Resonance is a phenomenon in which a small-amplitude driving force could produce large-amplitude motion.

Standing waves cause a string to resonate or vibrate at its natural frequency or resonant frequency. Since velocity is constant for a given medium, the equation V=fλ can be used to find the resonant frequency for any given wavelength that creates a standing wave. 

All mechanical structures have natural frequencies at which they resonate. If an outside driving force is applied to a structure at the resonant frequency, the structure will experience maximum vibration and maximum displacement amplitudes. The condition where the natural frequency and the driving frequency are equal is also called resonance. The resonating string reveals the driving force to be the reflected wave. In a non-ideal situation, energy is lost to damping at the resonant frequency and must be replaced by some outside driving force at the same frequency.

In the case of a string, one or both ends may be fixed to a point. For a pipe, one or both ends may be open. If a string is fixed at both ends (e.g. as on a guitar), the ends of the string will be the nodes. Those points are held in place and cannot vibrate. For a string secured at only one end, an antinode will be located at the unfixed end because, unlike a fixed point, this point will be in motion. If a standing wave has been established, no point on the string will have a larger displacement than the unfixed end, so its displacement will be equal to the amplitude of the wave.

Pipes create longitudinal waves, whereas string generates transverse waves. Transverse waves are used in pipe diagrams to more easily illustrate the motion of particles. At regions with nodes, the particles in the standing wave have zero displacement. The regions with antinodes contain particles with maximum displacement. A pipe emitting must be open because disturbances in the fluid outside the air particles at the open end or ends must be moving with maximum displacement. A longitudinal wave travels by the vibrations of constituent particles. To imagine the propagation of a longitudinal wave, think about the particles like people in a crowded subway car. Some people are jostled more than others when the car shakes. The vibrations will not displace the people leaning against a wall at the end of a car because they are caught between a rigid surface and a crowd.

When a pipe is closed at one end, the molecules adjacent to that end cannot oscillate. If an incoming wave displaces nearby molecules, those will collide with the molecules against the end of the pipe. The molecules against the end will rebound against the end and once again collide with the nearby molecules reversing the direction of the wave. For that reason, the closed end of a pipe is the node for particle displacement.

Practice Questions

 

MCAT Official Prep (AAMC)

Physics Online Flashcards Question 1

Physics Question Pack Question 41

 

Key Points

• Resonance is a phenomenon in which a small-amplitude driving force could produce large-amplitude motion.

• Standing waves cause a string to resonate or vibrate at its natural frequency or resonant frequency.

• The condition where the natural frequency and the driving frequency are equal is also called resonance.

• In the case of a string, one or both ends may be fixed to a point. For a string secured at only one end, an antinode will be located at the unfixed end. If a standing wave has been established, no point on the string will have a larger displacement than the unfixed end, so its displacement will be equal to the amplitude of the wave.

• Pipes create longitudinal waves, whereas string generates transverse waves.

• A longitudinal wave travels by the vibrations of constituent particles.

• When a pipe is closed at one end, the molecules adjacent to that end cannot oscillate. The closed end of a pipe is the node for particle displacement.

Key Terms

Resonance: A phenomenon in which an external force or a vibrating system forces another system around it to vibrate with greater amplitude at a specified frequency of operation.

Natural frequency/resonant frequency: The frequency at which a system tends to oscillate in the absence of any driving or damping force.

Resonant Node: Where no displacement of material occur.

Antinode: Where maximum displacement of material occurs.

Standing wave: A vibration of a system in which some particular points remain fixed while others between them vibrate with the maximum amplitude.

Amplitude: The maximum extent of a vibration or oscillation, measured from the position of equilibrium.

Longitudinal wave: A wave vibrating in the direction of propagation.

Transverse wave: A wave vibrating at right angles to the direction of its propagation.