Very High Frequency Modes

When the added harmonic's frequency is a very high multiple of the main driving frequency it is seen that the bubble still responds to relative phase changes between both driving components (FIG. 5). The second frequency is changed like $n*f,~n=10,~20,~40$, $f=23.4kHz$. FIG. 5 shows the Bjerknes and buoyancy forces and the stable bubble positions of a bubble with ambient radius of $4\mu m$ as a function of temporal phase difference. The spatial phase shift is 0deg. Fig.5a corresponds to Fig. 2a, where two allmost symmetric stability lines exist, whose symmetry is broken due to buoyancy. With increasing order of the added harmonic a change in response is seen: the phase interval during which the position changes shrinks at high frequency. When the $40*f$ harmonic is added to the driving, the bubble almost digitally switches between an upper and lower position. This may be attributed to the increased interaction of the harmonic with the afterbounce frequency which is almost equal to the linear resonance frequency of the $4\mu m$ bubble [20]. The radial dynamics i.e. the minimum radius at collapse is not effected by adding very high harmonics.

Figure 5: Bjerknes forces and equilibrium positions as a function of temporal phase shift between the two driving components of a bubble undergoing very high frequency harmonic $f+n*f$ driving, $f=23.4kHz$: (a) $n=10$, (b) $n=20$, (c) $n=40$. The driving amplitudes are 1.4bars for $f$ and $0.07bars$ for the $n*f$ component.

a)

b)

c)