Stationary Waves MDCAT Quiz with Answers

A stationary wave is otherwise called a standing wave. It’s a wave that remains in a constant position with points of constructive and destructive interference, which don’t travel. Unlike progressive waves, which propagate through a medium, stationary waves appear to be “standing still.” They form when two waves of the same frequency, amplitude, and wavelength travel in opposite directions and interfere with each other. Understanding of stationary waves is very important for MDCAT students, especially in the part dealing with acoustics, vibration, and resonance through different media.

Formation of Stationary Waves
When two waves of identical shape and traveling in opposite directions superpose, they form stationery waves. Such waves may be produced as the result of reflection of a wave at a boundary. A wave traveling along a string or in a tube reflects off a fixed end, and the reflected wave combines with the incident wave to produce a standing wave. Points of constructive interference are called antinodes; those of destructive interference are nodes.

Nodes: These are the points where there is no displacement in the wave since the two interfering waves cancel each other out. At these points, the amplitude of the standing wave is always zero.

Antinodes: These are the points where the displacement is maximum, as the two interfering waves reinforce each other. The amplitude of the wave at these points is maximum.

Characteristics of Stationary Waves
No Energy Transfer: Unlike progressive waves, stationary waves do not transfer energy from one place to another. Instead, the energy oscillates between the nodes and antinodes, which makes stationary waves unique.

Amplitude Variation: The amplitude of a stationary wave varies from zero at the nodes to a maximum at the antinodes. The oscillation is one of the prominent features of stationary waves.

Waveform: The wave pattern seems to be locked in space, and the points of maximum and zero displacement don’t move. This creates a “standing” appearance where the wave doesn’t travel.

Types of Stationary Waves
Stationary Waves in a String When a vibrating string, such as in a guitar or a violin, is fixed at both ends, stationary waves are produced. The string vibrates in specific modes, with nodes at the ends where it is fixed and antinodes in between. The fundamental frequency of the string corresponds with the lowest frequency for which a stationary wave can be produced.

Stationary Waves in Air Columns: Stationary waves can also be produced in air columns, such as in pipes, tubes, and organ pipes. The air at the open ends of the pipe vibrates freely and forms antinodes while that at the closed end reflects back to form a node.

Overtones and Harmonics: When a standing wave is produced on a string or in an air column, it can oscillate at a number of frequencies, giving rise to higher-order standing waves, called overtones or harmonics. The fundamental frequency is the first harmonic, and the overtones are the higher harmonics. These harmonic frequencies are integer multiples of the fundamental frequency.

Mathematical Description of Stationary Waves
A standing wave can be mathematically described by the superposition of two traveling waves moving in opposite directions. For a standing wave on a string, the displacement
????
(
????
,
????
)
y(x,t) can be written as:

????
(
????
,
ℳ
)
=
2
????
sin
⁡
(
????
????
)
cos
⁡
(
????
∀
)
y(x,t)=2Asin(kx)cos(ωt)
where:

????
A is the amplitude of the individual waves,
????
k is the wave number,
∠∠
x is the position along the string,
????
ω is the angular frequency, and
????
t is time.
This equation demonstrates the sinusoidal nature of the wave in both space (in the
????
x-direction) and time. The term
sin
⁡
(
????
????
)
sin(kx) describes the spatial distribution of the wave, whereas
cos
⁡
(
᳜
????
)
cos(ωt) describes the time dependence. The sum of these two gives the typical form of a standing wave.

Applications of Stationary Waves
Musical Instruments: Many musical instruments are based on the principle of stationary waves. For stringed instruments, for instance, guitars and violins, the stationary waves are the ones produced by the vibrating strings. The stationary waves created by the air column inside a pipe give musical notes in wind instruments such as flutes and organ pipes.

Acoustics: Stationary waves play an important role in the acoustics of rooms and concert halls. The interference patterns of sound waves within a space can lead to regions of constructive interference (loud spots) and destructive interference (quiet spots), which affect the quality of sound.

Resonance: Stationary waves are intimately associated with the phenomenon of resonance, whereby an object oscillates with increasing amplitude when its natural frequency is in accord with that of an applied force. In instruments, this is the principle by which certain frequencies are amplified, resulting in the louder and clearer sounds produced.

MDCAT Quiz: Stationary Waves

MDCAT students may find questions on the formation and properties of stationary waves, including calculation of nodes and antinodes, fundamental frequency, and harmonic overtones. Other types of questions can be solving for wavelength and frequency of stationary waves in strings or air columns and their applications in musical instruments and acoustics.

Free Flashcards for Stationary Waves
MDCAT students can strengthen their understanding of wave properties, such as the formation of nodes and antinodes, and the relationship between frequency, wavelength, and harmonic numbers, using free flashcards on stationary waves. It may also include examples from real applications in musical instruments, which would enable students to visualize these concepts in a better way. Regular review of these flashcards will enhance students’ knowledge of stationary waves and, therefore, will be well-prepared for the MDCAT Quiz.