Optical Determination of Sound Velocity in Liquids

The project aims at designing an experimental setup to measure the speed of the sound in different liquids. The principle of the measurement is based on the production of standing wave in the liquid using an ultrasonic generator with tuned frequency. This standing wave changes the characteristics of light propagation of a laser beam, which gives a visible pattern in form of fringes projected on a screen. The distance between fringes is related to the wavelength of sound wave. Knowing the frequency of the sound wave produced by the ultrasonic generator, the speed of sound can be calculated. Two optical principles are used to produce such a visible pattern:

- Projection of the standing wave on a screen by divergent light.
- Diffraction of the light on a grating formed of standing waves.

The experimental setup should be applied for three different liquids of the choice of each student.

Project submission and report format

Each student should submit individual report for the project.

The report should be of Microsoft Word or PDF format. It should be about 10 pages with figures and tables, of format of MS Word or PDF. Student can add as many appendices as he judges necessary to develop ideas and clarifications. Optional topics (if any) should be included in the appendices.
The report should contain the following:

- Summary
- Introduction
- Development of the study in topics with separate title for each one
- Conclusion
- List of References

Figures, tables and references should be numbered and referenced in the text. For example (Fig. 1), (Table 1) or (Ref. 1). For reference you can use carets like [1] instead of (Ref. 1)
The reference should mention:

- The title of the article or book.
- Author if available
- For books:
o Book title
o Editor
o Pages of the article
o Year of edition
- For article in journal or revue
o The name of the journal
o Volume and number
o Pages
- For Internet site
o Website name
o Date of publication
o URL address
Due Time

Reports should be submitted before Tuesday 6 December 2016.

Topics to be treated

Part 1: Theory

- Standing wave in a liquid:
o Propagation of sound wave in a liquid and description of this wave as displacement of the liquid's particles or change of pressure.
o Reflection of the sound wave on the vessel walls and formation of standing wave.
o Description of nodes and antinodes of the standing wave for:
   - Particles displacement wave
   - Change of pressure wave
Use figures and drawing to explain the phenomena with focus on relative position of displacement nodes and antinodes with regard to pressure nodes and antinodes. Show the displacement and pressure graphs as functions of the position at different moments (0, T/4, T/2, 3T/4) where T is the sound wave period.

o Change of density in the liquid as function of position and time.
o Change of the optical index of refraction as function of position and time.

- Effect of the standing wave on the light propagation

o Projection of the standing wave on a screen
- Show that some part of standing waves (nodes or antinodes) are considered as transparent to the light while other parts are considered as opaque.
- Explain the phenomena using the deflection of light on regions of different density and refractive index.
- Show the consequence of the distribution of light intensity and the formation of fringes if the standing wave is illuminated by divergent light beam. Consider the light is propagating as straight-line rays.
- Relation between the distance separating two adjacent bright fringes and the sound wavelength.
o Diffraction of the light on standing wave grating.
- Show that standing wave is considered as an optical grating.
- Determination of the grating's period and relation to the sound wavelength.
- Diffraction of a laser beam (considered as parallel beam) on the grating and the angular position of bright fringes of diffracted light.
- Projection of diffraction pattern on a screen using a lens, relation of the distance between bright fringes and the sound wavelength.

Part 2: Design of Experimental Setup

- Choice of liquids: Choose three different liquids:
o Transparent to visible light
o Safe to use
o Known speed of sound

- Vessel: Characterize the vessel which contains the liquids:
o Material
o Thickness of the walls
o Geometry and dimension

- Ultrasonic generator: Characterize the ultrasonic generator used to generate the acoustic wave in the liquid:
o Power

o The range of frequencies (frequency not necessary in the audible range it can be in the ultrasonic range).

- Laser: Characterize the laser to be used:
o Type (gas, liquid or solid)
o Wavelength
o Power
o Diameter of the laser light beam (Consider cylindrical beam)

Projection Method

- Production of divergent beam:
o Explain how to produce a divergent light beam using the laser source with a convergent lens.
o Give the relation between the focal length of the lens and the divergence angle of the beam.
o Calculate the dimension of the light spot formed at distance s form the lens.

- Experimental setup sizing: Choose the following dimensions and values for the experimental setup:
o f: Focal length of the lens
o s'₁!: Distance between the lens and the center of liquid vessel.
o s₂: Distance between the center of liquid vessel and the screen.
o v: Frequency of the acoustic sound produced by the transducer. These dimensions and values should satisfy the following conditions:
o Production of standing wave in the liquid.
o The laser divergent beam at the level of the vessel should cover many nodes and antinodes (5-6 for example)
o The projected pattern on the screen should permit direct measurement of the distance between fringes.

- Procedure and equations:
o Describe the steps that students should apply, the values that he should register and the measurement he should make.
o Give the equations to use to calculate the speed of the sound in the liquid.

o Make an error analysis of the final result as function of the error of the different parameters used in the equations.
o Give estimation of the different parameters and results for the three liquids of your choice.

Diffraction Method

- Enlarge the laser beam:
o Show that two lenses with different focal lengths f₁ and f₂ illuminated by a parallel beam with diameter D₁ produce a parallel beam with diameter

D2 = f₂/f₁.D₁.

- Projection of diffraction pattern on the screen

o The light beams diffracted in different orders are considered as parallel beams. To form a diffraction pattern with sharp lines on the screen, use a lens with large focal point f₃.
o Show that the screen should be at distance f₃ from the lens.
o Give the position of diffracted line d from the central point, as function of the diffracted angle θ.

- Experimental setup sizing: Choose the following dimensions and values for the experimental setup:
o f₁ and f₂: Focal length of the lenses to enlarge the laser beam.
o f₃: Focal length of the projection lens.
o v: Frequency of the acoustic sound produced by the transducer. These dimensions and values should satisfy the following conditions:
o Production of standing wave in the liquid.
o The enlarged laser beam should cover many nodes and antinodes (10 at least)
o The projected pattern on the screen should permit direct measurement of the distance between the lines of the diffracted pattern.

- Procedure and equations:
o Describe the steps that students should apply, the values that he should register and the measurement he should make.
o Give the equations to use to calculate the speed of the sound in the liquid.
o Give estimation of the different parameters and results for the three liquids of your choice.

Optional topics

The treatment of any additional topic related to the subject of the project can leads to eventual bonus in project evaluation up to 10% of total project mark (up to 15% if a good physical implementation is presented matching report calculation). In the following examples of subjects

- Estimation of the experimental setup cost

List all equipment needed for the experimental setup with characteristics and prices.

- Acousto-optics technology

The interaction between light waves and sound waves can be described in general term of "Acousto optics" which is a branch of physics.

Describe briefly "acousto-optics" branch with main devices and applicatioins

- Implementation of the experimental setup

Get the requested devices and make a physical implementation of the experimental setup matching your calculation and characteristics (a physical implementation could be made by groups of 4 students maximum each. Each student has to submit an individual report). A physical implementation can lead to a bonus up to 15% of the project mark for each student.