Acoustic Principles: Exercises & Solutions for Exam Preparation

[Sarah_L]

:shy: Any help will be apreciated. These are some pre-exam exercises that were given to my daughter I would like to be able to check the answers and help her with any difficulties.

1) A string, whose density is 0.01 kg/m, and whose length is of 60
cm is stretched with a tension of 576 Newtons.
a) What is the frequency of the third harmonic?
b) At what point was the string plucked?

2) A tuning fork sounds the note A (440 Hz). What would you expect
to happen if it was struck near the mouth of the closed tube (closed
at one end).
Explain why this happens. Would the sound of a tuning fork tuned an
octave higher have the same effect?

3)If you hold down (without playin) the G on the piano, and then
strike the lower D sharply, you will hear a note corresponding to
the upper D. This can't come from the string that was hammered, as
it stops vibrating as soon as you release the key, so where does it come from? Explain this, from what you know about string vibrations.
Both this phenomenon, and the one discussed in question 2 are
examples of a particular acoustic mechanism. What is it?

Thanks a lot!

 

[Andrew Mason]

1) A string, whose density is 0.01 kg/m, and whose length is of 60 cm is stretched with a tension of 576 Newtons.
a) What is the frequency of the third harmonic?
b) At what point was the string plucked?

The string will vibrate with a fundamental frequency:
$$\lambda \nu = \sqrt{T/(m/l)}$$

Since $\lambda/2 = 60 cm$ for the fundamental frequency:
$$\nu = (2/.6)*\sqrt{576/(.01/.6)} = 619.67 Hz$$
How many times the fundamental is the third harmonic?

In order to excite the third harmonic you cannot have a node (place of 0 motion) in the middle of the string, so it would have to be plucked in the middle I would think.

2) A tuning fork sounds the note A (440 Hz). What would you expect
to happen if it was struck near the mouth of the closed tube (closed
at one end).
Explain why this happens. Would the sound of a tuning fork tuned an
octave higher have the same effect?

How long is the tube? It matters.
The sound will travel down the tube and reflect back from the closed end and constructively interfere with the incident wave. If the tube is 1/4 of a wavelength, it will reflect back from the opening and constructively interfere with both the wave from the tuning fork and the wave reflected from the closed end and you will have resonance (standing wave) - loud sound.

3)If you hold down (without playin) the G on the piano, and then
strike the lower D sharply, you will hear a note corresponding to
the upper D. This can't come from the string that was hammered, as
it stops vibrating as soon as you release the key, so where does it come from? Explain this, from what you know about string vibrations.
Both this phenomenon, and the one discussed in question 2 are
examples of a particular acoustic mechanism. What is it?

Resonance.

AM

 

[soeren]

[..]
3)If you hold down (without playin) the G on the piano, and then
strike the lower D sharply, you will hear a note corresponding to
the upper D. This can't come from the string that was hammered, as
it stops vibrating as soon as you release the key, so where does it come from? Explain this, from what you know about string vibrations.
Both this phenomenon, and the one discussed in question 2 are
examples of a particular acoustic mechanism. What is it?

Thanks a lot!

Hard to say in english ;-)
But i'll try it.

When you press an key on a piano, you will hear all tunes, which length of the waves is a multiple of the tune you hit sharp.

Octave - > double frequence
Quint -> 3/2 frequence
Quart -> 4/3 frequence

so when you hit g', you must hear g'', d'', g'', and so on...


greets
soeren