and L is the length of the material. � = F/A [1.6.2] ✏ = � L/L [1.6.3] The equation below and its proof, by consideration of the dimensional formula of its variables, shows that it can be used to derive the longitudinal compressive wave velocity in a material for a known Young’s modulus E and density ⇢ V = LT � 1 , E = M L � 1 T � 2 , ⇢ = M L � 3 LT � 1 = q M L � 1 T � 2 L 2 T � 2 LT � 1 = p L 2 T � 2 LT � 1 = LT � 1 Hence, v = p E/ ⇢ [1.6.4] is valid. 5 1.7 Poisson Ratio The poisson ratio of a material ( ⌫ ) is the application of the ’Poisson E ↵ ect’, measuring the deformation of a given material from a compressive or tensile force in the axis perpendicular to the action of the force. For a given axial strain (deformation in length of material) and subsequent lateral strain (deformation of cross-sectional area), so the poisson ratio describes a ratio of the deformations (11) , as shown by the following equation (12) ; where ✏ is a measure of strain in any direction in the material - and in any direction perpendicular to said direction. ⌫ = � d ✏ trans ✏ axial = � d ✏ y ✏ x = � d ✏ z ✏ x [1.7.1] Figure 6: A diagram showing the outward deformation of a material in response to a vertical compressive force 1.8 Lam ́ e Parameters, Strain Tensors, and Hooke’s Law The Lam ́ e parameters are a pair of material intrinsic properties given by the vari- ables � (Lam ́ e’s first parameter) and μ (Lam ́ e’s second parameter - synonymous with shear modulus) (13) . They describe Hooke’s law (scaling of force required to deform material with increasing distance) when considering 3-dimensional space by the formula shown below; where � is the stress, ✏ is the strain tensor, I is the identity matrix of size relating to the material, and tr is the trace function (sum of elements along a diagonal of ✏ ). � = 2 μ ✏ + � tr ( ✏ ) I [1.8.1] A strain tensor is an algebraic element used to define a set of directional strain vectors (14) in the x, y, z / i, j, k directions. Strain and stress are tensor quanti- ties, which means they follow the standard coordinate transformation qualities of tensors (15) In 3D space and tensor notation, strain can be written as the following: ✏ = " ✏ i ✏ ij ✏ ik ✏ ji ✏ jj ✏ jk ✏ ki ✏ jk ✏ kk # [1 8 2] 6 Equation [1.8.1] can be rewritten in tensor notation as follows; where � ij is the stress tensor, ✏ ij is the strain tensor, � is the first parameter, μ is the second parameter, and � ij is the Kronecker delta (function of two variables which equals 1 only if the two variables are equal - similar to the Dirac delta function). � ij = 2 μ ✏ ij + �� ij ✏ kk [1.8.3] This helps us understand that the compression forces on a sample from an oscil- lating transducer producing ultrasonic signals can be considered as directional. 1.9 Bulk and Shear Moduli The shear modulus is a measure of the elastic (able to return to original shape) shear sti ↵ ness of a material, and essentially is a ratio of the shear stress to the shear strain. The formula defining the shear modulus of a material is given by the following equation; where E is the Young’s modulus of the material and v is the Poisson ratio of the material. G = E 2(1+ ⌫ ) [1.9.1] The bulk modulus of a material is a measure of its resistance to a uniform compressive force (such as pressure) (16) It is defined by a ratio of the ob- jects decrease in volume when subjected to a uniform compressive force, to the magnitude of the force (pressure). The formula defining the bulk modulus of a material is given by the following equation; where E is the Young’s modulus of the material and ⌫ is the Poisson ratio of the material. K = E 3(2 � 2 ⌫ ) [1.9.2] The velocity of longitudinal compressive waves in a medium can be expressed as a relationship of the Bulk and Shear moduli, and the density of the material. The following equation expresses this, where K is the bulk modulus, G is the shear modulus, and ⇢ is the density of the material. v = q ( K +(4 / 3) G ) ⇢ [1.9.3] The bulk and shear moduli can also be used to express the Young’s modulus E of a material, as per the following equation (18) E = 9 KG 3 K + G [1.9.4] 1.10 Amplitude Attenuation in Materials The energy of waveforms propagating a solid will reduce in amplitude (20) (25) This attenuation is due to imperfections in the solids structure causing thermal consumption of the energy, acoustic scattering, energy absorption by the trans- ducer, and the intrinsic attenuation value of the material (19) The equation describing the voltage amplitude at n return pulses from a sample is as follows; where V n is the voltage amplitude at n returns, V 0 is the amplitude of the initial voltage, ↵ is the attenuation coe � cient (relevant for analysis), and D is the total distance travelled by the wave. V n = V 0 e � ↵ D [1.10.1] 7 1.11 Snell’s Law and Refraction The refractive index of a material is a ratio of the phase velocity of light and the speed of light, which tells us