Problem 41 - University Physics with Modern Physics

Question

Two organ pipes, open at one end but closed at the other, are each 1.14 m long. One is now lengthened by 2.00 \(\mathrm{cm} .\) Find the frequency of the beat they produce when playing together in their fundamental.

Step-by-Step Solution

Answer

The beat frequency is approximately 1.28 Hz.

Step 1: Identify the properties of the organ pipes

Both organ pipes are open at one end and closed at the other, meaning they produce sound at odd harmonics only. Each pipe originally has a length of 1.14 m.

Step 2: Determine the fundamental frequency of the first pipe

The fundamental frequency of a pipe open at one end and closed at the other is given by \(f = \frac{v}{4L}\), where \(v\) is the speed of sound (approximately 343 m/s at room temperature) and \(L\) is the length of the pipe. For the first pipe: \(f_1 = \frac{343 \text{ m/s}}{4 \times 1.14 \text{ m}}\).

Step 3: Calculate the new length of the second pipe

The second pipe is lengthened by 2.00 cm, which is 0.02 m. Thus, the new length of the second pipe is \(1.14 \text{ m} + 0.02 \text{ m} = 1.16 \text{ m}\).

Step 4: Determine the fundamental frequency of the second pipe

Using the modified length for the second pipe in the frequency formula: \(f_2 = \frac{343 \text{ m/s}}{4 \times 1.16 \text{ m}}\).

Step 5: Find the beat frequency

The beat frequency is the absolute difference between the two frequencies: \(f_{beat} = |f_1 - f_2|\).

Key Concepts

Harmonics

When we talk about harmonics, we refer to the multiples of a fundamental frequency. In simple terms, harmonics are frequencies at which a system naturally vibrates. These are integral multiples of a fundamental frequency, the lowest frequency that can sustain a standing wave pattern. Let's break it down a bit further to clear up any confusion.

• The first harmonic, also known as the fundamental, is the simplest vibration the system can produce.
• Higher harmonics are simply higher integer multiples of the fundamental frequency.
• For example, if the fundamental frequency is 100 Hz, the second harmonic will be 200 Hz, the third 300 Hz, and so on.

So, when you hear a musical note, you're not just hearing the fundamental frequency. You're hearing a rich mix of harmonics, which gives musical instruments their unique sounds. When organ pipes are considered, they typically produce sound at odd harmonics when one end is open and the other closed, because of the particular standing wave pattern they support.

Fundamental Frequency

The fundamental frequency is the lowest frequency produced by a vibrating object and is often considered the basic frequency of that object. Think of it like the base rate of vibration. Imagine holding a guitar string under tension. When you pluck the string, it vibrates in multiple modes, creating what we perceive as a single note. This base vibration is essential because:

• It determines the pitch of the sound. Lower fundamental frequencies produce lower pitches, while higher ones produce higher pitches.
• Understanding it helps us predict other harmonic frequencies and the complete sound wave pattern.

In our exercise's context, the fundamental frequency is determined by the formula \(f = \frac{v}{4L}\) for a pipe with one end closed. Here, \(v\) is the speed of sound, typically around 343 m/s in air, and \(L\) is the length of the pipe. This formula reflects how longer pipes produce lower fundamental frequencies, supporting the idea of larger instruments having deeper tones.

Organ Pipes

When dealing with organ pipes, especially those open at one end and closed at the other, it's important to understand how they produce sound. These organ pipes rely on the principles of resonance and standing waves. A few key points about how they work:

• Such pipes support standing waves with a node at the closed end and an antinode at the open end.
• Only certain wavelengths and thus frequencies can resonate in these pipes, resulting in discrete harmonics.
• Given the condition, the simplest sound they produce is the fundamental frequency, with possible odd harmonics following: e.g., 1st, 3rd, 5th, etc.

In the situation given in the exercise, both pipes are nearly identical in length, which means they should have very similar frequencies. When the length of one pipe is changed slightly, as in the example with one pipe lengthened by 2 cm, their fundamental frequencies differ slightly, leading to what's called a "beat frequency." This is perceived as periodic variations in the loudness of sound, a fascinating phenomenon explained by the interaction of two closely spaced frequencies.