Method for determining the sound velocity in a basic material, particularly for measuring the thickness of a wall

0001The invention relates to a method for determining the speed of sound Cb in a base material using an ultrasound test head which has a transmission oscillator, a reception oscillator and a lead body, the lead body a) having a coupling surface with which the test head can be coupled to the base material, b) receives the receiving transducer and the transmitting transducer and c) has a speed of sound Cv, the transmitting oscillator and the receiving oscillator are each aligned obliquely to one another and obliquely to the coupling surface at an entry angle, so that a main transmission direction of the transmitting oscillator and a main receiving direction of the receiving oscillator intersect below the coupling surface, the transmitting oscillator and receiving oscillator have a center distance K from one another, the transmitting oscillator has a center distance Ds from the coupling surface and the receiving oscillator has a distance De from the coupling surface, in which method an ultrasound pulse is generated by the transmitting oscillator, runs through the lead body into the base material, causes a creeping wave there and a part of it via the lead body Reception vibrator reached.

0002The determination of the speed of sound Cb is a prerequisite for being able to determine the wall thickness of the base material. Although it is known to determine the wall thickness of a base material by multiple reflection of an impulse at an entry surface and a rear surface of the base material, this method requires sufficiently reflective and thus smooth surfaces, in particular a sufficiently smooth rear surface, so that it runs back and forth several times comes in the base material. This method cannot be used for rough rear surfaces, rather one has to rely on a one-time back and forth run. The wall thickness can then be determined via the speed of sound Cb.

0003Out <patcit id="pcit0001" dnum="US6035717A"><text>US 6,035,717</text></patcit> a method and a device for determining the thickness of a coated base material are known. In this method, the uncoated base material is first measured for the determination of the speed of sound Cb of the base material; an impulse is transmitted from a transmitter oscillator through the lead body into the base material, where a creeping wave is generated, a portion of which decouples and is received by the receiver transducer . In this respect, there is agreement with the invention.

0004The path of this impulse is according to <patcit id="pcit0002" dnum="US6035717A"><text>US 6,035,717</text></patcit> but now accepted as fixed. It appears that the inventor of this US patent was well aware that this assumption of a geometrically fixed path along the main rays brings certain inaccuracies in the determination of the speed of sound Cb. He therefore suggests in practical instructions that the distances between the two transducers from the coupling surface be kept as small as possible. Indeed, this makes the determination of the speed of sound in the base material more precise, in other words, the inaccuracy is reduced. However, a test head with a short sound path of the lead body has the disadvantage that only a little material of the lead body is available for the wear of the lead body that occurs during every practical test, so the test head must be replaced earlier than a test head with a larger lead distance.

0005This is where the invention begins. It has set itself the task of proceeding according to the<patcit id="pcit0003" dnum="US6035717A"><text>US 6,035,717 A</text></patcit> to further develop in such a way that the speed of sound Cb in the base material is determined more precisely, so that the thickness of a layer on this base material can also be determined more precisely, and that a test head can be used which can have a sufficiently long lead section for practical use.

0006This problem is solved on the basis of the features mentioned at the outset and including them by measuring the shortest sound propagation time Ttot that occurs with the different sound paths that differ in the entry angle, and the speed of sound Cb in the base material is determined by the path that exists between the transmitter oscillators and receive transducer delivers the shortest total running time Ttot depending on the speed of sound.

0007This method takes into account the fact that the path that the impulse takes through the lead body, along the surface of the base material (as a surface wave) and back through the lead body is influenced in addition to the known variables K, Dv and Cv (K = center-to-center distance of the contact surfaces of the transducers, Dv = center-to-center distance of the contact surface of a transducer from the coupling surface) by the speed of sound Cb. If this is relatively large in comparison to the speed of sound Cv in the leading body, the proportion of the running distance Sb along the surface of the base material also becomes relatively large. If, on the other hand, the speed of sound Cb in the base material is relatively low, the distance Sb of the surface wave in the base material becomes relatively short, whereas the distance Sv within the lead body becomes longer. Similar conditions also exist for the refraction of light between different optical media, for example water and air. In this case too, the geometrically shortest path is not the shortest path in time for a light pulse.

0008The achievement of the invention is to have recognized that the detection of the shortest transit time Ttot of the ultrasonic pulse and the optimization of all possible sound paths towards the sound path that delivers the shortest total run time as a function of Cb provide a precise statement about the speed of sound Cb im Supplies basic material. The invention is therefore based on the actual paths that a sound pulse travels. It makes no assumptions about the way like this in the <patcit id="pcit0004" dnum="US6035717A"><text>US 6,035,717 A</text></patcit> the case is. The errors of this known measuring method and the corresponding device are therefore avoided according to the invention.

