4.5.3 Boundary Conditions for Standing Waves

• Stationary waves can form on strings or in pipes
• In both cases, progressive waves travel in a medium (i.e. the string or air) and superimpose with their reflections
• The number of nodes and antinodes that fit within the available length of medium depends on:
  • The frequency of the progressive waves
  • The boundary conditions (i.e. whether the progressive waves travel between two fixed ends, two free ends or a fixed and a free end)

Standing Waves on Stretched Strings

• When guitar strings are plucked, they can vibrate with different frequencies
• The frequency with which a string vibrates depends on:
  • The tension, which is adjusted using rotating 'tuning pegs'
  • The mass per unit length, which is the reason why a guitar has strings of different thicknesses

Standing wave on a guitar string

• For a string, the boundary condition can be
  • Fixed at both ends
  • Free at both ends
  • One end fixed, the other free

• At specific frequencies, known as natural frequencies, an integer number of half wavelengths will fit on the length of the string
  • As progressive waves of different natural frequencies are sent along the string, standing waves with different numbers of nodes and antinodes form

Standing Waves in Pipes 

• When the air within a pipe vibrates, longitudinal waves travel along the pipe
• Simply blowing across the open end of a pipe can produce a standing wave in the pipe
• For a pipe, there is more than one possible boundary condition, theses are pipes that are:
  • Closed at both ends
  • Open at both ends
  • Closed at one end and open on the other

Nodes & Antinodes

• When a progressive wave travels towards a free end for a string, or open end for a pipe:
  • The reflected wave is in phase with the incident wave
  • The amplitudes of the incident and reflected waves add up
  • A free end is a location of maximum displacement - i.e. an antinode

Standing wave inside a pipe open at both ends

• When a progressive wave travels towards a fixed end for a string, or closed end for a pipe:
  • The reflected wave is in anti-phase with the incident wave
  • The two waves cancel out
  • A fixed end is a location of zero displacement - i.e. a node
  • The open end is therefore a location of maximum displacement - i.e. an antinode

Standing wave inside a pipe open at both ends