Question

A narrow beam of ultrasonic waves reflects off the liver tumor in the figure below. If the speed of the wave is 11.0% less in the liver than in the surrounding medium, determine the depth of the tumor (in cm).

A narrow beam of ultrasonic waves reflects off the liver tumor in the figure below. If the speed of the wave is 11.0percent less in the liver than in the surrounding medium, determine the depth of the tumor (in cm).

Answer 1

Concepts and reason

The concepts used to solve this problem are reflection and refraction of the waves travelling from one medium to another.

For the solution , first find the angle of refraction by using the Snell’s law .Afterwards , use trigonometry to find the depth of the tumor.

Fundamentals

The expression for Snell’s law is,

Here, and is the refractive indices of the two media, is the angle of reflection in one media and is the angle of refraction in another media.

Snell’s law is also expressed as,

Here and is the velocities of the wave in the media one and two respectively.

Let v be the speed of incident wave and be the speed of the wave in the tumor.

The Snell’s law is expressed as,

Substitute for , for and for in the last expression as follows:

Rearrange the above expression in terms of as follows:

The expression for the depth d of the tumor is,

Here, h is the half of the distance between the points on the surface where the light enters and leaves the liver.

Substitute for h and for .

Ans:

The depth of the tumor is .