The model described above does not predict damping of the acoustic 
modes. Yet damping is experimentally observed, and it would therefore be 
desirable to be able to calculate damping times of nanoparticles embedded in 
a homogeneous medium. The parameter that determines the acoustic inter- 
action of a sphere and a matrix is the acoustic impedance of both materials 
Z = v L, where  is the material density and v L is the longitudinal sound 
velocity of the material. The acoustic impedance governs the reflection of 
sound waves at a material interface. If the impedance of a sphere equals 
the impedance of a matrix in which it is embedded, sound waves are not re- 
flected but propagate out of the sphere immediately, and the mode will be 
highly damped. If there is a large impedance mismatch between the sphere 
and the matrix, the sound waves are strongly reflected and will remain con- 
fined in the sphere much longer. In this case, the acoustic mode will have a 
very low damping.