An organ pipe open at both ends and another organ pipe closed at one end will resonate with each other, if their lengths are in the ratio of (a) 1: 1 (b) 1: 4 (c) 2: 1 (d) 1: 2

To understand how an open organ pipe and a closed organ pipe resonate with each other, we need to look at the fundamental frequencies of both types of pipes and how their lengths relate to these frequencies. Organ pipes produce sound through the vibration of air columns, and the way they resonate depends significantly on whether they are open at both ends or closed at one end.

Frequency and Length Relationship

The frequency of the sound produced by a pipe is inversely proportional to its length. This means that shorter pipes will produce higher frequencies, while longer pipes produce lower frequencies. For an organ pipe that is open at both ends, the fundamental frequency can be determined by the formula:

• f_open = v / 2L

where f_open is the fundamental frequency, v is the speed of sound in air, and L is the length of the pipe. On the other hand, for a closed pipe (closed at one end), the fundamental frequency is given by:

• f_closed = v / 4L

Resonance Condition

For two pipes to resonate together, their frequencies must be in a simple ratio. Given that the frequency of the open pipe is proportional to 1/L and that of the closed pipe is proportional to 1/(4L), we can set up a relationship based on their lengths.

Finding the Ratio

Let’s denote the length of the open pipe as L_open and the length of the closed pipe as L_closed. The frequencies can thus be expressed as follows:

• f_open = v / (2 * L_open)
• f_closed = v / (4 * L_closed)

For resonance, we can set the ratio of the frequencies equal to a simple ratio:

• f_open / f_closed = n / m

Substituting the frequency equations gives us:

• (v / (2 * L_open)) / (v / (4 * L_closed)) = n / m

After simplifying this, we find:

• 2 * L_closed / L_open = n / m

Evaluating the Options

Now, we can evaluate the given options to see which ratio satisfies the condition for resonance:

• (a) 1:1
• (b) 1:4
• (c) 2:1
• (d) 1:2

From our derived relationship, we need to find a ratio where:

• 2 * L_closed = L_open

This means that the length of the open pipe should be double that of the closed pipe, leading us to the conclusion that the correct ratio is:

The Answer

The ratio of lengths should be 1:2, which corresponds to option (d). This means that if you have an open pipe that is twice as long as a closed pipe, they will resonate together. This phenomenon is a beautiful example of how physical properties of sound and resonance work in harmony.