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Does anyone know offhand whether an ultrasonic transducer will respond efficiently to harmonics of the fundamental? For example, will a 40kHz transmitter also transmit with reasonable efficiency at (say) 20kHz or 80kHz?

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A standard piezoelectric ultrasonic transducer will not work efficiently at sub harmonics or harmonics - it's a 2nd order resonant mechanical device with high Q - there aren't different modes of operation as you might find with some crystals (for instance).

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A transmitter is unlikely to do much when driven with a sine wave at anything other than its fundamental operating frequency, at least if the amplitude is low enough to avoid overdrive distortion. A transmitter driven with a combination of frequencies that includes its resonant frequency will, in the absence of overdrive distortion, resonate in response to the portion of the input signal which is at the resonant frequency. For example, since a 1Khz square wave contains a 3Khz signal at 1/3 the amplitude, driving a 3Khz transducer with a 1Khz square wave may cause it to behave as through it was driven by a 3khz signal whose amplitude was 1/3 that of the original.

Since transducers are often driven with square waves, they may thus response to being driven at 1/3 or 1/5 of their resonant frequency. Note, however, that the behavior of a transducer in response to a combination of signals will only match the sum of the behaviors that would be produced by the individual signals if the combined signal's amplitude never gets so high as to overdrive the transducer. Overdriving a transducer may cause the internal generation of additional harmonics, some of which may effectively cancel out those which were present in the original signal.

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