Waves

The One Thing

A wave transfers energy and information through oscillations without producing a net transport of matter. Its behaviour is determined by the direction of particle vibration, the wave equation and speed, and the effects of reflection and superposition.

Reactions, Processes and Experiments

Key Terms

• Wave: A disturbance that travels through a medium or space, transferring energy without overall transfer of the medium.
• Mechanical wave: A wave that requires a material medium for propagation, such as sound or a wave on a string.
• Transverse wave: A wave in which particles of the medium vibrate perpendicular to the direction in which the wave travels.
• Longitudinal wave: A wave in which particles of the medium vibrate parallel to the direction of wave propagation.
• Crest: The point of maximum positive displacement in a transverse wave.
• Trough: The point of maximum negative displacement in a transverse wave.
• Compression: A region of high pressure and density in a longitudinal wave.
• Rarefaction: A region of low pressure and density in a longitudinal wave.
• Amplitude: The maximum displacement of a particle from its mean position; it is related to the energy carried by a wave.
• Wavelength: The distance between two consecutive points in the same phase, such as two successive crests or compressions.
• Frequency: The number of complete oscillations made per second, measured in hertz.
• Time period: The time taken by a particle to complete one oscillation.
• Wave velocity: The speed at which a particular phase of the wave travels through the medium.
• Phase: The state of vibration of a particle at a given instant, represented by its position and direction of motion.
• Progressive wave: A wave that travels continuously through a medium and transfers energy from one region to another.
• Principle of superposition: When two or more waves overlap, the resultant displacement is the algebraic or vector sum of the individual displacements.
• Constructive interference: Interference in which waves reinforce each other, producing a larger resultant amplitude.
• Destructive interference: Interference in which waves oppose each other, producing a smaller or zero resultant amplitude.
• Reflection of a wave: The return of a wave into its original medium after striking a boundary.
• Fixed-end reflection: Reflection from a rigid boundary in which a transverse pulse is inverted and undergoes a phase change of $\pi$ radians.
• Free-end reflection: Reflection from a freely movable boundary in which a transverse pulse is not inverted and has no phase change.
• Stationary wave: A wave pattern formed by the superposition of two identical waves traveling in opposite directions, producing fixed nodes and antinodes.
• Node: A point in a stationary wave where the displacement is always zero.
• Antinode: A point in a stationary wave where the displacement has maximum amplitude.
• Beat: A periodic variation in intensity or loudness produced when two waves of nearly equal frequencies interfere.
• Beat frequency: The number of beats heard per second, equal to the absolute difference between the two frequencies.

Easily Confused

• Transverse and longitudinal waves: Transverse particles vibrate perpendicular to propagation, whereas longitudinal particles vibrate parallel to propagation.
• Crest and compression: A crest is a point of maximum positive displacement in a transverse wave, whereas a compression is a high-pressure, high-density region in a longitudinal wave.
• Trough and rarefaction: A trough is a point of maximum negative displacement in a transverse wave, whereas a rarefaction is a low-pressure, low-density region in a longitudinal wave.
• Fixed-end and free-end reflection: A fixed-end reflection is inverted and has a phase change of $\pi$, whereas a free-end reflection is not inverted and has no phase change.
• Progressive and stationary waves: A progressive wave transfers energy continuously from one region to another, whereas a stationary wave has fixed nodes and antinodes and does not transport energy along the medium on average.
• Node and antinode: A node has permanently zero displacement, whereas an antinode has maximum amplitude.
• Frequency and time period: Frequency is the number of oscillations per second, while time period is the time for one oscillation; $f=1/T$.
• Wave speed and particle motion: Wave velocity describes the speed of propagation of a particular phase, whereas particles of the medium oscillate about equilibrium positions.
• Mechanical and electromagnetic waves: Mechanical waves require a material medium, whereas electromagnetic waves can travel through a vacuum.
• Beat and beat frequency: A beat is one variation in loudness or intensity, whereas beat frequency is the number of beats per second, $f_{\text{beat}}=|f_1-f_2|$.
• Open and closed pipes: An open pipe has $f_n=nv/(2L)$, whereas a pipe closed at one end supports only odd harmonics, $f_n=(2n-1)v/(4L)$.

What Gets Asked

• Define or classify waves from the direction of particle vibration. The mark-losing error is confusing perpendicular vibration in transverse waves with parallel vibration in longitudinal waves.
• Use the basic relations $v=f\lambda$, $f=1/T$, $\omega =2\pi f$, and $k=2\pi /\lambda$. A common slip is to confuse wave velocity with frequency or time period.
• Write progressive-wave equations: $y(x,t)=A\sin (kx-\omega t+\varphi )$ for the positive $x$-direction and $y(x,t)=A\sin (kx+\omega t+\varphi )$ for the negative $x$-direction. The sign before $\omega t$ identifies the direction.
• Calculate phase differences using $\Delta \varphi =2\pi \Delta x/\lambda$ or $\Delta \varphi =2\pi \Delta t/T$. The relevant separation must be identified as a distance or a time interval.
• Apply superposition and interference conditions. Constructive interference corresponds to path difference $n\lambda$, while destructive interference corresponds to $(n+\frac{1}{2})\lambda$.
• Describe reflection, stationary waves, pipe harmonics, or beats. The specific slips are reversing the fixed-end and free-end observations, omitting that a closed pipe has only odd harmonics, and using $f_1+f_2$ instead of $f_{\text{beat}}=|f_1-f_2|$.