Calculating Velocity of Wave in Wood Stick and Resonance Tube

[Fleet]

Homework Statement

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)

Homework 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.

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.

 

[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.

 

[kerimek]

I would approach this problem by first identifying the key variables and equations that are relevant to calculating the velocity of the wave in the wood stick. The given information states that the velocity of sound in air is 343 m/s, and we are also given the length of the resonance tube relative to the wood stick. This suggests that we may need to use the equation for the velocity of a wave in a medium, which is v = λf, where v is the velocity, λ is the wavelength, and f is the frequency.

Next, I would consider the given equations for a standing wave in a string and a half-open resonance tube. These equations involve the length of the string or air column (L), the number of the partial tone (n), and the wavelength (λ). We are given the length of the resonance tube relative to the wood stick, but we do not know the actual lengths of either the stick or the tube. Therefore, we cannot directly use these equations to solve for the velocity of the wave in the wood stick.

However, we can use the given information and equations to set up a system of equations and solve for the unknown variables. For example, we can use the equation for a half-open resonance tube (L = (2n-1)λ/4) and the given information that the resonance tube is four times as long as the wood stick to set up the following equation:

4L_wood = (2n-1)L_tube

We can also use the equation for a standing wave in a string (L = nλ/2) and the known frequency of the standing wave in the wood stick (n=1) to set up the following equation:

L_wood = nλ_wood/2

From these two equations, we can solve for the wavelength of the standing wave in the wood stick (λ_wood) in terms of the wavelength of the standing wave in the resonance tube (λ_tube):

λ_wood = 8λ_tube/15

Now we can use this relationship to substitute into the equation for the velocity of a wave (v = λf) and solve for the velocity of the wave in the wood stick:

v_wood = λ_woodf = (8/15)λ_tubef

We know the velocity of sound in air (343 m/s) and the frequency of the standing wave in the wood stick (f), so we can solve for