How Do Drumhead Strike Points and String Tension Affect Musical Acoustics?

[jon9986]

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.

Jonathan

 

[Aaron Barnard]

Answer:
1. The modes that will be strong are the symmetric modes, which are modes with nodes at the two points of contact. The modes which will be absent are the anti-symmetric modes which have a node in between the points of contact.
2. Doubling the tension of a string increases its fundamental frequency by half an octave because doubling the tension causes the same amount of stretch to occur over half the original length of the string. This raises the pitch by 12 semitones.
3. For a midrange string on a piano, it is likely that the effective length of string contacted by the hammer is about 1/4 of the total length of the string. This would mean that the harmonic number above which it would be a poor approximation to pretend that the hammer makes a simple point contact is about 16th harmonic (or 8th overtone).

 

[Graeme Robinson]

,

Musical acoustics can be a complex and fascinating subject, and I'm happy to offer some help with your questions.

When striking a drumhead at two symmetrically located points, the modes that will be strong are the ones that have a node (point of no vibration) at the two striking points. This includes the fundamental mode (with one node in the center) and any higher modes with nodes at those points. The modes that will be absent are the ones with antinodes (points of maximum vibration) at the striking points, as they will cancel out.

Doubling the tension on a string will increase its fundamental frequency by one octave, not half an octave. This is because the frequency of a string is inversely proportional to its length, and doubling the tension effectively halves the length. This raises the pitch by 12 semitones, or one octave.

For a midrange string on a piano, the effective length of string contacted by the hammer is typically around 1/8 to 1/10 of the total length of the string. This means that the hammer is making contact with a length of string that is around 8-10 times shorter than the entire string. This would correspond to the 7th or 8th harmonic, with a frequency that is 7-8 times higher than the fundamental frequency.

I hope this helps with your understanding of musical acoustics. If you have any further questions, don't hesitate to ask. Good luck with your studies!