Wave Parameters, Graphical Representations, and Wave Equations

Wave Frequency

• Definition: Frequency is the number of vibrations produced per second by a source.
• Symbols: Denoted by the letters $f$, $n$, or $\nu$ (nu).
• S.I. Unit:
  • second$^{-1}$ (symbol $s^{-1}$)
  • hertz (symbol $Hz$)
• Formula:     $f = \frac{1}{T}$
• Derivation of Relationship with Time Period:
  • In time $T\,s$, the number of vibrations = $1$.
  • In $1\,s$, the number of vibrations = \frac{1}{T} = f.
  • Mathematical formulation: $f = \frac{1}{T}$
• Intrinsic Characteristics:
  • The frequency of a wave is strictly equal to the frequency of vibration of its source.
  • It is an inherent characteristic of the source which produces the disturbance.
  • It does not depend on the amplitude of vibration.
  • It does not depend on the nature of the medium in which the wave propagates.

Wavelength

• Definition: Wavelength is the distance travelled by a wave in one time period of vibration of the particle of the medium.
• Symbol: Denoted by the Greek letter $\lambda$ (lambda).
• S.I. Unit: metre ($m$).
• Medium Dependence: The value of wavelength depends on the specific medium in which the wave travels.
• Definition in Longitudinal Waves:
  • The distance between two consecutive compressions is equal to one wavelength ($\lambda$).
  • The distance between two consecutive rarefactions is equal to one wavelength ($\lambda$).
• Definition in Transverse Waves:
  • The distance between two consecutive crests is equal to one wavelength ($\lambda$).
  • The distance between two consecutive troughs is equal to one wavelength ($\lambda$).

Wave Velocity

• Definition: The distance travelled by a wave in one second is called its wave velocity or wave speed.
• Physical Nature:
  • It is the speed with which energy is transferred from one place to another place by wave motion.
  • It is not the velocity of an individual particle vibrating about its mean position.
• Symbol: Denoted by the letter $V$.
• S.I. Unit: metre per second ($m\,s^{-1}$).
• Properties and Medium Determinants:
  • Wave velocity is constant for a given medium.
  • It depends directly on the elasticity and the density of the medium.
  • The velocity changes when the wave passes from one medium to another medium.

Graphical Representations of Waves

• Displacement-Time Graph:
  • Shows the variation of displacement with time for a single particle of the medium at a given position as a wave propagates through the medium.
  • The maximum displacement (amplitude) is represented by the letter $a$.
  • The time period of vibration is represented by the letter $T$.
  • Each particle of the medium goes through such motion sequentially (one after another as the wave moves in the medium), rather than simultaneously.
• Displacement-Distance Graph:
  • Shows the displaced positions of particles of the medium from their mean positions at a single instant when a wave propagates through the medium.
  • Also referred to as a snapshot of a wave.
  • Illustrates a wave (e.g., a transverse wave) at a specific point in time.
  • The amplitude of the wave is shown by the letter $a$.
  • The wavelength is shown by the letter $\lambda$

Relationship Between Wavelength, Wave Velocity, and Frequency

• Defined Variables:
  • Wave velocity = $V$
  • Time period = $T$
  • Frequency = $f$
  • Wavelength = $\lambda$
• Derivation Steps:
  • By definition of wavelength:         $\text{Wavelength } \lambda = \text{Distance travelled by wave in one time period}$
  • Expressed in terms of wave velocity and time period:         $\text{Wavelength } \lambda = \text{Wave velocity} \times T$
  • Substituting symbols:         $\lambda = V \times T$
  • Rearranging terms:         $V \times T = \lambda$
  • Since $T = \frac{1}{f}$:         $V = f \times \lambda$