phy C4 standing/stationary waves

standing/stationary waves

• formed by the superposition of two waves that are:

  • same frequency, amplitude, speed, type (FAST)

  • opposite direction

  • results from progressive wave being reflected antiphase at a boundary → superposition of incident and reflected wave

• nodes: points that always show no displacement (due to destructive interference)

• antinodes: points that reach maximum amplitude of displacement (due to constructive interference)

• distance between consecutive nodes (or antinodes) = λ/2

• all particles between two nodes are in phase

standing vs travelling waves

• phase

  • standing wave: all points between two nodes are in phase

  • travelling wave: all particles along a wavelength have different phase

• amplitude

  • standing wave: all points on the wave have fixed amplitudes. max amplitude is fixed between zero (at nodes) and double the amplitude of component waves (at antinodes)

  • travelling wave: all points on the wave have the same amplitude

• wavelength

  • standing wave: 2 x distance between adjacent nodes (or antinodes)

  • travelling wave: shortest distance between two points in phase

• energy

  • standing wave: does not transmit energy, but there is an energy associated with it

  • travelling wave: transmits energy

• wave pattern

  • standing wave: wave does not move

  • travelling wave: wave moves

boundary conditions

• The number of nodes and antinodes that fit within the available length of medium depends on:

  • The frequency of the travelling waves

  • The boundary conditions (i.e. whether the travelling waves travel between two fixed ends, two free ends or a fixed and a free end) -. standing waves that meet these conditions is a possible resonant mode

• for stretched strings

  • two fixed ends→ so the standing wave created must have a node at the fixed end.

  • travelling waves meeting a fixed boundary will be reflected anti-phase → forms standing wave

why open end of tube has standing wave?

pressure difference due to compressions/rarefactions → partially reflected in phase at free end since denser to less dense → forms standing wave

resonant frequencies

• aka harmonics. frequencies of standing waves.

  • f_n = n f₁

• fundamental frequency aka first harmonic: standing wave with lowest possible frequency

• timbre/quality: depends on relative amplitude of different harmonics produced by the instrument

first harmonic of a 1.0m long stretched string is 650Hz, what will its first harmonic be if its length is shortened to 80cm, keeping tension constant?

• first harmonic = 1.0m for stretched string (two nodes) → ½ λ

• λ = 1.0 × 2 = 2.0m

• v = fλ = 650 × 2 = 1300 → tension constant means v constant

• λ_new = 0.8 × 2 = 1.6

• f₁ = v/λ = 1300/1.6 = 812.5 Hz

resonance

• resulting amplitude of system becomes maximum when driving frequency of external driving force = natural frequency of system. there is a maximum transfer of energy from the driving force to driven system.

  • natural frequency f₀ depends on dimensions and nature of material

• useful applications

  • musical instruments

  • radio receivers: tuner adjusted so that frequency of electrical oscillations in circuit is same as that of radio waves → amplifies signals of that radio wave + diminish radio waves of other frequencies

  • microwave oven: microwaves with frequency similar to natural frequency of vibration of water molecules → resonate, absorb energy from microwaves, heat up water in food (containers don’t heat up because they don’t have water molecules)

  • magnetic resonance imaging: strong varying radio frequency EM fields are used to cause oscillations in atomic nuclei → resonance, molecules absorb energy → analyse pattern of energy absorption, computer generates image

• not useful applications

  • shattering of glass

  • earthquakes: buildings forced to oscillate in resonance with seismic waves, increased energy transfer causes more damage (solution: dampeners)

  • human internal organs resonate in response to external frequencies (usually < 10Hz) → high levels of vibration can damage heart/lungs/intestines/brain

damping

• process where energy is taken from an oscillating system due to dissipative forces

frequency response graph

• amplitude of oscillation against driving frequency

• as driving frequency increases,

  1. f < f₀: amplitude of driven oscillating system increasing

  2. f = f₀: resonance, max amplitude

  3. f increased further, f > f₀, amplitude decreases

• at resonance, theoretically infinite amplitude if there was no damping, because continuous input of energy

  • damping: amplitude/energy increases until rate of energy transfer = rate of dissipation

effect of damping on frequency response graph

• amplitude of peak oscillation decreases

• peak becomes broader, spreads over wider range of frequencies

• resonance occurs at frequency smaller than f₀

types of damping

• light damping

  • oscillations are maintained about equilibrium position after system has been displaced. amplitude of oscillations decreases over a long time.

  • occurs for most oscillations

• critical damping

  • no oscillations occur, motion is brought to rest in the shortest possible time

  • eg car suspension system

• heavy/over-damping

  • no oscillations occur but system takes long time to return to equilibrium position compared to critically damped system, because damping force > critical amount

  • eg anti-slam doors

phase difference depending on f vs f₀ (out of syllabus?)

• f << f₀ → in phase

• resonance, f = f₀ → phase difference π/2

• f >> f₀ —> anti-phase, phase difference π