Organ pipes, open at one end, resonate best at their first resonant length. Two pipes have length 23.0 cm and 30.0 cm respectively.

What is the wavelength of the sound emitted by each pipe?
What are the respective frequencies if the speed of sound is 341 m/s?
What is the air temperature in this church?

I'm mot confused by the first question. Does it mean, that the sound will sound best at 1/4 of the pipe's length, or that the pipe has been made so that it's already a quarter of the sound's wavelength?

The "first resonant wavelength" is four times the (open) tube length. That is supposedly the "best" sound that the pipe makes.

The pipe lengths therefore tell you the wavelengths. Divide the sound speed by the wavelength to get the frequencies.

The equation for sound speed vs temperature can be used to determine the temperature.

In the context of organ pipes open at one end, the first resonant length refers to the length of the pipe that produces the fundamental frequency, which is the lowest frequency and the one that resonates most strongly. To determine the wavelength of the sound emitted by each pipe, we need to consider that the fundamental frequency of a pipe open at one end is given by the equation:

f = (v / 2L)

Where:
f is the frequency,
v is the speed of sound, and
L is the length of the pipe.

Let's calculate the wavelength and frequency for each pipe:

For the first pipe with a length of 23.0 cm:
1. Convert the length to meters: 23.0 cm = 0.23 m
2. Calculate the frequency using the given speed of sound (v = 341 m/s):
f1 = (341 / (2 * 0.23)) = 741.74 Hz
3. Calculate the wavelength (λ1) using the formula:
λ1 = v / f1 = 341 / 741.74 = 0.46 m

For the second pipe with a length of 30.0 cm:
1. Convert the length to meters: 30.0 cm = 0.3 m
2. Calculate the frequency using the given speed of sound:
f2 = (341 / (2 * 0.3)) = 567.33 Hz
3. Calculate the wavelength (λ2) using the formula:
λ2 = v / f2 = 341 / 567.33 = 0.60 m

So, the wavelength for the first pipe is approximately 0.46 m, and for the second pipe, it is approximately 0.60 m.

Next, to find the respective frequencies for each pipe using the given speed of sound (v = 341 m/s), we can use the formula mentioned earlier:

For the first pipe:
f1 = (341 / (2 * 0.23)) = 741.74 Hz

For the second pipe:
f2 = (341 / (2 * 0.3)) = 567.33 Hz

Thus, the respective frequencies for each pipe are approximately 741.74 Hz for the first pipe and 567.33 Hz for the second pipe.

Finally, from the given information, we cannot determine the air temperature in the church, as it is not provided. Air temperature affects the speed of sound, but without that specific data, we cannot determine the exact temperature.

In the context of the question, "first resonant length" refers to the fundamental frequency or the first harmonic of the pipe. For an open pipe, such as an organ pipe, the fundamental frequency corresponds to a wavelength that is twice the length of the pipe.

To find the wavelength of the sound emitted by each pipe, we need to use the formula:

λ = 2L

Where λ represents the wavelength and L is the length of the pipe.

For the pipe with a length of 23.0 cm, the wavelength is:

λ₁ = 2 * 23.0 cm

For the pipe with a length of 30.0 cm, the wavelength is:

λ₂ = 2 * 30.0 cm

To convert the wavelengths to meters, we divide by 100:

λ₁ = 2 * 23.0 cm / 100 = 0.46 meters
λ₂ = 2 * 30.0 cm / 100 = 0.60 meters

Now, let's calculate the frequencies of the sound emitted by each pipe. The formula to find the frequency is:

f = v / λ

Where f represents the frequency, v is the speed of sound, and λ is the wavelength.

Given that the speed of sound is 341 m/s, we can calculate the frequencies as follows:

For the pipe with a wavelength of 0.46 meters:
f₁ = 341 m/s / 0.46 meters

For the pipe with a wavelength of 0.60 meters:
f₂ = 341 m/s / 0.60 meters

Finally, to determine the air temperature in the church, more information is needed. The air temperature affects the speed of sound, and without that information, we cannot calculate the temperature accurately. The speed of sound in air is dependent on factors such as temperature, humidity, and air composition.