Vibrations
- Vibration: Mechanical phenomenon where oscillation occurs around an equilibrium point
- Free Vibration: When a force is applied to any object, then that object is allowed to vibrate at its natural frequency
- Forced Vibration: When a force is applied to an object and it is forced to vibrate at a particular frequency
- Natural frequency: frequency at which an object vibrates when it is not disturbed by an outside force
Resonance
- Resonance: Phenomenon that occurs where an object is forced to vibrate at its natural frequency
- Energy is transferred from the forced oscillator to the receiver with maximum efficiency, and the amplitude of the vibration is increased greatly
Modes of Vibration
- The different ways in which a system vibrates naturally/oscillates
- The frequency of oscillation is termed as modal frequency. The shape made by the system is called mode shape/harmonic pattern
- The modes of vibration and the different harmonic frequencies depend on the physical nature and dimensions of the system; aka boundary conditions
- Fundamental Frequency: The lowest frequency at which a system will resonate is called the fundamental frequency. Here, the system vibrates in its 1st mode
- Harmonic: Positive integer multiple of the fundamental frequency of a system
- Overtones: allowed resonant modes of vibration above the fundamental mode of vibration
- Nodes: In a standing wave, there are moments when the 2 identical waves travelling in opposite directions are 180˚ out of phase; here a complete destructive interference occurs, and there is no particle displacement
- Anti-nodes: There are also moments when 2 identical waves are in phase and so complete constructive interference occurs and there is maximum particle displacement; here the particles vibrate about the equilibrium position but do not propagate in any direction
Closed Pipes (clarinets)
- One end is open and one end is closed
Displacement
- At the open end, the particle displacement is at a maximum, and is thus an anti-node
- At the closed end, the particle displacement is at a minimum, and is thus a node
- Here the particles always collide with the wall and the air particles that are directly next to the wall do not move at all (thus the displacement is O at this point) and so this is a displacement node
Pressure-Distance
- At the open end, the pressure at the end of the pipe cannot oscillate. Instead, it is fixed at the ambient pressure of the surrounding air. So at open ends there is a pressure node
note: for all equations below:
- n is harmonic number
- L is the length of the pipe
- In is the internodal distance, i.e. distance between 2 nodes
If “n” is the harmonic number
Open Pipes (Flutes)
Displacement-Distance
- An open pipe is one where both the ends of the pipe are open to the outside pressure
- At the open ends of a pipe a compression reflects as a rarefaction and a rarefaction reflects as a compression and so the wave is inverted.
- This occurs as when a compression exits the open pipe a region of low pressure is left relative to the outside and so air particles rush in: thus this a point of maximum particle displacement: displacement antinode.
Pressure-Distance
- At the end of a pipe open to the air, the pressure at the end of the pipe cannot oscillate.
- Instead, it is fixed at the ambient pressure of the surrounding air. This is completely analogous to the fixed end of a string; in other words, at the open end of the pipe, the standing sound wave pattern must have a node.
If “n” is the harmonic number
Strings
- Strings are always fixed at both ends and so at these ends the string has 0 displacement and thus has nodes