0009Further advantages and features of the invention emerge from the remaining claims and the following. Description of non-restrictive exemplary embodiments of the invention, which are explained in more detail below with reference to the drawing, the method according to the invention also being explained. The drawing shows:<dl id="dl0001"><dt>Figure 1:</dt><dd>A basic representation in side view of a test head with two transducers, which is coupled to a base material, the individual sections of the total path are shown,</dd><dt>Figure 2:</dt><dd>the representation according to <figref idref="f0001">Fig. 1</figref> with marked routes, sound speeds etc.,</dd><dt>Figure 3:</dt><dd>a representation like <figref idref="f0001">Figure 1</figref>, however, the base material now additionally has a thin layer (a coating), for example a color, a metal coating or a plastic coating, and</dd><dt>Figure 4:</dt><dd>a representation similar <figref idref="f0001">Figure 1</figref>, but now with two additional transducers for a wall thickness measurement.</dd></dl>

0010The in <figref idref="f0001">Figure 1</figref> The test head shown has a specially shaped, essentially prismatic lead body 20. This has a flat coupling surface 22, also called an active surface, and opposing bevels, on which a transmission oscillator 24 or a reception oscillator 26 are held, in particular cemented on. Both oscillators 24, 26 are identical. They are arranged at an angle to one another and also at an angle to the coupling surface 22. This arrangement is discussed in more detail below.

0011A perpendicular line, i.e. a line running at right angles to the contact surface of the vibrator with the lead body 20 and through the center of this contact surface of the vibrator runs at a certain angle to the coupling surface 22, this angle is (90 ° -α_v) and is the same for both transducers 24, 26. Furthermore, the respective perpendicular bisectors lie in the same plane, namely in the plane of the<figref idref="f0001">Fig. 1</figref>.

0012This can also be expressed differently: the two oscillators 24, 26 are arranged in a folding symmetry with respect to a plane of symmetry 32. They are so inclined to the coupling surface 22 that a surface wave 35 is generated in a base material 34 to which the lead body 20 is coupled by suitable means known per se, which will be discussed in more detail below.

0013A separating layer 36 provided essentially along the plane of symmetry 3 2 ensures that direct crosstalk (cross talk) between the transmit oscillator 24 and the receive oscillator 26 is prevented.

0014The specified perpendiculars usually coincide with a main beam, ie a main transmission beam 38 and a main reception beam 40.

0015The Schallge speed Cv in the lead body 20 is known. The distance K between the surface centers of the two oscillators 24, 26 is also known. Finally, the distance of the surface center of the transmitter oscillator 24 from the coupling surface 22 and the distance of the center of the reception oscillator 26 from the coupling surface 22 can be determined and are thus known. Because of the symmetry, both have the value Dv. With the help of only these specifications, it is now possible to determine the speed of sound Cb in the base material 34. In a further step, the thickness, that is to say the wall thickness Db, of the base material 34 can then be determined.

0016If the speed of sound Cb in the base material 34 is approximately as high as the speed of sound of steel, the shortest path of a sound pulse from the transmitter oscillator 24 to the receiver oscillator 26 is as follows: the pulse runs along the main transmission beam, then as a surface wave 35 in the base material 34 and finally along again of the main receive beam 40 into the receive transducer 26. This path is in <figref idref="f0001">Fig. 1</figref> drawn in dashed lines, it runs along the main transmission beam 38 and the main reception beam 40.

0017However, if the speed of sound Cb in the base material 34 is lower than that of steel, the sound path will use as much distance as possible within the lead body 20, the length of the distance Sb, which is realized by the surface wave 38 in the base material 34, will be short. This case is in<figref idref="f0001">figure <b>1</b></figref> represented by a dotted path 42.

0018Conversely, if the speed of sound Cb in the base material is greater than that of steel, the sound path Sv within the lead body 20 becomes short in favor of a longer path Sb than surface wave 35. This case is in <figref idref="f0001">Figure 1</figref> represented by the dash-dotted sound path 44.

0019To simplify the illustration, in <figref idref="f0001">Figure 1</figref> only the complete sound path, which runs along the main rays 38, 40, is shown in dashed lines. It can be seen that the running distance Sb of the surface wave 35 is a function of the speed of sound Cb in the base material 34 and also depends on the constant quantities K, Cv and Dv. According to the invention, the speed of sound Cb in the base material 34 is obtained by optimizing the associated sound path. It is based on the sound path that provides the shortest total running time Ttot as a function of the speed of sound Cb to be determined.

