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

Verified step by step guidance

1. Identify the type of pipe: Since the pipe is closed at one end and open at the other, it is a half-open pipe. In such pipes, the standing wave patterns form with a node at the closed end and an antinode at the open end.
2. Understand the harmonics in a half-open pipe: The fundamental frequency (first harmonic) has one node and one antinode. The first overtone (third harmonic) has three nodes and two antinodes. The second overtone (fifth harmonic) has five nodes and three antinodes.
3. Calculate the positions of the nodes: For a half-open pipe, the nodes occur at positions given by \( x_n = \frac{n \lambda}{4} \) for odd values of n (1, 3, 5, ...). Here, \( \lambda \) is the wavelength of the standing wave, which can be found using the length of the pipe and the harmonic number.
4. Calculate the positions of the antinodes: The antinodes occur at positions given by \( x_a = \frac{(2n-1) \lambda}{4} \) for odd values of n (1, 3, 5, ...).
5. Apply the formulas for each harmonic: For the fundamental frequency, calculate the position of the node and antinode using the respective formulas. Repeat the process for the first and second overtones, ensuring to use the correct harmonic number to find the wavelength for each case.

Key Concepts

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. In a pipe, these waves create fixed points called nodes, where there is no displacement, and antinodes, where the displacement is maximum. The pattern of these nodes and antinodes depends on the boundary conditions of the pipe, such as whether it is open or closed at its ends.

Boundary Conditions

Boundary conditions refer to the constraints applied at the ends of a medium where waves are present. For a pipe closed at one end and open at the other, the closed end must be a displacement node (no movement), while the open end must be an antinode (maximum movement). These conditions dictate the wavelengths and frequencies of the standing waves that can exist in the pipe.

Harmonics

Harmonics are the different frequencies at which a system can oscillate, with the fundamental frequency being the lowest. In a closed-open pipe, the fundamental frequency has one node at the closed end and one antinode at the open end. The first overtone (first harmonic) has one additional node and antinode, and the second overtone (second harmonic) has two additional nodes and antinodes, affecting the locations of displacement and pressure nodes along the pipe.