Amplitude, wavelength, frequency and period Knowledge-Based Study

Study notes

1. What is a Wave?

   A mechanical wave is a disturbance that travels through a medium, transferring energy and momentum from one location to another without transporting matter. As the wave propagates, individual particles of the medium oscillate about their stable equilibrium positions.

   Crests, Troughs, and Equilibrium

   In a periodic transverse wave, particles oscillate perpendicularly to the direction of wave travel:

   • Equilibrium Position: The undisturbed position of the medium in the absence of a wave.
   • Crest: The point of maximum positive displacement above the equilibrium position.
   • Trough: The point of maximum negative displacement below the equilibrium position.

   Amplitude ($A$)

   The amplitude of a wave is the maximum magnitude of displacement of any particle from its equilibrium position. It is measured in meters (m) in the SI system.

   Key Note: Amplitude measures the distance from the equilibrium line to a crest, or from the equilibrium line to a trough. It is not the peak-to-peak distance (the full vertical distance from trough to crest is $2A$). In mechanical waves, the energy transported is directly proportional to the square of the amplitude ($E \propto A^2$).

   Wavelength ($\lambda$)

   A periodic wave repeats itself identically in both space and time. The spatial repetition of a wave is quantified by its wavelength.

   Wavelength ($\lambda$) is the minimum distance between two successive points on a wave that are in the same phase of oscillation. Its SI unit is the meter (m).

   Identifying Wavelength

   Wavelength can be determined by measuring the distance between:

   • Two consecutive crests,
   • Two consecutive troughs, or
   • Any two successive points moving in the same direction with identical displacement.

   In longitudinal waves, such as sound waves in air, wavelength corresponds to the distance between the centers of two successive compressions or two successive rarefactions.

   Period ($T$)

   The temporal repetition of a wave is described by its period. The period ($T$) is the time required for one complete oscillation or cycle of a particle in the medium, or the time taken for one full wavelength to pass a fixed point in space. The SI unit of period is the second (s).

   Frequency ($f$)

   The frequency ($f$) represents the number of complete oscillations or wave cycles that occur per unit of time:

   $$f = \frac{1}{T}$$$$T = \frac{1}{f}$$

   Units of Frequency

   Frequency is measured in hertz (Hz), where $1\text{ Hz} = 1\text{ s}^{-1}$. A wave with a frequency of 50 Hz completes 50 full cycles every second.

   Important Property: The frequency of a wave is determined exclusively by the source generating the wave and remains constant when the wave travels from one medium to another.

   Wave Speed ($v$)

   The speed at which a wave disturbance travels through a medium is termed the wave speed ($v$). Wave speed depends primarily on the physical and mechanical properties of the medium (such as tension, density, and elasticity), not on the amplitude or frequency of the wave.

   The Fundamental Wave Relationship

   Because a wave travels a distance equal to one wavelength ($\lambda$) in a time interval equal to one period ($T$), the speed can be expressed as:

   $$v = \frac{\lambda}{T}$$Since $f = \frac{1}{T}$, substituting frequency gives the standard wave equation:$$v = f\lambda$$

   • $v$: wave speed in meters per second ($\text{m/s}$)
   • $f$: frequency in hertz ($\text{Hz}$)
   • $\lambda$: wavelength in meters ($\text{m}$)

   Conceptual Summary

   When a wave enters a new medium:

   • The frequency ($f$) remains unchanged because it is set by the source.
   • The wave speed ($v$) changes due to the new medium's properties.
   • The wavelength ($\lambda$) adjusts proportionally to satisfy $v = f\lambda$.