0020Now the amplitude of the sound pressure decreases, the larger the angle to the main beam. However, if one only measures the signal with the shortest total transit time Ttot, one is independent of the amplitude of the received signal within certain limits. It would be ideal if the transducers 24, 26 were spherical emitters, but this is not the case. Within the sound velocities that occur in practice, the influence of the non-spherical radiation of the vibrators 24, 26 is not so noticeable that one would have to take this into account and evaluate it specifically. The alignment of the oscillators 24, 26 is ideally done for an average value of the speed of sound Cb (for example for steel Cb about 6000 m / s).

0021The respective path that the sound pulse takes with the shortest total running time Ttot is therefore a function of the speed of sound Cb and furthermore depends on the known values ​​K, Dv and Cv. In the case of a structure that is not symmetrical about the fold, the different center-to-center distances of the transmit oscillator 24 and the receive oscillator 26 from the coupling surface 22 must be taken into account.

0022According to <figref idref="f0001">Figure 1</figref> The ultrasound propagates from the transmitter oscillator 24 via a first path section Sv to the base material 34, for which it requires the time Tv. A creeping wave 35 is generated there. She has the in<figref idref="f0001">Figure 2</figref> specified length Sb. This length is run through in time Tb. A portion of the creeping wave reaches the receiving oscillator 26 via a path which, due to the symmetry, has the length Sv and for which the time Tv is required.

0023The search is for the distance Sb between the start point and the end point of the creeping wave or surface wave 35 for longitudinal waves:<ul id="ul0001" list-style="none" compact="compact"><li>General formulas for sound propagation are set out below. Only the shortest total running time Tto-t is taken into account. In order to determine this, it is necessary to take into account the entire running distance from the transmitting oscillator 24 to the receiving oscillator 26. At least constant over time for the short duration of the measurement, and also assumed to be known, K = distance from the center of the test heads 24, 26; Dv = center distance of the transducers 24, 26 from the coupling surface 22 and Cv = speed of sound in the leading body.</li></ul>

0024An ultrasound pulse emitted by the transmitter oscillator 24 causes not only a surface wave 35 in the base material 34, but also further waves, the longitudinal surface wave 35 has the shortest running distance and also the shortest total running time Ttot.

0025According to <figref idref="f0001">Figure 2</figref> applies: <maths id="math0001" num="(1)"><math display="block"><msub><mi>S</mi><mi>b</mi></msub><mo>=</mo><mi>K</mi><mo>-</mo><mn>2</mn><mo>⁢</mo><msub><mi>K</mi><mi>b</mi></msub><mo>;</mo></math><img file="EP1525430B1_D0001.tif" /></maths><maths id="math0002" num="(2)"><math display="block"><mi>tan</mi><msub><mi>α</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>K</mi><mi>b</mi></msub><msub><mi>D</mi><mi>v</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>K</mi><mi>b</mi></msub><mo>=</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><msub><mi>α</mi><mi>v</mi></msub><mo>;</mo></math><img file="EP1525430B1_D0002.tif" /></maths><maths id="math0003" num="(3)"><math display="block"><msub><mi>S</mi><mi>b</mi></msub><mo>=</mo><mi>K</mi><mo>-</mo><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><msub><mi>α</mi><mi>v</mi></msub><mo>;</mo><mspace width="3em" /><mfenced><mn>1</mn><mo>/</mo><mn>2</mn></mfenced></math><img file="EP1525430B1_D0003.tif" /></maths>

