A piano hammer strikes the string and then remains in contact with it for a certain time duration (before being thrown off by forces propagating in the string).
During the time the hammer is in contact with the string, the string is “loaded” by the hammer impedance and wave scattering occurs.
Figure 12:
A string loaded by a mass and spring.
A lossless three-port scattering junction, as shown in Fig. 13, can be developed that defines the energy transfer between the hammer and the two sections of string.
Figure 13:
A three-port hammer/string scattering junction.
At the junction, the velocities of the string sections and the hammer spring must all be equal,
In addition, the sum of the forces at the junction must be zero, since it is a massless point,
Using these expressions and the previous wave variable definitions, the scattering equations become: