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Decoupling between first sound and second sound in $^3$He - superfluid $^4$He mixtures

Bulk superfluid helium supports two sound modes: first sound is an ordinary pressure wave, while second sound is a temperature wave, unique to inviscid superfluid systems. These sound modes do not usually exist independently, but rather variations in pressure are accompanied by variations in temperature, and vice versa. We studied the coupling between first and second sound in dilute $^3$He - superfluid $^4$He mixtures, between 1.6 K and 2.2 K, at $^3$He concentrations ranging from 0 to 11 %, under saturated vapor pressure, using a quartz tuning fork oscillator. Second sound coupled to first sound can create anomalies in the resonance response of the fork, which disappear only at very specific temperatures and concentrations, where two terms governing the coupling cancel each other, and second sound and first sound become decoupled.