Hot chocolate effect. Tap a metal spoon inside amug of water and note the frequency fi you hear. Then add aspoonful of powder (say, chocolate mix or instant coffee) and tapagain as you stir the powder. The frequency you hear has a lowervalue fs because the tiny air bubbles released by the powderchange the water’s bulk modulus. As the bubbles reach the watersurface and disappear, the frequency gradually shifts back to itsinitial value. During the effect, the bubbles don’t appreciablychange the water’s density or volume or the sound’s wavelength. Rather, they change the value of dV/dp—that is, the differentialchange in volume due to the differential change in the pressurecaused by the sound wave in the water. If fs /fi=0.333, what is theratio (dV/dp)s/(dV/dp)i?

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Shock Waves

Shock waves are common in continuum mechanics, and their dominance is attributable to the fact that they are stronger (or even less) compressible. Substantial perturbations propagated as sudden dynamic changes of a compressible medium supersonically. Shock waves are all around us in our daily lives. Lightning, earthquakes, volcanic eruptions, and meteorite impact all produce them in nature. Even the earth, which is protected by its magnetic field, generates a shock wave as in solar wind that travels around 20 Earth radiuses. These shock waves might be either constant, linked to the body, or unsteady, i.e., moving around with time. All of the shocks mentioned above can be damaging, so precautions must be taken to reduce their impact.