0026Different sound paths differ in the entry angle α_v. This is obtained assuming the shortest possible total term Ttot:<maths id="math0004" num="(4)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mn>2</mn><mo>⁢</mo><msub><mi>T</mi><mi>v</mi></msub><mo>+</mo><msub><mi>T</mi><mi>b</mi></msub><mo>;</mo></math><img file="EP1525430B1_D0004.tif" /></maths><maths id="math0005" num="(5)"><math display="block"><msub><mi>C.</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>S</mi><mi>v</mi></msub><msub><mi>T</mi><mi>v</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>T</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>S</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>v</mi></msub></mfrac><mo>;</mo></math><img file="EP1525430B1_D0005.tif" /></maths><maths id="math0006" num="(6)"><math display="block"><msub><mi>C.</mi><mi>b</mi></msub><mo>=</mo><mfrac><msub><mi>S</mi><mi>b</mi></msub><msub><mi>T</mi><mi>b</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>T</mi><mi>b</mi></msub><mo>=</mo><mfrac><msub><mi>S</mi><mi>b</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo></math><img file="EP1525430B1_D0006.tif" /></maths><maths id="math0007" num="(7)"><math display="block"><mi>cos</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>D</mi><mi>v</mi></msub><msub><mi>S</mi><mi>v</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>S</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>D</mi><mi>v</mi></msub><mrow><mi>cos</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>;</mo></math><img file="EP1525430B1_D0007.tif" /></maths><maths id="math0008" num="(8)"><math display="block"><msub><mi>T</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>D</mi><mi>v</mi></msub><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><mi>cos</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>;</mo><mspace width="3em" /><mfenced><mn>5</mn><mo>/</mo><mn>7</mn></mfenced></math><img file="EP1525430B1_D0008.tif" /></maths><maths id="math0009" num="(9)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub></mrow><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><mi>cos</mi><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>K</mi><mo>-</mo><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mspace width="3em" /><mfenced><mn>4</mn><mo>/</mo><mn>8</mn><mo>/</mo><mn>3</mn><mo>/</mo><mn>6</mn></mfenced></math><img file="EP1525430B1_D0009.tif" /></maths><maths id="math0010" num="(10)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mfrac><mi>K</mi><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>+</mo><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub><mo>⁢</mo><mfenced><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><mi>cos</mi><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced><mo>;</mo><mspace width="3em" /><mfenced><mn>9</mn></mfenced></math><img file="EP1525430B1_D0010.tif" /></maths>

0027We are looking for a minimum of the linear function Ttot (αv). This can be done, for example, using the first derivative based on the angle (α_v are determined, the first derivative must be α for a certain angle_v be zero, the second derivative must be positive: <maths id="math0011" num="(11)"><math display="block"><mfrac><mrow><mo>∂</mo><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mfenced><msub><mi>α</mi><mi>v</mi></msub></mfenced></mrow><mrow><mo>∂</mo><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>=</mo><mn>0</mn><mo>!</mo></math><img file="EP1525430B1_D0011.tif" /></maths><maths id="math0012" num="(12)"><math display="block"><mfrac><mrow><mo>∂</mo><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mfenced><msub><mi>α</mi><mi>v</mi></msub></mfenced></mrow><mrow><mo>∂</mo><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>=</mo><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub><mo>⁢</mo><mfenced><mfrac><mrow><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><msup><mi>cos</mi><mrow><mn>2</mn><mspace width="1em" /></mrow></msup><mo>⁢</mo><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>b</mi></msub><mspace width="1em" /><msup><mi>cos</mi><mrow><mn>2</mn><mspace width="1em" /></mrow></msup><mo>⁢</mo><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac></mfenced><mo>=</mo><mn>0</mn><mo>;</mo><mspace width="3em" /><mfenced><mn>10</mn></mfenced></math><img file="EP1525430B1_D0012.tif" /></maths><maths id="math0013" num="(13)"><math display="block"><mfrac><mrow><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><msub><mi>C.</mi><mi>v</mi></msub></mfrac><mo>-</mo><mfrac><mn>1</mn><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>=</mo><mn>0</mn><mo>;</mo><mo>→</mo><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>α</mi><mi>v</mi></msub><mo>=</mo><mi>arcsin</mi><mfenced><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced><mo>;</mo><mspace width="3em" /><mfenced><mn>12</mn></mfenced></math><img file="EP1525430B1_D0013.tif" /></maths>

0028If one now takes into account (3), it can be seen that the sound path Sb is dependent on the two sound velocities Cv and Cb: <maths id="math0014" num="(14)"><math display="block"><msub><mi>S</mi><mi>b</mi></msub><mo>=</mo><mi>K</mi><mo>-</mo><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><mfenced><mi>arcsin</mi><mfenced><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mfenced><mo>;</mo><mspace width="3em" /><mfenced><mn>3</mn><mo>/</mo><mn>12</mn></mfenced></math><img file="EP1525430B1_D0014.tif" /></maths>

0029Since K, Dv and Cv can be assumed to be constant, this means: Sb = f (Cb). The following equation (15) now describes the relationship between the measured total time Ttot and the sound speed Cb to be determined:<maths id="math0015" num="(15)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mfrac><mi>K</mi><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>+</mo><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub><mo>⁢</mo><mfenced><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><mi>cos</mi><mfenced><mi>arcsin</mi><mfenced><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mfenced></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>tan</mi><mfenced><mi>arcsin</mi><mfenced><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mfenced></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced><mo>;</mo><mspace width="3em" /><mfenced><mn>9</mn><mo>/</mo><mn>12</mn></mfenced></math><img file="EP1525430B1_D0015.tif" /></maths>

0030This creates a clear connection between the total running time Ttot and the sound velocity Cb to be determined. All other quantities in equation (15) are known and constant.

