# Velocity of wave

1. Nov 16, 2009

### Fleet

1. The problem statement, all variables and given/known data
A certain music instrument consists of a stick of wood placed horizontally and a resonance "tube" (see attached picture, which is from the original assignment-paper) placed vertically under the stick of wood. When the wood-stick is hit, is creates a standing wave, which is amplified in the resonance "tube". The resonance "tube" is half-open (closed in botton, open in top). The music instrument is seen below:

I have to calculate the velocity of the transverse wave v_stick in the wood stick.
Information I am given:
Velocity of sound: v_sound=343 m/s

The resonance tube is four times as long as the stick of wood (the distance between the triangles on the picture)

2. Relevant equations
For a string with a standing wave we have that:
$L=n*\frac{\lambda}{2}$

For a half-open resonance "tube" we have that:
$L=(2n-1)\frac{\lambda}{4}$

L is the length of the string (wood stick) or the air "pillar" in which the standing wave exists, n is the number of the partial-tone and lambda is the wave length.

3. The attempt at a solution

$4L_{stick}=L_{tube}$

I'm unsure of what i know of the wave in the resonance tube, but if I can say that the standing wave in the wood stick has n=1 and the one in the resonance tube has n=8 (see the attached picture), I get:

$4\frac{lambda_{stick}}{2}=\frac{15\lambda_{tube}}{4} \Leftrightarrow 2\lambda_{stick}=\frac{15}{4}\lambda_{tube}$

I know I can insert the wave eqaution v=lambda*frequency, but what I have just seems so wrong and I have thought very long time about it. I hope you are willing to help me. I'm so stuck.

Best regards.

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Last edited: Nov 16, 2009
2. Nov 16, 2009

### Delphi51

Wow, the two n's make it complicated - many different solutions depending on their values, I think.

I would have begun with the fact that the
frequency on wood = frequency in tube
and put in the conversion to wavelength on each side and then convert the wavelength to L's using those two formulas. I get an expression for the velocity that is (2n-1)*343/8m where n is the air tone number and m the wood tone number. This may be the same as you have. I don't know how you will choose m and n.