# Resonant Lengths in open air column question

In summary, the fundamental frequency of a pipe with open ends is related to the length of the pipe and the speed of sound in air.

## Homework Statement

An organ pipe 1.2m long and open at both ends produces a note with the fundamental frequency. If the speed of sound in air is 345 m/s, what is the fundamental frequency?

## Homework Equations

Wave equation (f = v/lambda)

## The Attempt at a Solution

My textbook solves the problem like so:

My question is: why do they use the full wavelength here? As I understand it, the fundamental frequency is only half a wavelength. This would be more like the second resonant length, which was not what the question asks.

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the fundamental frequency is only half a wavelength
First, a frequency is not a length.
Fundamental frequency means the lowest note the pipe can produce. Being open at both ends, the pipe will only contain half a wavelength. Thus, as you imply, the fundamental frequency here corresponds to a half wavelength. Hence the λ=2L.
But to find the frequency that you hear you must divide the speed of sound by the full wavelength.

haruspex said:
First, a frequency is not a length.
Fundamental frequency means the lowest note the pipe can produce. Being open at both ends, the pipe will only contain half a wavelength. Thus, as you imply, the fundamental frequency here corresponds to a half wavelength. Hence the λ=2L.
But to find the frequency that you hear you must divide the speed of sound by the full wavelength.

So the wavelength has to be twice the length of the pipe because the pipe is open on both sides? So the fundamental frequency corresponds to the actual length of the pipe?

How would this be solved if the pipe were closed on one end?

How would this be solved if the pipe were closed on one end?
In that case, how much of a wavelength (at fundamental frequency) would fit in the pipe?

haruspex said:
In that case, how much of a wavelength (at fundamental frequency) would fit in the pipe?

1/4 wavelength?

Also, was the rest of the conclusion earlier correct?

1/4 wavelength?
Yes.

the fundamental frequency corresponds to the actual length of the pipe?
It might be clearer to think in terms of the "fundamental wavelength", i.e. the longest wavelength which can be produced. That is related, on the one hand, to the pipe length and its number of closed ends, and on the other to the fundamental frequency by the speed of sound in air.

## 1. What is the concept of resonant lengths in an open air column?

Resonant lengths refer to the specific lengths at which sound waves can resonate or vibrate within an open air column, producing a noticeable increase in sound intensity. This phenomenon is based on the physics of standing waves and is commonly observed in musical instruments such as flutes or organ pipes.

## 2. How do resonant lengths affect the quality of sound produced?

The resonant lengths of an open air column can greatly impact the quality of sound produced. When the length of the column is at a resonant length, the sound waves can reinforce and amplify each other, creating a louder and more sustained sound. However, at non-resonant lengths, the sound waves may cancel each other out, resulting in a weaker and less defined sound.

## 3. What factors determine the resonant lengths of an open air column?

The resonant lengths of an open air column are primarily determined by its physical dimensions, such as length and diameter. The material, temperature, and shape of the column can also play a role in determining the resonant lengths. Additionally, the frequency and wavelength of the sound waves being produced can also affect the resonant lengths.

## 4. How can resonant lengths be manipulated to produce different pitches of sound?

By adjusting the length or diameter of an open air column, the resonant lengths can be manipulated to produce different pitches of sound. Shorter lengths and wider diameters typically produce lower pitches, while longer lengths and narrower diameters produce higher pitches. This is why musical instruments with open air columns, such as flutes or organ pipes, are designed with different lengths and diameters to produce a range of pitches.

## 5. Can resonant lengths only be observed in open air columns?

No, resonant lengths can also be observed in other types of columns, such as closed air columns (e.g. bottles or tubes with one end closed). However, the concept of resonant lengths is most commonly associated with open air columns, as they allow for a more noticeable and pronounced effect on sound waves.

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