0031Thus, the determination of the speed of sound Cb in the base material is clearly possible over the total running time Ttot. With this knowledge, the wall thickness Db of the base material can now be determined.

0032<figref idref="f0002">Figure 3</figref> shows the arrangement from the previous figures, but there is now additionally a layer 46, a so-called coating on the base material 34. This layer has a thickness Ds. It should be determined using the sound speed Cs of the layer. Since this is also unknown, the speed of sound is first determined as before.

0033<figref idref="f0002">Figure 3</figref> shows again only the sound path with the shortest total running time Ttot. There are still other propagations, for example a surface wave is also generated on the surface of the coating 46 facing the coupling surface 22, but this should arrive in the base material 34 after the surface wave 35. This means that the speed of sound Cb in the base material 34 must be sufficiently greater than the speed of sound Cs in the layer 46. In practice, this is mostly the case. The base material is typically a metal, the speed of sound is 4500 to 7000 m / s. Layer 46 is typically a plastic, paint, and the like, and the speeds of sound are typically 2000 to 3000 ms. In the event that the speed of sound Cs in the layer 46 is relatively high, for example the layer is a metal coating on a base material made of plastic, the layer consists of a metal of higher speed of sound than the base material, e.g. B. Coating in Ag, base material in Au, must be worked according to the previous considerations by simply using the layer 46 as the base material.

0034In the following, the in <figref idref="f0002">Figure 3</figref> shown sound path as the shortest path. The entry angle α_v is changed in layer 4 6 to αs. The Laufstrec ken in the volume of layer 46 result from<figref idref="f0002">Figure 3</figref>, they are Ss. The associated sound propagation time is Ts.

0035The following then applies to the shortest total term: <maths id="math0016" num="(16)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mn>2</mn><mo>⁢</mo><mfenced><msub><mi>T</mi><mi>v</mi></msub><mo>+</mo><msub><mi>T</mi><mi>s</mi></msub></mfenced><mo>+</mo><msub><mi>T</mi><mi>b</mi></msub><mo>;</mo></math><img file="EP1525430B1_D0016.tif" /></maths><maths id="math0017" num="(17)"><math display="block"><msub><mi>C.</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>S</mi><mi>v</mi></msub><msub><mi>T</mi><mi>v</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>T</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>S</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>v</mi></msub></mfrac><mo>;</mo></math><img file="EP1525430B1_D0017.tif" /></maths><maths id="math0018" num="(18)"><math display="block"><mi>cos</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>D</mi><mi>v</mi></msub><msub><mi>S</mi><mi>v</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>S</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>D</mi><mi>v</mi></msub><mrow><mi>cos</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>;</mo></math><img file="EP1525430B1_D0018.tif" /></maths><maths id="math0019" num="(19)"><math display="block"><msub><mi>T</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>D</mi><mi>v</mi></msub><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><mi>cos</mi><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>;</mo><mspace width="3em" /><mfenced><mn>17</mn><mo>/</mo><mn>18</mn></mfenced></math><img file="EP1525430B1_D0019.tif" /></maths>

