Halliday, Resnick & Walker — Waves–II: Problem 8

Hot chocolate effect. Tap a metal spoon inside a mug of water and note the frequency \( f_i \) you hear. Then add a spoonful of powder (say, chocolate mix or instant coffee) and tap again as you stir the powder. The frequency you hear has a lower value \( f_s \) because the tiny air bubbles released by the powder change the water’s bulk modulus. As the bubbles reach the water surface and disappear, the frequency gradually shifts back to its initial value. During the effect, the bubbles don’t appreciably change the water’s density or volume or the sound’s wavelength. Rather, they change the value of \( dV/dp \)—that is, the differential change in volume due to the differential change in the pressure caused by the sound wave in the water. If \( f_s/f_i = 0.333 \), what is the ratio \( (dV/dp)_s / (dV/dp)_i \)?