how fast light propagates in a medium relative to c . Shown below is this relationship; where n is the refractive index, c is the speed of light, and v is the phase velocity of the light. n = c/v [1.11.1] v = c/n [1.11.2] For a material with refractive index 1 5, light will propagate 1 5 times slower in that medium. The optical form for snells law relates the angle of incidence to the angle of refraction by a ratio of the refractive indexes of the two mediums. Shown below is the formula relating these two angles; where n 1 , n 2 are the refractive indices of the two mediums, and ✓ 1 , ✓ 2 are the angles of the light ray measured from the normal of the surface of the mediums. n 1 sin ✓ 1 = n 2 sin ✓ 2 [1.11.3] Given that acoustic waves are similar to electromagnetic waves (22)(23) , we can equate the refractive index in equation [1.11.3] to the relationship in equation [1.11.2] giving us Snell’s law in a form pertaining to acoustics; where v 1 , v 2 are the velocities of the acoustic waves propagating the two mediums. v 1 sin ✓ 1 = v 2 sin ✓ 2 [1.11.4] Rearranging equation [1.11.4] into a form pertaining to transducer wave creation allows us to evaluate the angle in which the acoustic signals from a transducer will propagate a medium given an angle from the transducer. sin ✓ sample = sin ✓ transducer ⇥ v sample v transducer [1.11.5] 8 2 Experiments 2.1 Longitudinal Compressive Wave Velocity A type A transducer coupled to a driver unit was set up to analyse a set of material samples with an oscilloscope as shown below, with coupling gel applied to the lower transducer plate. Figure 7: A diagram of the experimental setup Through the use of equation [1.1.3] the longitudinal compressive velocity of the materials was calculated. Material Measured Velocity Literature Value Brass 4402 68 ms � 1 ± 38 79 ms � 1 4372 ms � 1 Copper 4693 39 ms � 1 ± 41 36 ms � 1 4759 ms � 1 Aluminium 6450 32 ms � 1 ± 56 85 ms � 1 6374 ms � 1 Perspex (acrylic) 2636 64 ms � 1 ± 23 32 ms � 1 2700 ms � 1 Nylon 2423 25 ms � 1 ± 21 35 ms � 1 2680 ms � 1 Mild Steel 5737 97 ms � 1 ± 50 56 ms � 1 5960 ms � 1 Shown over are graphs that show the relationship between the Young’s Modulus and density values (taken from www.engineeringtoolbox.com) of the materials and the longitudinal compressive wave velocity they exhibit. The velocity was also plotted as a function of relationship given by equation [1.6.4] to visualise any potential trends. 9 Figure 8: Graph of relationships between Young’s modulus, density and velocity of longitudinal compressive waves We can see from this trend that an increased rigidity corresponds to an increase in longitudinal compressive wave velocity as discussed in section 1.6. By using equation [1.9.3] we receive a more empirical display of the linear relationship rigidity has on the wave propagation speed. Figure 9: Linear e ↵ ect of density and rigidity of velocity 10 2.2 Transverse Shear Wave Velocity A pair of type E (45 o ) angle beam transducers in a transmitter/receiver setup were used, along with a long cuboid of aluminium. The experiment was set up as shown below, with coupling gel used to bridge the gap between the transducers and the aluminium. Figure 10: Experimental setup to measure transverse shear wave velocity in aluminium The position of the receiver transducer was variated along the aluminium sam- ple, beginning next to the transmitter transducer and increasing distance. The positions that received a signal were noted down. The velocity of the wave was calculated through the following method; where x is the distance between re- ceived signals, T is the thickness of the material sample, and � t is the period of the signals. v transverse = 2 p 0 5 2 x + T � t [2.2.1] The experimental data was analysed using equation [2.2.1] and produced a trans- verse shear wave velocity of 3047 56 ms � 1 ± 80 99 ms � 1 , while the literature value for this speed is 3111 ms � 1 . The absence of longitudinal compressive wave propagation was also analysed by consideration of the critical angle value for aluminium (di ↵ racted wave does not produce wave type and energy is con- verted) using the following equations (derived from equation [1.11.5]); where ✓ critical 1 st is the angle of incidence that produces a critical angle for this wave type, v longitudinal t and v longitudinal s are the longitudinal compressive wave ve- locities of the transducer (perspex) and the sample (in this case aluminium) respectively. ✓ critical 1 st = sin � 1 ( sin ( ✓ = 90 o ) ⇥ v longitudinalt v longitudinals [2.2.2] ✓ critical 1 st = 90 ⇥ v longitudinalt v longitudinals [2.2.3] From this formula, we find that longitudinal compressive waves reach a critical angle at 24 12 o Since the angle of incidence required to reach a critical an- gle for the propagation of longitudinal compressive waves is less than the angle produced by the transducer (37 50 o ) this type of wave will not propagate in