0036The conditions in layer 46 are corresponding, for layer 46 the following applies: <maths id="math0020" num="(20)"><math display="block"><msub><mi>T</mi><mi>s</mi></msub><mo>=</mo><mfrac><msub><mi>D</mi><mi>s</mi></msub><mrow><msub><mi>C.</mi><mi>s</mi></msub><mspace width="1em" /><mi>cos</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub></mrow></mfrac><mo>;</mo><mspace width="3em" /><mfenced><mn>19</mn></mfenced></math><img file="EP1525430B1_D0020.tif" /></maths><maths id="math0021" num="(21)"><math display="block"><msub><mi>T</mi><mi>b</mi></msub><mo>=</mo><mfrac><msub><mi>S</mi><mi>b</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo></math><img file="EP1525430B1_D0021.tif" /></maths><maths id="math0022" num="(22)"><math display="block"><msub><mi>S</mi><mi>b</mi></msub><mo>=</mo><mi>K</mi><mo>-</mo><mn>2</mn><mo>⁢</mo><mfenced><msub><mi>K</mi><mi>b</mi></msub><mo>+</mo><msub><mi>K</mi><mi>s</mi></msub></mfenced><mo>;</mo></math><img file="EP1525430B1_D0022.tif" /></maths><maths id="math0023" num="(23)"><math display="block"><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>K</mi><mi>s</mi></msub><msub><mi>D</mi><mi>v</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>K</mi><mi>s</mi></msub><mo>=</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub><mo>;</mo></math><img file="EP1525430B1_D0023.tif" /></maths><maths id="math0024" num="(24)"><math display="block"><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub><mo>=</mo><mfrac><msub><mi>K</mi><mi>b</mi></msub><msub><mi>D</mi><mi>s</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>K</mi><mi>b</mi></msub><mo>=</mo><msub><mi>D</mi><mi>s</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub><mo>;</mo></math><img file="EP1525430B1_D0024.tif" /></maths><maths id="math0025" num="(25)"><math display="block"><msub><mi>S</mi><mi>b</mi></msub><mo>=</mo><mi>K</mi><mo>-</mo><mn>2</mn><mo>⁢</mo><mfenced><msub><mi>D</mi><mi>s</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub><mo>+</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mfenced><mo>;</mo><mspace width="3em" /><mfenced><mn>22</mn><mo>/</mo><mn>23</mn><mo>/</mo><mn>24</mn></mfenced></math><img file="EP1525430B1_D0025.tif" /></maths><maths id="math0026" num="(26)"><math display="block"><msub><mi>T</mi><mi>b</mi></msub><mo>=</mo><mfrac><mrow><mi>K</mi><mo>-</mo><mn>2</mn><mo>⁢</mo><mfenced><msub><mi>D</mi><mi>s</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub><mo>+</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mfenced></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mspace width="3em" /><mfenced><mn>21</mn><mo>/</mo><mn>25</mn></mfenced></math><img file="EP1525430B1_D0026.tif" /></maths>

0037All elements for Ttot are now available: <maths id="math0027" num="(27)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mn>2</mn><mo>⁢</mo><mfenced><mfrac><msub><mi>D</mi><mi>v</mi></msub><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><mi>cos</mi><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>+</mo><mfrac><msub><mi>D</mi><mi>s</mi></msub><mrow><msub><mi>C.</mi><mi>s</mi></msub><mspace width="1em" /><mi>cos</mi><msub><mi>α</mi><mi>s</mi></msub></mrow></mfrac></mfenced><mo>+</mo><mfrac><mrow><mi>K</mi><mo>-</mo><mn>2</mn><mo>⁢</mo><mfenced><msub><mi>D</mi><mi>s</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub><mo>+</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mfenced></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mspace width="3em" /><mfenced><mn>16</mn><mo>/</mo><mn>19</mn><mo>/</mo><mn>20</mn><mo>/</mo><mn>26</mn></mfenced></math><img file="EP1525430B1_D0027.tif" /></maths><maths id="math0028" num="(28)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub></mrow><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><mi>cos</mi><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>s</mi></msub></mrow><mrow><msub><mi>C.</mi><mi>s</mi></msub><mspace width="1em" /><mi>cos</mi><msub><mi>α</mi><mi>s</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mi>K</mi><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>-</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>s</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>-</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msub><mi>D</mi><mi>v</mi></msub><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mspace width="3em" /><mfenced><mn>27</mn></mfenced></math><img file="EP1525430B1_D0028.tif" /></maths><maths id="math0029" num="(29)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mfrac><mi>K</mi><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>+</mo><mn>2</mn><mo>⁢</mo><mfenced><msub><mi>D</mi><mi>v</mi></msub><mo>⁢</mo><mfenced><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>v</mi></msub><mspace width="1em" /><mi>cos</mi><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced><mo>+</mo><msub><mi>D</mi><mi>s</mi></msub><mo>⁢</mo><mfenced><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>s</mi></msub><mspace width="1em" /><mi>cos</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mspace width="1em" /><mi>tan</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mfenced><mo>;</mo><mspace width="3em" /><mfenced><mn>28</mn></mfenced></math><img file="EP1525430B1_D0029.tif" /></maths>

