There are two ways of transfer of energy;

1. By the transfer of matter.

2. Without the transfer of matter.

Second method is a wave process. In this process, disturbance carries energy from one place to another without the transfer of matter. The disturbance is called a **wave** and the process is called **wave motion**.

Thus, *a wave is defined as a disturbance which propagates from one place to another without net transport of matter and the process of transfer of the disturbance is called wave motion. *Sound, light, water ripples, etc. are the examples of wave.

## Characteristics of Wave Motion

- Wave motion is a disturbance propagating in a medium.
- It transfers energy as well as momentum from one point to another.
- When it propagates in a medium, medium particles execute vibration about their mean position i.e. their motion is SHM.
- Wave motion has finite and fixed speed given by \[v=fλ\] Where, $v$ is velocity, $f$ is frequency and $λ$ is wavelength of the wave.
- The wave speed is different from particle velocity in a medium.
- When a wave travels in a medium, there is a continuous phase difference among the successive medium particles.
- The vibrating particles of the medium possess both kinetic energy and potential energy.
- It follows reflection, refraction, interference and diffraction but polarization is only for transverse wave.

## Types of Wave

There are three types of wave motion;**1. Electromagnetic wave or non-mechanical wave**

The wave which does not require a medium for its propagation is known as electromagnetic wave. Light, heat, radio waves etc. are electromagnetic waves.**2. Mechanical wave**

The wave which requires a medium for its propagation is known as mechanical wave. Sound wave, water waves, waves on pipes and strings, seismic waves, etc. are mechanical waves.**3. Matter wave**

The wave associated with a moving matter is known as matter wave. It is explained on the basis of quantum mechanics.

### Wavelength (λ)

The distance travelled by a wave in one time period is called **wavelength** $(λ)$. It is the distance between any two nearest points which are in the same phase.

### Frequency (f)

The number of oscillations/vibrations/cycles per second is called **frequency** $(f)$. It may also be defined as the number of waves passing through a point per unit time. \[f=\frac{\text{Number of cycles}}{\text{time}}\] Its unit is $\text{Hertz}$ $\text{(Hz)}$. $1\;\text{Hz} = 1 \;\text{cycle/second}$.

When a sinusoidal wave passes through a medium, every particle in the medium undergoes SHM with the same frequency. The frequency of the wave is the frequency of the vibrating source.

### Time Period (T)

The time for one complete cycle is called **time period** $(T)$. \[T=\frac{1}{f}\]

### Wave Speed

The linear distance covered per unit time by the wave is called **wave speed** $(v)$.

The distance travelled by a wave in time period $T\;\text{sec} = λ$

The distance travelled by a wave in $1\;\text{sec} = \frac{λ}{T}$ \[v=\frac{λ}{T}\] \[v=fλ\] The value of $v$ depends on the elastic and inertial properties of the medium i.e. it is constant for a given medium but changes according to the nature of the medium.

### Particle Speed

During the propagation of mechanical wave, the medium particles undergo up and down motions around their mean positions. The speed of these medium particles is known as **particle speed**. The displacement of medium particle in time $t$ is given by \[y=A\sin(ωt-kx)\] Differentiating with respect to time $t$, \[\frac{dy}{dt}=A\cos(ωt-kx)\] \[v=A\cos(ωt-kx)\] This gives the particle speed of the medium. For maximum speed, $\cos(ωt-kx)=1$, \[v_{\text{max}}=ωA\] Similarly, acceleration of the medium particle is given by \[a=\frac{d^2y}{dt^2}=-ω^2A\sin(ωt-kx)=-ω^2y\]

### Amplitude

The maximum distance through which a medium particle moves either side of the mean position is called **amplitude** of the wave. The amplitude at the crest and the trough are identical.

### Wave Number (k)

The number of waves per unit distance is called **wave number** $(k)$. Its unit is $\text{m}^{-1}$. Mathematically, \[\text{Wave Number}\;(k)=\frac{2π}{λ}\]

### Phase of a Wave

The argument of the sine/cosine in a given wave equation is called **phase**. The equation of the progressive wave is, \[y=a\sin(ωt-kx)\] In this equation, $(ωt-kx)$ is phase denoted by $Φ$. \[Φ=ωt-kx\] The phase changes with both time $(t)$ and space coordinate $(x)$. When one wave is ahead of another by some angle then the difference in angle is called **phase difference**.

### Relation between Phase Difference and Path Difference

Let us consider a complete wave of wavelength $λ$ which starts from $A$ and reaches $B$. If another wave starts from $B$ then the path difference between these two waves will be $λ$ and phase difference will be $2π$ radian.

For path difference $=λ$, phase difference $= 2π$

For path difference $= 1$, phase difference $= \frac{2π}{λ}$

For path difference $= x$, phase difference $= \frac{2π}{λ}x$

Thus, \[\text{Phase Difference} \; (Φ)= \frac{2π}{λ}x\]

**More on Wave And Wave Motion**