Animations of Instantaneous Intensity
Thomas Burns, Ph.D., P.E.
Principal, Engineering Research at Starkey Labs, Inc., Eden Praire, MN
The animations below were created as a supplement to accompany Tom Burn's 1995 Ph.D. Thesis in Acoustics:


Thomas H. Burns, Measurement and Visualization of Instantaneous Power Flow In Steady-State Acoustic Fields, Ph.D. Thesis in Acoustics, The Pennsylvania State University (1995).

Abstract
The objective of this study is to investigate the time and spatial dependence of power flow in monochromatic, steady-state acoustic fields. Knowledge of the instantaneous acoustic intensity vector Iinst(r,t) at discrete points throughout a three-dimensional (3D) space allows one to visualize, simultaneously, the time and spatial variation of power flow. The method of computing Iinst(r,t) at multiple points in rectangular coordinates is accomplished by:

  1. directly measuring the time-averaged active intensity <Iactive> and the potential energy density V(r) on a plane of discrete points arranged in a spatial grid,
  2. calculating the relative phase distribution φ on that plane by spatially integrating the ratio of <Iactive> to V(r); this ratio is proportional to ∇φ(r),
  3. using V(r) and φ to calculate the complex pressure on that plane,
  4. reconstructing the single-frequency complex pressure and complex particle velocity on multiple parallel planes throughout the 3D radiated free-field using Nearfield Acoustic Holography,
  5. calculating the time-averaged active intensity <Iactive> and the maximum amplitude Q(r) of the instantaneous reactive intensity at each discrete point in the reconstructed 3D field, and
  6. substituting <Iactive>, Q(r), and V(r) into an equation for the steady-state instantaneous intensity at a fixed point in space:
    Iinst = < I >(r) + < I >(r) cos 2(ωt t - φ(r)) + Q(r)sin 2(ω t - φ(r))
A display of multiple (spatially -fixed) instantaneous intensity vectors that are "sequenced" in time allows one to visualize, simultaneously, the time and spatial dependence of power flow in the form of an animation. This technique is used to analyze the canonical problems of a monopole source and point-dipole source, and also to study the instantaneous power flow in the radiated nearfield of multiple-driver loudspeaker systems.



Defining Intensity

Intensity is conceptually defined as the amount of energy per unit time (power) that passes through surface area. Intensity is a vector quantity whose magnitude indicates the amount of energy a sound wave is carrying, and whose direction inticates the direction of energy flow. Intensity is calculated from the product of pressure and particle velocity. The sound field radiated by a sound source usually has a near-field region (where the pressure and particle velocity of the medium are roughly 90o out of phase with each other) and a far-field region (where the particle velocity and pressure are in phase).

Active Intensity is the product of the pressure and the in-phase component of the particle velocity. The time-average of the active intensity is non-zero, the direction is perpendicular to the sound wavefronts, and it is identified with the flow of sound energy.
Reactive Intensity is the product of the pressure and the 90o out-of-phase component of particle velocity. The direction of the reactive intensity is opposite to the pressure gradient, and the time average of the reactive intensity is zero. The reactive intensity is associated not with the radiation of sound energy, but with the local motion of the medium.


Instantaneous Intensity Field Radiated by a Monopole Sound Source

This set of three animations shows the active intensity (left), the reactive intensity (middle) and the total intensity (right) for an oscillating simple source, or monopole. Notice that the active intensity and the intensity look the same; this is because the reactive intensity does not contribute to the net flow of energy from the source, but only involves the local motion of the fluid in the near-field of the source. The total intenstiy is directed outward away from the source at all distances.

Long red arrows have the greatest intensity, and short dark blue arrows have the smallest intensity with dark green arrows between. Click on the animation to download a Quicktime movie.




Instantaneous Intensity Field Radiated by a Dipole Sound Source

This set of three animations shows the active intensity (left), the reactive intensity (middle) and the total intensity (right) for a dipole source. The Top and bottom of the dipole source radiate sound with opposite phase. The active intensity vectors all point away from the source since the active intensity represents the flow of energy away from the source. The reactive intensity shows significant circulation with the arrows changing direction back and forth, indicating that energy is being stored locally in the medium around the source. The total intensity shows a clear distinction between the near-field where the intensity vectors rotate (incidating some circulation of sound energy) and the far-field where the intensity vectors continually point away from the source, indicating the radiation of sound energy.

Long red arrows have the greatest intensity, and short dark blue arrows have the smallest intensity with dark green arrows between. Click on the animation to download a Quicktime movie.




Instantaneous Intensity Field Radiated by a Boxed Loudspeaker

This pair of animations illustrate the radiation of sound from a boxed loudspeaker. The animation on the left shows the total intensity. The intensity vectors exhibit strong circulation in the near-field close to the source, while the vectors in the far-field point away from the speaker without much circulation. The animation on the right shows the active and reactive intensities at the same time. The red active intensity vectors always points away from the source, indicating that the speaker is radiating sound energy. The yellow reactive intensity vectors swap directions each half-cycle, indicating that the air near the loudspekaer is being moved back and forth.




Instantaneous Intensity Field Radiated by a Two-speaker system

This set of four animations illustrates the sound field radiated by a two loudspeaker system at several frequencies. All four animations shows the total intensity vectors as a function of time. Evidence of near field (where the intensity vectors rotate) and farfield (where the intensity vectors mostly point away from the source) are clear. Notice that for the 500 Hz and 1000 Hz animations, the sound field is not symmetric to the center of the woofer. The pressure maximum is located on the left side of the woofer, and teh kinetic energy density maximum is located on the right side. As a result, the intensity vector field is not symmetric.

Long red and yellow arrows have the greatest intensity, and short dark blue arrows have the smallest intensity with dark green arrows between. Click on the animation to download a Quicktime movie.

Left: 500 Hz.         Middle Left: 1000 Hz         Middle Right: 2000 Hz         Right: 3000 Hz




Instantaneous Intensity Field Radiated by a Two-speaker system, alternate arrangment

This set of animations also shows a boxed two-loudspeaker system, but the woofer has been moved closer to the center of the box.

Long red arrows have the greatest intensity, and short dark blue arrows have the smallest intensity with dark green arrows between. Click on the animation to download a Quicktime movie.

Left: 500 Hz.         Middle Left: 1000 Hz         Middle Right: 2000 Hz         Right: 3000 Hz




Instantaneous Intensity Field Radiated by a Three-speaker system, (two-woofers plus tweeter)

This set of animations shows the radiation of sound from a three-speaker system with two woofers and a tweeter. At 500 and 1000 Hz, the two woofers are acoustically coupled together so that the pressure in the region between them adds constructively.

Long red arrows have the greatest intensity, and short dark blue arrows have the smallest intensity with dark green arrows between. Click on the animation to download a Quicktime movie.

Left: 500 Hz.         Middle Left: 1000 Hz         Middle Right: 2000 Hz         Right: 3000 Hz
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