Ultrasonics
Introduction
Ultrasonics is the term used to describe those sound waves whose frequency is above the
audible range of human ear upward from approximately 20kHz to several MHz. The
ultrasonics in this experiment is generated by a transducer made of piezoelectric crystal. The
crystal is set into oscillation by the oscillatory electric field produced by the electronic
oscillator. The Phywe generator used in this experiment has two modes of operation:
continuous wave (CW) at around 800kHz and repetitive pulses.
Experimental Setup
The equipment setup is shown schematically below. Note that the transducer should be
immersed in water at all times to prevent it from being overheated.
Error Analysis
The first experiment is suitable to demonstrate the normal distribution of data, and to
refresh your knowledge of error analysis. See appendix below for Standard deviation
and Standard error of the mean. While the analysis can be easily calculated by a handheld calculator, you are required to explain the concept as well as the mathematics of
standard deviation, and to calculate the propagation of errors.
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Experiment 1
Velocity Measurement
The frequency f, wavelength λ , and velocity v of a sound beam, are related by v = λ f .
From this relationship, knowing the frequency and wavelength the velocity of sound in water
can be deduced.
The ultrasonic frequency of the oscillator (in CW) driving the transducer can be determined
by an oscilloscope, or better still by a frequency meter. The wavelength is determined by the
lines of stationary bubbles (cavitations at the pressure minima) formed in the water. In order
that the bubbles are stationary, the standing wave condition must be first established. For best
result adjust the aluminium reflector for a strong standing wave, which is indirectly visible as
parallel lines of fine bubbles perpendicular to the direction of the ultrasonic beam. Typically
the aluminium reflector is position at a distance of less than 15 cm from the transducer.
The separation between the bubbles is 1 2 λ . A travelling microscope with digital readout is
used for measuring the separation. To facilitate better viewing of the bubbles a table lamp
should be used to illuminate the bubbles.
1.
Take at least 25 readings at different parts along the ultrasonic beam. Plot the
distribution to verify that you have enough statistics for a normal distribution. Find
the standard deviation by: (a) hand-held calculator, and (b) from your normal
distribution graph. Verify that they agree according to error analysis theory .
2.
Work out the mathematics for error propagation. Estimate the standard deviation and
standard error of the mean of the velocity. What is the sample size needed to have a
95% confidence level?
3.
It is well know that the velocity of sound is basically determined by the
modulus of elasticity, K, of water. The expression can easily be found with the aid of
Newton's second law as:
v = K /ρ
where ρ is the density of water and for our purpose may be taken as 1g/cm3
Calculate the standard deviation of your K value.
4.
Explain why a standing wave is a necessary condition in this experiment, and the
steps you have taken to ensure that the condition is met.
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Experiment 2
SONAR
Use the pulse mode of the ultrasonic generator for a demonstration of distance measurement
with SONAR (Sound, Navigation and Ranging).
Place the aluminium reflector at various distances from the transducer. Align the receiving
sensor so that it is facing the incoming reflected pulses. The sensor should be rigidly clamped
and pick-up signal, which is weak, sent to the oscilloscope set at high gain input. The
ultrasonic generator also gives a reference pulse as marker of time frame. Connect the
reference pulse to the other channel of the oscilloscope. For a stable display without jitters,
adjust the triggering level to bring the ultrasonic oscillator sweeping rate and the scope into
synchronisation.
1.
Use a digital camera to take the traces of original and reflected (echoes) pulses for a
number of different distances.
2.
Using the oscilloscope, measure accurately the time separation between the
original pulse and the reflected echoes, hence deduce the distance travelled.
3.
Confirm your results by comparing them with a ruler measurement, taking care to
allow for a slightly skew path of the pulses. Plot a graph of the ultrasonic
measurement versus ruler measurement. Discuss the accuracy of your measurement.
4.
Give the reason(s) that SONAR is employed in underwater by submarines and
fishing boats rather than the highly accurate electromagnetic waves.
Figure 1
Original and reflected (echoes)
pulses for 0.3m reflection
Figure 2
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Original and reflected (echoes)
pulses for 0.2m reflection
Appendix
Appendix - Standard Error of the Mean
When trying to work out the mean of a sample, we may have to answer one of two questions:
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If there is no control over the size of the sample, what is the confidence of the mean.
If the size of the sample can be controlled, how big a sample should be taken to ensure that
the confidence interval is within the required limits.
The solution is based on the standard error of the mean. The distribution of sample means is
normally distributed, regardless of the distribution of the population from which the sample was
drawn. The standard error of the mean is the standard deviation of the sample mean. The formula is:
It follows from this equation, that the larger the sample size, the smaller the likely error. This is one of
those cases where statistics tells you something that is intuitively obvious.
In practice we don't always know the standard deviation of the population (or a value calculated from
a very large sample), so we may have to estimate the value from the sample.
The confidence limits of the sample mean are:
Example - Confidence Interval for the Mean.
The standard deviation of the weight of individual chilies from this years crop was 2.0 grams. The
average weight of a sample of 16 chilies was 10 grams. Thus the standard error of the mean is:
StErr = 2/√16 = 0.5
Therefore the 95% confidence interval for the mean is:
Confidence interval = 10 ± 1.96 * 0.5 = 10 ± 0.98 = 9.02 to 10.98
Sample Size
The formula for the standard error can be re-arranged to estimate the required sample size:
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Thus if we know the confidence level and the interval size, we can work out the standard error to plug
into the equation:
Example - Sample Size
Back to our crop of chilies, say, we need to be 95% certain that our estimate of the mean has an error
of plus or minus 0.50 grams, how big should the sample be.
Step 1 - Calculate the standard error
StErr = interval/1.96 = 0.5/1.96 = 0.26
Step 2 - Work out the sample size
n = (StDev/StErr)2 = (2.00/0.26)2 = 7.692 = 59.1
In practice, we would take the next integer value which is greater then 59.1, i.e. 60 as the required
sample size.
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