0038The total running time Ttot is now not only (as in (15)) a function of the entry angle αv, but also a function of the entry angle αs and d can be represented as follows: <maths id="math0030" num="(30)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mfenced><msub><mi>α</mi><mi>v</mi></msub><msub><mi>α</mi><mi>s</mi></msub></mfenced><mo>=</mo><mfrac><mi>K</mi><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>+</mo><mn>2</mn><mo>⁢</mo><mfenced><msub><mi>f</mi><mn>1</mn></msub><mfenced><msub><mi>α</mi><mi>v</mi></msub></mfenced><mo>+</mo><msub><mi>f</mi><mn>2</mn></msub><mfenced><msub><mi>α</mi><mi>s</mi></msub></mfenced></mfenced><mo>;</mo><mspace width="3em" /><mfenced><mn>29</mn></mfenced></math><img file="EP1525430B1_D0030.tif" /></maths>

0039If the function Ttot (αv, αs) has a minimum, this can also be determined using the first derivative after the two angles. The first derivatives must be 0:<maths id="math0031" num="(31)"><math display="block"><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>1</mn></msub><mfenced><msub><mi>α</mi><mi>v</mi></msub></mfenced></mrow><mrow><mo>∂</mo><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>=</mo><mi mathvariant="bold">O</mi><mo>!</mo></math><img file="EP1525430B1_D0031.tif" /></maths><maths id="math0032" num="(32)"><math display="block"><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>2</mn></msub><mfenced><msub><mi>α</mi><mi>s</mi></msub></mfenced></mrow><mrow><mo>∂</mo><msub><mi>α</mi><mi>s</mi></msub></mrow></mfrac><mo>=</mo><mi mathvariant="bold">O</mi><mo>!</mo></math><img file="EP1525430B1_D0032.tif" /></maths><maths id="math0033" num="(33)"><math display="block"><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>1</mn></msub><mfenced><msub><mi>α</mi><mi>v</mi></msub></mfenced></mrow><mrow><mo>∂</mo><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>D</mi><mi>v</mi></msub><mo>⁢</mo><mfenced><mfrac><mrow><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><mrow><msub><mi>C.</mi><mi>v</mi></msub><mo>⁢</mo><msup><mrow><mspace width="1em" /><mi>cos</mi></mrow><mn>2</mn></msup><mo>⁢</mo><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>b</mi></msub><mo>⁢</mo><msup><mrow><mspace width="1em" /><mi>cos</mi></mrow><mn>2</mn></msup><mo>⁢</mo><msub><mi>α</mi><mi>v</mi></msub></mrow></mfrac></mfenced><mo>=</mo><mn>0</mn><mspace width="3em" /><mfenced><mn>31</mn><mo>/</mo><mn>29</mn></mfenced></math><img file="EP1525430B1_D0033.tif" /></maths><maths id="math0034" num="(34)"><math display="block"><mfrac><mrow><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><msub><mi>C.</mi><mi>v</mi></msub></mfrac><mo>-</mo><mfrac><mn>1</mn><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>=</mo><mn>0</mn><mo>;</mo><mo>→</mo><mfrac><mrow><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub></mrow><msub><mi>C.</mi><mi>v</mi></msub></mfrac><mo>=</mo><mfrac><mn>1</mn><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mo>→</mo><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>v</mi></msub><mo>=</mo><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>α</mi><mi>v</mi></msub><mo>=</mo><mi>arcs</mi><mo>⁢</mo><mi mathvariant="bold">i</mi><mo>⁢</mo><mi mathvariant="normal">n</mi><mo>⁢</mo><mi /><mfenced><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced><mo>;</mo><mspace width="3em" /><mfenced><mn>33</mn></mfenced></math><img file="EP1525430B1_D0034.tif" /></maths><maths id="math0035" num="(35)"><math display="block"><mfrac><mrow><mo>∂</mo><msub><mi>f</mi><mn>2</mn></msub><mfenced><msub><mi>α</mi><mi>s</mi></msub></mfenced></mrow><mrow><mo>∂</mo><msub><mi>α</mi><mi>s</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>D</mi><mi>s</mi></msub><mo>⁢</mo><mfenced><mfrac><mrow><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub></mrow><mrow><msub><mi>C.</mi><mi>s</mi></msub><mo>⁢</mo><msup><mrow><mspace width="1em" /><mi>cos</mi></mrow><mn>2</mn></msup><mo>⁢</mo><msub><mi>α</mi><mi>s</mi></msub></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>b</mi></msub><mo>⁢</mo><msup><mrow><mspace width="1em" /><mi>cos</mi></mrow><mn>2</mn></msup><mo>⁢</mo><msub><mi>α</mi><mi>s</mi></msub></mrow></mfrac></mfenced><mo>=</mo><mn>0</mn><mspace width="3em" /><mfenced><mn>32</mn><mo>/</mo><mn>29</mn></mfenced></math><img file="EP1525430B1_D0035.tif" /></maths><maths id="math0036" num="(36)"><math display="block"><mfrac><mrow><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub></mrow><msub><mi>C.</mi><mi>s</mi></msub></mfrac><mo>-</mo><mfrac><mn>1</mn><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>=</mo><mn>0</mn><mo>;</mo><mo>→</mo><mfrac><mrow><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub></mrow><msub><mi>C.</mi><mi>s</mi></msub></mfrac><mo>=</mo><mfrac><mn>1</mn><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mo>→</mo><mi>sin</mi><mspace width="1em" /><msub><mi>α</mi><mi>s</mi></msub><mo>=</mo><mfrac><msub><mi>C.</mi><mi>s</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>;</mo><mo>→</mo><msub><mi>α</mi><mi>s</mi></msub><mo>=</mo><mi>arcs</mi><mo>⁢</mo><mi mathvariant="bold">i</mi><mo>⁢</mo><mi mathvariant="normal">n</mi><mo>⁢</mo><mi /><mfenced><mfrac><msub><mi>C.</mi><mi>s</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced><mspace width="3em" /><mfenced><mn>35</mn></mfenced></math><img file="EP1525430B1_D0036.tif" /></maths>

