Problem 25 - University Physics with Modern Physics

Question

Standing sound waves are produced in a pipe that is 1.20 \(\mathrm{m}\) long. For the fundamental and first two overtones, determine the locations along the pipe (measured from the left end) of the dis- placement nodes and the pressure nodes if (a) the pipe is open at both ends and (b) the pipe is closed at the left end and open at the right end.

Step-by-Step Solution

Answer

Nodes for open pipe: 0.60m (fundamental), 0.40m & 0.80m (first overtone), 0.30m, 0.60m & 0.90m (second overtone). Nodes for closed pipe: 0m & 1.20m (fundamental), 0m & 0.60m (first overtone), 0m, 0.24m & 0.72m (second overtone).

Step 1: Understanding the Problem for an Open Pipe

For a pipe open at both ends, standing waves form such that antinodes are at both ends. The fundamental frequency has one node between the ends of the pipe, the first overtone has two nodes, and the second overtone has three nodes.

Step 2: Analyzing the Fundamental Frequency in an Open Pipe

For the fundamental frequency, the pipe is one half-wavelength long. The length of the pipe is 1.20 meters, so the wavelength \( \lambda \) of the fundamental frequency is 2.40 meters. Displacement nodes at the fundamental are located at \( L/2 \), so the node is at 0.60 meters.

Step 3: Analyzing the First Overtone in an Open Pipe

The first overtone corresponds to a whole wavelength fitting inside the pipe. Thus, displacement nodes occur at distances of \( L/3 \) and \( 2L/3 \), which are 0.40 meters and 0.80 meters from the left end.

Step 4: Analyzing the Second Overtone in an Open Pipe

For the second overtone, 1.5 wavelengths fit into the pipe's length, thus nodes are located at \( L/4 \), \( 2L/4 \), and \( 3L/4 \), or 0.30 meters, 0.60 meters, and 0.90 meters from the left end.

Step 5: Understanding the Problem for a Closed Pipe

In a pipe closed at one end, there is a node at the closed end and an antinode at the open end. The fundamental frequency has one node at the left end, the first overtone adds a node at the halfway point, and the second overtone adds another node at \( 1/6 \) of the wavelength.

Step 6: Analyzing the Fundamental Frequency in a Closed Pipe

The fundamental mode has a node at the closed end (0 meters) and an antinode at the open end (1.20 meters). Displacement and pressure nodes overlap at 0 meters, and there is a pressure node also at 1.20 meters.

Step 7: Analyzing the First Overtone in a Closed Pipe

For the first overtone, three-quarters of a wavelength fit inside the pipe, creating nodes at the closed end (0 meters) and at one-third of the pipe length (0.60 meters) where a pressure node occurs.

Step 8: Analyzing the Second Overtone in a Closed Pipe

The second overtone fits 1.25 wavelengths in the pipe, creating nodes at the closed end, at one-fifth of its length (0.24 meters), and at three-fifths (0.72 meters), where pressure nodes are present.

Key Concepts

Open Pipe

An open pipe allows standing waves to form with antinodes at both ends. This is because the air is free to move, hence more displacement can occur at these points. In an open pipe, the fundamental frequency or first harmonic has one node placed midway between the two antinodes.
The first overtone involves two nodes here, and the second overtone involves three.

• **Fundamental Frequency:** The wavelength is equal to twice the length of the pipe. Since the length is 1.20 meters, the wavelength is 2.40 meters. A displacement node occurs at half the pipe's length, 0.60 meters.
• **First Overtone:** This brings about a full wavelength across the pipe's length, leading to nodes at 0.40 meters and 0.80 meters.
• **Second Overtone:** 1.5 wavelengths fit within the pipe, producing nodes at 0.30 meters, 0.60 meters, and 0.90 meters.

Closed Pipe

In a pipe closed at one end, standing waves have a node at the closed end and an antinode at the open end. The differences between nodes and antinodes play a crucial role in the formation of sound waves. Here, the configuration only allows odd harmonics.
For the fundamental frequency, there's one node at the closed end and an antinode at the open end.

• **Fundamental Frequency:** The node is at the closed end (0 meters), with an antinode at 1.20 meters.
• **First Overtone:** Three-quarters of a wavelength fits, creating an extra node at one-third the pipe's length (0.40 meters).
• **Second Overtone:** 1.25 wavelengths fit, leading to nodes at 0 meters, 0.24 meters, and 0.72 meters.

Displacement Nodes

Displacement nodes are points of no movement. In a pipe, these nodes mark areas where the air remains relatively still compared to other positions.
In an open pipe, displacement nodes are found between antinodes, and thus represent regions of minimum amplitude.
For example:

• In the fundamental frequency of an open pipe, the node is at the midpoint, 0.60 meters.
• In a closed pipe, displacement node at the closed end, with additional nodes occurring as overtones increase.

Pressure Nodes

Pressure nodes are points where pressure variation is minimal. They are the opposite of displacement nodes, as they occur where there is maximum movement.
In a closed pipe, the closed end is always a displacement and pressure node, since no air can move in or out, making pressure constant.

• **Open Pipe:** Pressure nodes occur around displacement antinodes. Primary example is in full wavelengths of overtones.
• **Closed Pipe:** Pressure nodes at the closed end and additional points as the overtones grow.

Overtones

Overtones are higher frequencies beyond the fundamental frequency. In physics, they are harmonics allowing more waves to fit into the given pipe length, causing varied patterns of nodes and antinodes.
Different configurations of open and closed pipes have unique overtone sequences.

• **Open Pipes:** Support a sequence involving every natural number multiple of the fundamental, i.e., whole number harmonics.
• **Closed Pipes:** Only support odd-numbered harmonics because the node-antinode pattern restricts even harmonics.

Overtones enrich the sound, creating complex wave patterns that build the characteristic tone of musical instruments.