Piano string excitation in the case of small hammer mass

When a hammer of mass m strikes a string of mass M, it is dependent on the string for the force which will finally cause it to rebound. They remain in contact for a finite time, and this makes a general solution for the resulting string motion rather complicated. In the limit m≪M for a very stiff hammer the solution is relatively simple, but has some features that seem not to have been noted before. If the original hammer kinetic energy is E0, then 0.865 E0 is ultimately transferred to the string. Most interesting is that the normal modes with antinodes at the striking point receive only about two‐thirds as much energy as certain other modes. Analytic expressions are also given for hammer masses which are finite but still small enough that the solution involves only the first reflection from the near end of the string. These apply to cases with m as large as 0.463 M, and show how the larger values of m affect the relative amplitude of the high‐frequency modes. The mode energy spectrum level rolls off at 6...