15. Ultrasonic Interferometer — Velocity of Sound and Elastic Constants of a Liquid (PG)¶
Module
Oscillations, Damping & Acoustic Methods (PG) · Module 4
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Explore this on the interactive simulator with real-time sliders and graphs: 🔬 Launch in Simulator
Aim¶
To determine the velocity of ultrasonic waves in a liquid using an ultrasonic interferometer, and hence to calculate the liquid's adiabatic compressibility and bulk modulus.
Theoretical Background¶
An ultrasonic interferometer generates high-frequency (megahertz-range) longitudinal sound waves in a liquid using a piezoelectric quartz crystal fixed at the base of a small measuring cell, driven by a radio-frequency oscillator. A movable, reflecting plate at the top of the liquid column can be raised or lowered very precisely using a micrometer screw, allowing the length of the liquid column between the crystal and the reflector to be adjusted continuously.
When the reflector is at a distance from the crystal equal to a whole number of half-wavelengths, the wave reflected back from it arrives in phase with the newly generated wave, setting up a stationary (standing) wave pattern in the liquid column; this resonance condition loads the crystal's electrical circuit in a way that produces a sharp, repeatable maximum in the anode (or drive) current of the oscillator, easily read off a built-in meter.
As the micrometer is turned steadily in one direction, successive current maxima occur every time the reflector moves through exactly half a wavelength, so the distance between consecutive maxima directly gives λ/2; averaging this spacing over many maxima (rather than trusting a single interval) gives a precise wavelength, and hence, since the driving frequency f is fixed and known, the velocity of sound v = fλ in the liquid.
Once v is known, and given the liquid's density ρ (measured separately, e.g. by a pyknometer), the liquid's adiabatic compressibility β = 1/(ρv²) and adiabatic bulk modulus K = ρv² follow immediately — these fundamental elastic constants, difficult to access by any purely mechanical (static) method for a liquid, are exactly what make the ultrasonic interferometer an important and widely used PG-level instrument in acoustics, chemical physics, and materials characterisation of liquids and solutions.
Governing Formulas¶
Wavelength from the spacing of successive current maxima:
Velocity of ultrasonic waves in the liquid:
Adiabatic compressibility and bulk modulus:
where \(f\) is the fixed crystal-driving frequency, \(\overline{\Delta d}\) the mean spacing between successive current maxima, and \(\rho\) the liquid's density.
Simulator Controls¶
| Control | Symbol | Range | Unit |
|---|---|---|---|
| Driving frequency | f | 1.0 – 4.0 | MHz |
| Liquid selection (density) | ρ | 780 – 1300 | kg/m³ |
| Number of maxima to average over | N | 5 – 30 | — |
Readouts:
| Quantity | Symbol | Unit |
|---|---|---|
| Mean spacing between successive maxima | Δd | mm |
Procedure¶
- Launch the Ultrasonic Interferometer panel; select a liquid and set the driving frequency f.
- Turn the (virtual) micrometer steadily and note the reflector positions at which the current readout passes through successive maxima.
- Record the total distance moved over N consecutive maxima (choosing N as large as practical) and divide by N to get the mean spacing Δd, as shown on the readout.
- Repeat for a different driving frequency f and, separately, for a different liquid; tabulate Δd, λ = 2Δd, and v = fλ for each case.
- Using the known liquid density ρ, calculate the adiabatic compressibility β and bulk modulus K for each liquid studied.
Observation Table¶
Table: Wavelength and velocity of ultrasonic waves for different liquids/frequencies
| S. No. | Liquid | f (MHz) | N (maxima averaged) | Δd (mm) | λ = 2Δd (mm) | v = fλ (m/s) |
|---|---|---|---|---|---|---|
| 1 | ||||||
| 2 | ||||||
| 3 | ||||||
| 4 | ||||||
| 5 |
Graph¶
Plot Total distance moved, D (y-axis) against Number of maxima counted, N (x-axis).
Expected trend: a straight line through the origin, of slope Δd = λ/2, confirming a constant spacing between successive maxima.
Total distance D vs N, expected to be a straight line through the origin; slope = Δd
Calculation¶
- From the slope of the D vs N graph, obtain the mean maxima spacing Δd = λ/2, hence λ = 2Δd.
- Compute v = fλ for each liquid/frequency combination studied.
- Using the known density ρ of each liquid, compute the adiabatic compressibility β = 1/(ρv²) and bulk modulus K = ρv².
Result¶
Table: Velocity of sound and elastic constants of the liquid(s) studied
| Liquid | ρ (kg/m³) | v (m/s) | β (m²/N) | K (N/m²) |
|---|---|---|---|---|
| 1 | ||||
| 2 |
Maximum Permissible Error¶
Averaging the maxima spacing over as many resonances as practical (large N) directly reduces the fractional error in Δd, and hence in v, β and K, since micrometer positioning error is essentially fixed per reading but is divided by N when averaged this way.
Precautions¶
- Turn the micrometer slowly and always in the same direction while approaching and passing through a current maximum, to avoid backlash error in the screw.
- Average the maxima spacing over as many resonances as conveniently possible rather than relying on just two consecutive ones, to reduce the fractional error.
- Keep the liquid free of air bubbles and at a steady temperature, since both dissolved gas and temperature noticeably affect the velocity of sound in a liquid.
Self-Check Quiz¶
1. The distance between two successive current maxima in an ultrasonic interferometer corresponds to a reflector movement of:
- A) One full wavelength, λ
- B) Half a wavelength, λ/2
- C) A quarter wavelength, λ/4
- D) Twice the wavelength, 2λ
Answer: B — Consecutive resonance (standing-wave) conditions occur each time the reflector moves through λ/2, since this brings the reflected wave back into phase with the source wave once more.
2. If the driving frequency f is doubled (all else fixed), the measured wavelength λ in the same liquid:
- A) Doubles
- B) Halves
- C) Stays the same, since v is a material property of the liquid
- D) Quadruples
Answer: B — Since v = fλ and v is fixed by the liquid (at a given temperature), doubling f must halve λ to keep the product v = fλ unchanged.
3. The adiabatic bulk modulus K of the liquid is calculated from the measured velocity v and density ρ as:
- A) K = v/ρ
- B) K = ρv
- C) K = ρv²
- D) K = v²/ρ
Answer: C — K = ρv² directly, following from the general relation between the speed of a longitudinal wave in a medium and its elastic modulus and density.
Viva-Voce Questions¶
- Explain how a standing wave in the liquid column produces a sharp maximum in the crystal's drive current.
- Why is it better to measure the total distance over many maxima (large N) and divide, rather than reading a single λ/2 spacing?
- Derive the relations for adiabatic compressibility and bulk modulus from the measured velocity and known density.
- Why does dissolved air or a temperature change affect the measured velocity of sound in a liquid?
- What advantage does the ultrasonic interferometer have over other methods for finding the elastic constants of a liquid?
- How would the observed velocity differ between a liquid and, say, air, and what does this reveal about the relative compressibility of liquids versus gases?