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Old 5th November 2009, 01:47 AM   #1
thadman is offline thadman  United States
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Default Wave transformation, Higher order modes?

Over the weekend, I was trying to visualize the wavefront expanding through a horn and had some ideas. I was wondering if this phenomenon is what Dr. Geddes refers to as Higher Order Modes.

There are 3 possible dimensions in which a wave can propagate (x,y,z).

Waves initially propagate in one dimensions (x).

If the wavefront is forced to expand, antiparallel wavefronts will be generated.

Visualize an oscillating piston located within a cylinder. Neglecting resistance at the boundaries (sides of the cylinder), the wave will propagate unimpeded. However, once the wave reaches the end of the cylinder (an impedance mismatch exists at this boundary), an abrupt transformation of the wavefront occurs. A secondary wavefront is created at this boundary, which propagates antiparallel to the original wavefront (ie a reflection). The primary wavefront is then forced to expand in 3 dimensions.

Let us now visualize an oscillating piston located within a waveguide. As the wavefront propagates down the waveguide, it is constantly expanding and thus secondary antiparallel wavefronts are constantly being generated. Are these secondary antiparallel wavefronts the "Higher Order Modes" that Dr. Geddes attempts to solve with a resistive element within the waveguide?

These secondary antiparallel waves interact with the wave source. At low frequencies, the effect is primarily reactive and is why transducers are very inefficient. At high frequencies, the effect is primarily resistive. The reactive force bends the initially flat wavefront into 3 dimensions (sphere) and is why transducers are omnidirectional at low frequencies.

The reactive force is similar to the effects of a capacitor in that it dissipates energy over time. A source which minimizes the reactive force and allows the resistive force to approach a constant value may reproduce a square wave most effectively.

Is this a correct observation? I would much appreciate feedback.
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Last edited by thadman; 5th November 2009 at 02:07 AM.
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Old 5th November 2009, 02:23 AM   #2
thadman is offline thadman  United States
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A source which provides the most gradual transformation to 3 dimensions may satisfy this requirement (minimize reactance, allow resistance to approach a constant value). We have two available functions which provide a constant rate of expansion (ie linear), a 2nd order function or an exponential function. The exponential function will never provide a vector which is orthogonal to the initial vector, thus a 2nd order function is required (ie roundover).
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Last edited by thadman; 5th November 2009 at 02:44 AM.
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Old 5th November 2009, 11:53 AM   #3
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A trip to the library might help. Check out Morse and Ingard, Theoretical Acoustics. Chapter 9.2, Higher (order) Modes in Ducts. Horns are just ducts of non constant cross section.
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