# For a resonance tube apparatus (open ended on one side) with a total tube length of 1.00 meter, on a day when the speed of sound is 340 m/s, how many resonance positions will be observed as the piston position is varied for a frequency of 500 Hz? and 1000 Hz?

Use the n as the variable, but it has to be an integer. How many integer solutions of n are between length L of less than one meter? I will be happy to critique your thinking.

## To determine the number of resonance positions observed in a resonance tube apparatus, we can use the equation:

L = (2n - 1) * λ/4

where L is the total tube length, n is the harmonic number (integer), and λ is the wavelength of the sound wave.

Given that the total tube length is 1.00 meter, we can rearrange the equation to solve for n:

n = (L * 4) / λ

Now, let's calculate the number of resonance positions for a frequency of 500 Hz:

First, we need to find the wavelength using the speed of sound formula:

v = λ * f

where v is the speed of sound (340 m/s) and f is the frequency (500 Hz).

λ = v / f

= 340 / 500

= 0.68 m

Substituting the values into the equation for n:

n = (1.00 * 4) / 0.68

= 5.88

Since n has to be an integer, we can round it down to the nearest whole number:

n = 5

Therefore, for a frequency of 500 Hz, there will be 5 resonance positions observed.

Now, let's calculate the number of resonance positions for a frequency of 1000 Hz:

Using the same process, let's first find the wavelength:

λ = v / f

= 340 / 1000

= 0.34 m

Substituting the values into the equation for n:

n = (1.00 * 4) / 0.34

= 11.76

Rounding it down to the nearest whole number:

n = 11

For a frequency of 1000 Hz, there will be 11 resonance positions observed.

In summary, for a resonance tube apparatus with a total tube length of 1.00 meter, there will be 5 resonance positions observed for a frequency of 500 Hz and 11 resonance positions observed for a frequency of 1000 Hz.