aluminium. If instead we use formula [2.2.3] to calculate the required angle of incidence to create critical angle for transverse shear waves ( v longitudinal s to v shear s ) we return the value of 59 90 o , showing that transverse shear waves do indeed propagate through aluminium at this angle. The allowed wave propaga- tion modes of the material brass were also analysed. The shear wave velocity in brass was calculated to be 2154 24 ms � 1 ± 55 73 ms � 1 , while the literature value is 2100 ms � 1 Using equation [2.2.2] the longitudinal compressive waves were found to reach a critical angle at 38 14 o , which means that some longitudinal compressive waves will propagate the medium, as 38 14 o > 37 50 o . The angle at 11 which shear waves reach the critical angle is invalid, meaning that the angle of the transducer would have to be above 90 o and would go into the surface of the material, hence making it impossible by means of this experimental setup (24) 2.3 Rayleigh SAW Velocity A pair of type D (65 o ) transducers were set up in the same fashion as shown in figure 10 with the same sample of aluminium from section 2.2. The position of the receiver transducer was variated in spaced increments away from the transmitter, with the length and time di ↵ erence between points being recorded. Once the values were recorded, a linear curve fit was performed to receive a value for the Rayleigh wave velocity via the relationship given in equation [1.1.1] ( v = d/t ). Figure 11: Velocity Curve fit for Rayleigh wave velocity The velocity for the Rayleigh waves in the aluminium was calculated to be 2867 64 ms � 1 ± 37 41 ms � 1 . The literature value for the Rayleigh wave velocity in aluminium is 2906 ms � 1 . Similar to the experiment in section 2.2, the allowed propagation of wave types for this setup was analysed using equation [2.2.3]. The longitudinal compressive waves still do not propagate as 24 12 o < 65 o . The transverse shear waves in aluminium do not propagate for this angle of trans- ducer, as 59 9 o < 65 o . This leaves only the Rayleigh wave mode of propagation. 2.4 Longitudinal Compressive Wave Attenuation The experimental apparatus was set up as shown in figure 7, using a sample of brass. Using the oscilloscope, the voltage amplitude of the diminishing pulse responses were recorded. The relationship given by equation [1.10.1] was rear- ranged into the following form using logarithmic identities, and a linear curve fit was used to find the attenuation in terms of decibelage. 1 D = ↵ ⇥ 1 � ln vn vo [2.4.1] 12 Figure 12: Curve fit for finding attenuation coe � cient The value derived from this fit for the attenuation was 4 64 dB/m ± 0 53 dB/m This value in nepers is calculated by the following conversion formula; where x is the attenuation in decibels. N p = x ⇥ 20 log 10 e [2.4.2] This gives an attenuation value of brass of in nepers of 0 5342 nepers/cm ± 0 0576 nepers/cm 3 Discussion 3.1 Experiments There are not immediate relationships apparent between the velocity of L-wave velocity and either the increasing Young’s modulus of materials, or the density of materials. Through analysis of the data in section 2.1, a linear relationship be- tween equation [1.6.4] ( v = p E/ ⇢ ) and the L-wave velocity is evident, showing that there is an increase in wave propagation speed with an increase in material rigidity. This relationship also tells us that with an increase in density ⇢ the wave propagation speed will decrease. This aligns with the model of L-wave velocity, as a more rigid medium will impart acoustic signals through moments within the material faster, while an increase in moments (through an increased density) will require the signal to cross more ’material’ to propagate the same distance in the medium 1 . The relationship given by equation [1.9.3] shows this e ↵ ect also, instead taking into account the bulk K and shear G moduli, which are elements of the elasticity modulus E shown by equation [1.9.4]. 1 This can be visualised with the spring/slinky in figure 4 13 The second experimental process yielded T-wave velocity in aluminium that was around half the value for the the L-wave velocity in aluminium. This is ex- pected as the shear modulus of the material is less than the bulk modulus. The disallowed transmission of L-waves waves was found to be due to the critical angle for the velocity of L-waves being less than that of the angle produced by the transmission of the ultrasonic signal into the medium via the perspex. The critical angle needed to remove T-wave propagation from the medium was not reached by the angle of the transducer, hence the T-waves were detected. For the sample of brass, since there is propagation of some longitudinal compressive waves, the signal measured by the transducer receiver was cluttered with noise, making it challenging to produce a value for the T-wave velocity. This poten- tially could have contributed to the