CALCULATING SNELL S LAW

Transcription

1 CALCULATING SNELL S LAW What is Snell s Law? Snell s Law defines angular relationships of sound waves crossing an interface. The calculations for Snell s Law will come into play when we are conducting Shear Wave Inspection. It is important to note at this point that Shear Wave is also referred to as Transverse Wave or Angle Beam Testing. The illustration below will help us understand how sound waves are affected when moving through two materials with different sonic velocities. In the illustration on the left, we see that when transmitting sound perpendicular to the surface of the material, we are primarily generating Longitudinal Waves ( L-Waves), Shear Waves are at a minimum or non-existent. The illustration on the right helps us understand and envision that when Longitudinal Waves are transmitted through an angled wedge, or through water in the case of immersion testing, the L- waves are refracted as they enter the second material. We are also now looking at a refracted Shear Wave that is produced in the test material. These angles can be calculated by using an equation that has been developed from Snell s Law. Snell s Law Equation The Sine of the Incident Angle divided by the Sine of the Refracted Angle is equal to the Velocity of Material Number One divided by the Velocity of Material Number Two By using this formula we can determine the angle of incidence and the angle of refraction for either the longitudinal wave or the shear wave.

2 Why do we need to calculate these angles? In order to conduct Shear Wave testing, the Longitudinal Wave must be adjusted so that it is reflected out of the test material, leaving only the shear wave inside the material to transmit usable data, as illustrated in the diagram below. In order to achieve this condition, we must know the Incident Angle, the angle of the sound beam entering the test material. Our three main uses for the Snell s Law equation will be: 1) to calculate the Incident Angle required to reflect the Longitudinal Wave out of the test material, and 2) to calculate the 2 nd Critical Angle for our Shear Wave 3) determine the Refracted Angle of the Shear Wave in the test material It is important to note that as the Angle of Incidence is Increased, the Refracted Angle of the sound beam will increase as well. Note the diagram below. In this example, a sound beam with an Incident Angle of 36 o will result in a Refracted Angle of 45 o. If the Angle of Incidence is increased to 52 o, the refracted Angle is increased to 70 o. These angles will be affected by the velocities of Material One and Material Two.

3 The Incident Angle The Incident Angle is defined as the angle that the sound beam is being transmitted into the test material. In order to select the proper wedge angle or to select a proper angle for the transducer in immersion testing, we can calculate the minimum Incident Angle required to reflect the Longitudinal Wave out of the test material. This is called the FIRST CRITICAL ANGLE. FIRST CRITICAL ANGLE As the angle of incidence is increased, the first critical angle is reached when the refracted longitudinal wave angle reaches 90 o. At this point, only shear waves exist in the test material. When selecting a contact shear wave angle beam transducer, or when adjusting an immersed transducer at an incident angle to produce shear waves, two conditions are considered. First, the refracted longitudinal wave must be totally reflected (its angle of refraction must be 90 o or greater) so that the penetrating beam is limited to shear waves only. Second, the refracted shear wave must enter the test object in accordance with the requirements of the test standard. In the immersion method of testing, the first critical angle is calculated to ensure that the sound beam enters the test material at the desired angle. Let s look at a sample problem. In our example, we will use a Plexiglas transducer wedge and a Steel test piece. We will calculate the First Critical Angle required to move the L-wave to a minimum of 90 o. SPECIAL NOTE: When using the Snell s Law Equation, you will always know three parts of the equation. You will only be calculating for one variable. You will use your Ultrasonic Reference Data Sheet and your Trigonometry Table to solve this problem. Starting with our formula,, let s substitute the information that we already know. By using our charts, we already know that the Longitudinal Velocity of Plexiglas is.11 and the Longitudinal Velocity of Steel is.23. We also know that the Refracted Angle for the L- Wave must be 90 o in order to achieve our desired result.

