An ultrasonic beam reflects off a tumor in an organ at an angle of 50 with a shift L = 7 cm.
If the speed of the wave is 26 % less in the organ than in the medium above, determine the depth of the tumor below the organ's surface
To determine the depth of the tumor below the organ's surface, we can use the concept of Snell's Law and the formula for calculating the shift of a wave reflected at an angle.
Snell's Law states that the ratio of the incident and refracted angles (with respect to the normal) is equal to the ratio of the velocities of the wave in the two different mediums.
Let's denote the angle of incidence as θ1, the angle of refraction as θ2, the velocity in the medium above (air) as v1, and the velocity in the organ as v2.
According to the problem, the incident angle is 50 degrees, and the speed of the wave in the organ is 26% less than the speed in the medium above. This means v2 = 0.74 * v1.
Using Snell's Law, we can write the equation:
sin(θ1) / sin(θ2) = v1 / v2
Since we are given θ1 = 50 degrees and v2 = 0.74 * v1, we can solve for θ2:
sin(50) / sin(θ2) = 1 / 0.74
sin(θ2) = sin(50) * 0.74
θ2 ≈ arcsin(sin(50) * 0.74)
θ2 ≈ 38.63 degrees
Now, to find the depth of the tumor below the organ's surface, we can use the formula for calculating the shift of a wave reflected at an angle:
shift (L) = 2 * depth * tan(θ2)
Since we are given the shift L = 7 cm, we can rearrange the formula to solve for the depth:
depth = L / (2 * tan(θ2))
depth = 7 / (2 * tan(38.63))
depth ≈ 5.0 cm
Therefore, the depth of the tumor below the organ's surface is approximately 5.0 cm.