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Acoustic Intensity Around A Tuning Fork (Daniel A. Russell and Justin Junell, 2015)
Aim: To know the vector acoustic intensity around a tuning fork by comparing theory and measured vector intensity.
Theory: When the tines of the tuning fork oscillate back and fork it radiates sound in near-field and far-field radiation pattern. The pressure of the sound wave carries away energy from the tuning fork. This acoustic energy is calculated by acoustic intensity per unit time and unit area. The intensity is defined as the product of particle velocity and sound pressure in which the sound wave travels. The vector component of time-averaged intensity is given by, I_?= – sin?(2?)/(r^2 ?(kr)?^2 ).

The figure shows the normalized time-averaged intensity vector in xy-plane. Here the longitudinal quadrupole radiates sound and is given by the equations as follows:
Ix = Ir cos ? – I? sin ? and
Iy = Ir sin ? + I? cos ?
The vector plot is plotted using the x and y component intensity VectorPlot function. From the figure along the y-direction, the time-averaged intensity is directed outwards which is away from the tuning fork. Whereas, along with the x-direction, the points are towards tuning fork at a distance closer than 0.18 m, however this change at longer distances as the points are directing outwards away from the tuning fork.
Fig: Normalized time-average acoustic intensity vectors
The measurements around the tuning fork are made by using a two-microphone technique, which consists of two identical half-inch-diameter microphones. Since the type of intensity used is p-p probe method.
Experimental setup: A tuning fork of 426.6 Hz is mounted on the stand which has the tines of 1.3 cm thick, 0.9 cm wide and they are 2.1 cm apart. The magnets are attached to each tine and the electromagnetic coil is placed on one of the magnets as the other magnet acts as a distributor for mass-loading. The output of the two microphones is recorded using a two-channel FFT analyzer. For every 80 seconds, a turntable is made which has a total of 320 measurements with a time average of 125 ms. The intensity is measured by increasing 2 cm from the central axis of the tuning fork at 9 cm.
Fig: Measurement of vector intensity in radial Fig: Measurement of vector intensity in axial
direction direction
The measurements are recorded separately as the probe needs to be rotated 900 at each distance for measuring intensity in radial and tangential components. The intensity data is then synchronized by using the LabVIEW program. The vector components data is then extracted by keeping every fifth data point. With ListVectorPlot function the polar vector components are translated to rectangular components.
Results: The figure shows the results of intensity which is measured 125 ms at turntable is rotating. The comparison between theoretical and practical measurements is such that the points are y-direction is same as theoretical measurements, that they are pointing away from the tuning fork. Whereas, for x-direction, the points are moving away from the tuning fork which is different compared to theoretical measurements. The measurements were made over 125ms while rotating turntable small tangential component shows up. Theory predicts that for 421 Hz the change should be about 18 cm, however, from practical it is seen that the change is about 16 cm. Since the difference between theory and practical data becomes apparent in the transition between near-field and far-field.
Fig: Measured time-average acoustic intensity
The tuning fork which is modeled as the longitudinal quadrupole source, the transition from near-field to far-field occurs at a farther distance. The vector plot shows the comparison of practical and theoretical results of sound radiation in far-field and circulation in the near-field.

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Fig: Comparison between theory and practical data.
Conclusion: By comparing theoretical and practical measurements the points are moving outwards in the perpendicular axis in near-field and far-field. Whereas, the axis along parallel to the fork for theoretical measurements the points are along the line, however, in practical measurements the points are distributed. From the transition between near-field and far-field, it can be seen that the time-averaged intensity fades away along the axis parallel to the fork tines.

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