Properties of Waves

Fundamental Relationship: The speed of a wave ($v$) is determined by the product of its frequency and its wavelength. This relationship is expressed by the universal wave equation: $v = f\lambda$

Physical Interpretation: For a wave traveling at a constant speed through a specific medium, frequency and wavelength are inversely proportional. If the frequency increases (more cycles per second), the wavelength must decrease (shorter distance between cycles) to maintain the same velocity.

Wave Speed ($v$): This represents the rate at which the wave front or energy propagates through the medium, typically measured in $m/s$.

Transverse Waves: In these waves, the particles of the medium oscillate perpendicularly to the direction of energy transfer. Examples include electromagnetic waves (like light) and vibrations on a guitar string; these waves consist of peaks (crests) and valleys (troughs).

Longitudinal Waves: In these waves, the particles oscillate parallel to the direction of energy transfer. This creates regions of high pressure called compressions and regions of low pressure called rarefactions. Sound waves in air are the most common example.

Polarization: A key distinction is that only transverse waves can be polarized, as their oscillations occur in a plane perpendicular to the direction of travel, whereas longitudinal oscillations are restricted to the axis of travel.

Displacement-Distance Graphs: These provide a 'snapshot' of the entire wave at a single moment in time. The distance between two consecutive peaks on this graph directly represents the wavelength ($\lambda$).

Displacement-Time Graphs: These track the movement of a single point in the medium over a duration of time. The time interval between two consecutive peaks on this graph represents the period ($T$).

Phase Relationships: Graphs allow for the visualization of phase, where points moving in the same direction with the same displacement are said to be 'in phase'.

Check the X-Axis: Always verify if a wave graph's horizontal axis represents distance (giving you $\lambda$) or time (giving you $T$). Misidentifying the axis is a frequent cause of calculation errors.

Unit Consistency: Ensure all values are converted to SI units (meters, seconds, Hertz) before using the wave equation. Be particularly alert for prefixes like millimeters ($mm = 10^{-3}m$) or kilohertz ($kHz = 10^{3}Hz$).

Equilibrium Reference: When measuring amplitude, always measure from the center line (equilibrium) to the peak, never from the trough to the peak (which would be $2A$).