# Resonance and standing waves

### Forced oscillation

• Produced when an external periodic driving force vibrates a system
• The system will vibrate at the same frequency as the applied force
• The system will continue to vibrate as force is applied

### Free oscillation

• When an object is struck, and then allowed to vibrate without further interference

### Natural frequency

• The frequency at which an object will naturally vibrate
• In the absence of a driving force, objects will tend to vibrate at these frequencies

### Resonance

• When forced frequency is similar to the natural frequency and applied in the same direction, resonance occurs
• The amplitude of the vibration increases greatly as a result
• The system hence oscillates with greater amplitude at some frequencies than others
• The resonant frequency is the same as the natural frequency

### Standing waves

• Waves that appear to be standing still
• Waves and reflected waves combine to form standing waves
• The relative positions of the nodes and antinodes are the same, but the wave still travels.
• Nodes are points of minimum displacement – half a wavelength apart
• Antinodes are points of maximum displacement – also half a wavelength apart

### Vibration in a string

• In a string, the frequencies at which standing waves are established are called the natural frequency
• Like all systems, it has more than one natural frequency, but prefers its fundamental
• If f is the frequency of the first harmonic (n = 1), and L is the length of the string,

λn = 2L/n

• Frequency can be calculated using the wave equation, or fn = nf
• The ends of the string are fixed so there is always zero displacement (i.e. nodes)

### Open ended pipe

• e.g. didgeridoo, flute
• At open ends, sound waves are reflected with a phase change
• This produces a pressure node at the ends of the pipe and a displacement antinodes

### Closed ended pipe

• e.g. clarinet
• At closed ends, it reflects sound with no change of phase
• At closed ends, there is a pressure antinode and a displacement node

Double wavelength of open end.

### Summary

##### Open end
• The air particles are free to move in or out
• Hence max displacement but constant average pressure
• Therefore displacement node and pressure antinode
##### Closed end
• Air particles are pressed against the closed end and cannot move
• Hence min displacement but max pressure
• Therefore displacement node and pressure antinode

When a displacement node occurs, a pressure antinode does too, and vice versa.

•  Loud sound occurs at pressure antinodes (displacement nodes)
• Soft sound occurs at pressure nodes (displacement antinodes)