0040The results (34) and (36) are now inserted into equation (29), which gives: <maths id="math0037" num="(37)"><math display="block"><msub><mi>T</mi><mi mathvariant="italic">dead</mi></msub><mo>=</mo><mfrac><mi>K</mi><msub><mi>C.</mi><mi>b</mi></msub></mfrac><mo>+</mo><mn>2</mn><mo>⁢</mo><mfenced><msub><mi>D</mi><mi>v</mi></msub><mo>⁢</mo><mfenced><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>v</mi></msub><mo>⁢</mo><mi>cos arcsin</mi><mfenced><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>tan arcsin</mi><mfenced><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced><mo>+</mo><msub><mi>D</mi><mi>s</mi></msub><mo>⁢</mo><mfenced><mfrac><mn>1</mn><mrow><msub><mi>C.</mi><mi>s</mi></msub><mspace width="1em" /><mi>cos arcsin</mi><mfenced><mfrac><msub><mi>C.</mi><mi>s</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>tan arcsin</mi><mfenced><mfrac><msub><mi>C.</mi><mi>s</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mrow><msub><mi>C.</mi><mi>b</mi></msub></mfrac></mfenced></mfenced><mo>;</mo><mspace width="3em" /><mrow><mfenced><mn>29</mn><mo>/</mo><mn>34</mn><mo>/</mo><mn>36</mn></mfenced><mo>)</mo></mrow></math><img file="EP1525430B1_D0037.tif" /></maths>

0041The shortest total running time Ttot can be measured using a suitable measuring instrument, for example the DMS 2 device from the applicant. It can be used to determine the thickness Ds of the layer 46 and / or the thickness Db of the base material 34 if the speeds of sound Cs and Cb are known. In this way, Cb can be determined at an uncoated point on the test specimen using (27). An additional pair of transducers for thickness measurement with the transducers 48, which are identical in construction, is used for the sound, see <figref idref="f0002">Figure 4</figref>. A back wall echo is generated, the duration of which is reduced by the duration of Ds. From this, the thickness Db of the base material is calculated at a known speed of sound Cb.

0042A second echo that follows in time is obtained from a rear wall 50 of the base material 34. The thickness Db can be determined from the time difference between the two echoes and the previously measured sound velocity Cb in the base material 34. However, the thickness Db can also be obtained as the difference between this echo of the rear wall 50 and the entrance echo, taking into account the speeds of sound, minus the thickness Ds of the layer 46.

0043The following relationships can be seen or become clear from equation (37):<ol id="ol0001"><li>1) <maths id="math0038"><math display="inline"><mfrac><msub><mi>C.</mi><mi>v</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></math><img file="EP1525430B1_D0038.tif" /></maths> and <maths id="math0039"><math display="inline"><mfrac><msub><mi>C.</mi><mi>s</mi></msub><msub><mi>C.</mi><mi>b</mi></msub></mfrac></math><img file="EP1525430B1_D0039.tif" /></maths> must be less than 1;</li><li>2) <i>C._s</i> can be smaller than C_v be.</li></ol>