measured value being significantly disparate to the literature value. It is interesting to note that the unique acoustic proper- ties of brass is perhaps why it is favoured for musical Instruments, as its slower speed of sound propagation due to its softness would be less likely to lead to resonances in the body of the instrument than other materials. It was found that there is no allowed angle of incidence by the transducer that would cause the T-waves in brass to reach a critical angle. The value for the Rayleigh wave velocity in aluminium was less than the T-wave velocity, and around half that of the L-wave velocity, as expected. The analysis of the required angle of incidence from the transducer to produce critical angles for L-waves and T-waves show why only the Rayleigh SAW waves can propa- gate, given the angle of the transducer of 65 o The attenuation coe � cient for L-waves in brass was found to be 0 5342 nepers/cm (by use of a curve fit to equation [2.4.1]), but sources describing the attenuation coe � cient for materials state that the value of ↵ is dependant on the frequency of the transmitted ultrasonic signals - which was not recorded during the exper- imental procedure. Nevertheless, this value is within the ballpark range for the attenuation in metals. A value for the attenuation of brass specifically could not be found to compare with the measured result. 3.2 Accuracy of values and Uncertainties The oscilloscope used to find the time di ↵ erence between return pulses did not have a very granular scale, meaning that the uncertainty for � t was greatly heightened. Despite this, some of the values for L-wave velocity (in the mate- rials analysed) when considering the range of their calculated uncertainty, do not eclipse the literature value for the velocity. This implies the presence of a systematic error, potentially from faults in the transducers used, the driver unit used, or any experimental apparatus. The T-wave and Rayleigh wave uncer- tainty also su ↵ ered from lack of granularity in the time scale of the oscilloscope, producing a larger uncertainty than was desired. An uncertainty value for the angle on incidence into a material was not calculated as these values only were analysed to prove the transmission (or non-transmission) of wave types in the material. The uncertainty produced for the value of the attenuation in brass did not have the problem of granularity, because the voltage amplitude scale in the y-axis in the oscilloscope gave a more specific reading. Overall, to improve the results acquired, a more high fidelity oscilloscope/setting should be used to reduce the time uncertainty, and the source of the systematic error should be determined (it is likely the transducer will absorb energy to cause 14 this). Also a range of attenuation values for di ↵ ering frequencies of ultrasonic waves should be recorded to observe the e ↵ ect altering frequencies has on the attenuation coe � cient. The coupling gel used in the experiments could also be a source of error, as trapped air bubbles could alter the results, a standardised quantity of gel might help reduce error in this case. 4 Bibliography Websites accessed November 23, 2021 till November 30, 2021 (1) en.wikipedia.org/wiki/Ultrasound (2) www.nde-ed.org/NDETechniques/Ultrasonics/immersion.xhtml (3) www.electronicdesign.com/power-management/article/21801833/what-is-the-piezoelectric-effect (4) en.wikipedia.org/wiki/Longitudinal_wave (5) novotest.biz/basics-of-acoustics-1-4-types-of-waves-and-laws-of-propagation-of\ -acoustic-waves-acoustic-field/ (6) www.britannica.com/science/shear-wave (7) www.homepages.ucl.ac.uk/~uceseug/Fluids2/Notes_Viscosity.pdf (8) en.wikipedia.org/wiki/Rayleigh_wave (9) www.engineeringtoolbox.com/young-modulus-d_417.html (10) www.birmingham.ac.uk/teachers/study-resources/stem/Physics/youngs-modulus (11) www.sciencedirect.com/topics/engineering/poissons-ratio (12) en.wikipedia.org/wiki/Poisson’s_ratio (13) en.wikipedia.org/wiki/Lame_parameters (14) www.continuummechanics.org/strain.html (15) serc.carleton.edu/NAGTWorkshops/mineralogy/mineral_physics/tensors.html (16) www.engineeringtoolbox.com/bulk-modulus-metals-d_1351.html (17) www.engineeringtoolbox.com/modulus-rigidity-d_946.html (18) en.wikipedia.org/wiki/Elastic_modulus (19) www.phase-trans.msm.cam.ac.uk/2000/amjad/b.pdf (20) www.sciencedirect.com/topics/earth-and-planetary-sciences/acoustic-wave (21) www.acs.psu.edu/drussell/demos/refract/refract.html (22) www.researchgate.net/publication/233601571_Snell’s_Law_of_Refraction_and_ Sound_Rays_for_a_Moving_Medium (23) hyperphysics.phy-astr.gsu.edu/hbase/phyopt/huygen.html (24) https://www.desmos.com/calculator/f5hhtz4bpg , (visualisation of angle within brass to produce transverse shear waves) (25) jontalle.web.engr.illinois.edu/uploads/473.F18/Lectures/Chapter_8.pdf (26)Kaye, G. and Laby, T., 1973. 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