Standing Waves

Standing (Stationary) Wave: a superposition of two waves moving in opposite directions, each having the same amplitude and frequency.

Interference

• Constructive Interference (in phase): the amplitudes of the traveling waves add up to create the amplitude of the standing wave
• Destructive Interference (out of phase: phase difference of 180 degrees): the amplitudes of the standing waves cancel out, hence the standing wave has zero amplitude (if amplitudes differ, total cancellation doesn't occur)

Antinodes: points where the amplitude is at a maximum (maximum displacement)

Nodes: points where the amplitude is zero

Characteristics of Standing Waves

• The nodes and antinodes do not move along the waves, and always stay in the same position
• No energy is transferred (unlike traveling/progressive waves)
• Standing waves result from interference between an incoming wave and its reflected counterpart
• Reflection from a hard boundary causes a 180° phase change (π radians)
• Node-to-node or antinode-to-antinode spacing = ½ wavelength

Standing Waves in Strings and Pipes

• Boundary Conditions: the conditions at the ends of a string or pipe
• First Harmonic: the longest possible wavelength
• nth Harmonic: the nth longest possible wavelength
• In an open-closed pipe (or a string with one fixed boundary and one free boundary), the harmonics are first, third, fifth, and so on (there are no even-numbered harmonics)
• Equation for the wavelength: λ_n = 4L/n, where n is the harmonic number (1st, 3rd, 5th, etc.), and L is the length (m)
• In an open-open or closed-closed pipe (or a fixed-fixed or free-free string), the harmonics are first, second, third, and so on (there are both even and odd harmonics)
• Equation for the wavelength: λ_n = 2L/n, where n is the harmonic number (1st, 2nd, 3rd, etc.), and L is the length (m)

Resonance

• Resonance: when the driving frequency of an oscillator matches the natural frequency of a system, causing the system to vibrate at maximum amplitude
• Natural Frequency: the frequency at which a system vibrates or oscillates when it is disturbed and allowed to oscillate without any external forces acting on it

Damping

Damping: techniques/forces used to restrain and reduce oscillations and vibrations, causing the amplitude of the system to slowly decrease until the system comes completely to rest (having lost all of its energy).

Types of Damping

• Light damping: There is only a small amount of damping. The system will continue to oscillate, but the amplitude of the oscillations decreases exponentially over time
• Heavy damping: There is a large amount of damping. The system gradually dissipates all its energy. It does not oscillate, but returns very slowly to its equilibrium state
• Critical damping: There is a very large amount of damping. The system returns to its equilibrium state as quickly as possible without any oscillations