Help With Musical Acoustics

If anyone could help me out with these problems, it would be greatly appreciated.

If one were to strike a drumhead simultaneously at two symmetrically located points, which modes will be strong and which would be absent from the vibration recipe?

If the tension applied to a string is doubled, its fundamental frequency is increased by half an octave because??? Show how many semitones this raises the pitch.

For a midrange string on a piano, look inside and estimate the effective length of string contacted by the hammer (the indentations in the felt are a good clue.) Express this as a fraction of the total length of the string. Use this to estimate the harmonic number (and its frequency) above which it would be a poor approximation to pretend that the hammer makes a simple point contact.


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