4 The resulting formula looks like this: Note: The velocities are both x Since they cancel each other out, we have simplified the equation by eliminating the extra step. The next step is to convert the sin of 90 o to a decimal, using your Trigonometry Table. The decimal value of 90 o is So our formula becomes: or Working the math, we get: Using our Trigonometry Table, we find that.4783 = approximately degrees This is the minimum Angle of Incidence that must be created in order for us to conduct Shear Wave Inspection. Important: In the above equation, we are making calculations related to our Longitudinal Wave, therefore, we use the Longitudinal Velocities for both Materials One and Two. We always use the Longitudinal Velocity for Material One since our transducer is always producing a Longitudinal Wave in the first material. Look back at the diagrams above. You will notice that we are producing a straight beam signal through material one, which is then refracted in Material Two, producing both a Longitudinal Wave and a Shear Wave.. In the following equation, we will be calculating for the Shear Wave in Material Two, so we will be using the Shear Velocity for the V 2 portion of the equation.

5 Second Critical Angle The Second Critical Angle is reached when the refracted Shear Wave reaches 90 o. At his point the longitudinal wave is being completely reflected out of the test material and the shear wave is basically on the surface of the test piece as illustrated by the diagram below. Most Shear Wave inspection is conducted between the First and Second Critical Angle. To calculate the Second Critical Angle, we would do the following: Consider the information that is known. V 1 =.11 V 2 =.13 (since we are calculating for the shear wave) The Refracted Angle of the Shear Wave is 90 o Our formula will be: The Sin of 90 o is 1, resulting in: or Working the math, we get: By using our Trigonometry Table, we find that our second critical angle is approximately 58 Degrees.

6 Knowing that shear wave inspection is conducted between the first and second critical angle, we now know, that our incident angle must be between 28 Degrees and 58 Degrees in order to conduct our inspection. (Using the formulas for First Critical Angle and Second Critical Angles) A Third Use for the Snell s Law Equation Now let s consider how we can use the equation to determine the actual refracted angle of the Shear Wave. For our purposes, let s use 28 o as our Angle of Incidence. We are still working with a Plexiglas Wedge (Material One) and a Steel Test Piece (Material Two), but this time we will use the Shear Velocity of Material Two in our calculation since we are trying to calculate the refracted angle for our Shear Wave. Here is our formula with our known information plugged into the correct places: Our next step is to convert 28 o into a decimal, using our Trigonometry Table. 28 o =.4695 The resulting equation is: By doing our normal math function of cross multiplication, we are left with the following formula: Working the math: Resulting in: By using our Trigonometry Table, we are able to determine that the Refracted Angle for our Shear Wave is approximately 34 Degrees.

7 PRACTICE PROBLEMS You will use your Ultrasonic Reference Data Sheet and your UT Trigonometry Chart for the following practice problems. 1) Calculate the 1st Critical Angle for the following: Material One is Plexiglas. Material Two is Aluminum. What is the First Critical Angle? 2) Calculate the 1st Critical Angle for the following: Material One is Water. Material Two is Steel. What is the First Critical Angle? 3) Calculate the 1st Critical Angle for the following: Material One is Water. Material Two is Aluminum. What is the First Critical Angle?

8 4) Calculate the 2nd Critical Angle for the following: Material One is Plexiglas. Material Two is Aluminum. What is the Second Critical Angle? 5) Calculate the 2nd Critical Angle for the following: Material One is Water. Material Two is Steel. What is the Second Critical Angle? 6) Calculate the 2nd Critical Angle for the following: Material One is Water. Material Two is Aluminum. What is the Second Critical Angle? 7) Calculate the Refracted Angle for the following: Material One is Plexiglas. Material Two is Aluminum. We also know that the Incident Angle is 26 o. What is the Refracted Angle for the Shear Wave?

9 8) Calculate the Refracted Angle for the following: Material One is Water. Material Two is Steel. We also know that the Incident Angle is 15 o. What is the Refracted Angle for the Shear Wave? 9) Calculate the Refracted Angle for the following: Material One is Water. Material Two is Aluminum. We also know that the Incident Angle is 13 o. What is the Refracted Angle for the Shear Wave?