Superposition (Stationary waves only) Flashcards
Define a wave.
A wave is a disturbance that travels through a medium or vacuum.
State the Principle of Superposition.
The Principle of Superposition states that when two or more waves of the same kind meet at a point in space, the resultant displacement at that point is equal to the vector sum of the displacements of the individual waves at that point.
Define stationary waves.
Stationary wave is the result of the superposition of two progressive waves of the same type, frequency, amplitude and speed, travelling along the same line but in the opposite directions. (Meet head-on) It is also known as a standing wave.
Define an antinode and a node. State whether the component waves arrive in-phase or anti-phase at the antinodes and nodes.
An antinode is a point in a stationary wave where the amplitude is the maximum. The component waves always arrive in phase at the antinodes. Their positions are marked by A.
A node is a point in a stationary wave where the amplitude is zero. The component waves always arrive anti-phase at the nodes (crest meets trough). Their positions are marked by N.
State the properties of a stationary wave.
The resultant wave has the following features:
1) The wave profile does not propagate
2) The particles of the wave (except for those at the nodes) oscillate about their respective equilibrium positions with the same frequency but different amplitudes (different displacement-time graphs for each particle). The frequency is the same as that of the component waves.
3) An antinode is a point in a stationary wave where the amplitude is the maximum. The component waves always arrive in phase at the antinodes. Their positions are marked by A.
4) A node is a point in a stationary wave where the amplitude is zero. The component waves always arrive anti-phase at the nodes (crest meets trough). Their positions are marked by N.
5) Between two adjacent nodes, all particles oscillate in phase (i.e. they reach their respective maxima, minima and equilibrium positions at the same instant.
6) Distance between two adjacent nodes or antinodes is half a wavelength
7) Particles in neighbouring segments vibrate 180 degrees or pi rad out of phase with each other.
Describe the fundamental mode of vibration.
In a string that is stretched and fixed at the two ends such that nodes will always be produced at these points, It is the simplest mode of vibration where the wave pattern is a single loop and the frequency at which the standing wave is vibrating at is known as the fundamental frequency (lowest frequency that can be generated by a string L).
As the frequency of the string is slowly increased, at a particular frequency, the string exhibits the second mode of vibration/2nd harmonic/1st overtone whereby the wave pattern has two loops.
State the formula used to calculate the wavelength and frequency for stationary waves in strings.
nth harmonic = (n-1)th overtone
Wavelength = 2L/n
Frequency = n(v/2L)
Explain why the end of the string that is attached to the mechanical oscillator is considered a node.
The amplitude of oscillations is small compared to that of the antinode. Hence, the boundary conditions are still valid.
State the relationship between displacement-distance and pressure variation-distance graphs for longitudinal waves.
They are 90 degrees out of phase with each other.
Displacement node = Pressure antinode
Displacement antinode = Pressure node (as they are exposed to the air, there is 0 excess pressure)
State the formula used to calculate the wavelength and frequency for stationary waves in air columns or pipes
For a closed pipe:
Fundamental mode = 1st harmonic
1st overtone = 3rd harmonic
2nd overtone = 5th harmonic
(n-1) overtone = (2n-1)th harmonic
Wavelength = 4L/(2n-1) Frequency = (2n-1)(v/4L)
For an open pipe (same as string):
Fundamental mode = 1st harmonic
1st overtone = 2nd harmonic
2nd overtone = 3rd harmonic
(n-1) overtone = nth harmonic
Wavelength = 2L/n Frequency = n(v/2L)
Describe what happens when a sound wave is sent into a closed pipe.
A closed pipe is one where one end of the pipe is closed. When a sound wave is sent into the closed pipe, the wave propagates to the end of the pipe and is reflected. The reflected wave superposes with the incident wave and a stationary wave is formed. A displacement node is formed at the close end of the pipe while a displacement antinode is formed at the open end of the pipe.
Explain how the end correction comes about.
The displacement antinode at the open ends of the pipes are actually located slightly outside the pipe, resulting in a small end correction c to be included the calculation of wavelength.
Always add c to L
