JEE Main & Advanced Physics Wave Mechanics Important Terms Regarding Wave Motion

(1) Amplitude (a) : Maximum displacement of a vibrating particle of medium from it's mean position is called amplitude.

(2) Wavelength \[(\lambda )\] : It is equal to the distance travelled by the wave during the time in which any one particle of the medium completes one vibration about its mean position.

(i) Or distance travelled by the wave in one time period is known as wavelength.

(ii) Or is the distance between the two successive points with the same phase.

(3) Frequency (n) : Frequency of vibration of a particle is defined as the number of vibrations completed by particle in one second.

It is the number of complete wavelengths traversed by the wave in one second. Units of frequency are hertz (Hz) and per second.

(4) Time period (T) : Time period of vibration of particle is defined as the time taken by the particle to complete one vibration about its mean position.

It is the time taken by the wave to travel a distance equal to one wavelength

Time period = 1/Frequency Þ T = 1/n

(5) Wave pulse : It is a short wave produced in a medium when the disturbance created for a short time.

(6) Wave train : A series of wave pulse is called wave train.

(7) Wave front : A wave front is a line or surface on which the disturbance has the same phase at all points. If the source is periodic, it produces a succession of wave front, all of the same shape. Ripples on a pond are the example of wave fronts.

(8) Wave function ; It is a mathematical description of the disturbance created by a wave. For a string, the wave function is a displacement for sound waves. It is a pressure or density fluctuation where as for light waves it is electric or magnetic field.

Now let us consider a one dimensional wave travelling along x-axis. During wave motion, a particle with equilibrium position x is displaced some distance y in the direction perpendicular to the x-axis. In this case y is a function of position (x) and time (t).

i.e. \[y=f(x,\,t)\]. This is called wave function .

Let the wave pulse is travelling with a speed v, after a time t, the pulse reaches a distance vt along the +x-axis as shown. The wave function now can be represented as \[y=f(x-vt)\]

If the wave pulse is travelling along \[-x-\]axis then \[y=f(x+vt)\]

If order of a wave function to represent a wave, the three quantities x, v, t must appear in combinations \[(x+vt)\] or \[(x-vt)\]

Thus \[y={{(x-vt)}^{2}},\,\,\sqrt{x-\upsilon t},\,A{{e}^{-B{{(x-vt)}^{2}}}}\] etc. represents travelling waves while \[y=({{x}^{2}}-{{v}^{2}}{{t}^{2}}),\,(\sqrt{x}-\sqrt{vt})\], \[A\sin \text{ }(4{{x}^{2}}\text{ }9{{t}^{2}})\] etc. doesn't represents a wave.

(9) Harmonic wave : If a travelling wave is a sin or cos function of \[(x\pm vt)\] the wave is said to be harmonic or plane progressive wave.

(10) The wave equation : All the travelling waves satisfy a differential equation which is called the wave equation. It is given by \[\frac{{{\partial }^{2}}y}{\partial {{t}^{2}}}={{v}^{2}}\frac{{{\partial }^{2}}y}{\partial {{x}^{2}}}\]; where \[v=\frac{\omega }{k}\]

It is satisfied by any equation of the form \[y=f(x\pm vt)\]

(11) Angular wave number or propagation constant (k) : Number of wavelengths in the distance \[2\pi \] is called the wave number or propagation constant i.e. \[k=\frac{2\pi }{\lambda }\].

It is unit is rad/m.

(12) Wave velocity (v) : It is the distance travelled by the disturbance in one time period. It only depends on the properties of the medium and it independent of time and position. \[v=n\lambda =\frac{\lambda }{T}=\frac{\omega }{2\pi }=\frac{\omega }{k}\]

(13) Group velocity \[({{v}_{g}})\]: The velocity with which the group of waves travels is known as group velocity

or the velocity with which a wave packet travels is known as group velocity \[{{v}_{g}}=\frac{d\omega }{dk}\].

(14) Phase \[(\phi )\] : The quantity which express at any instant, the displacement of the particle and it's direction of motion is called the phase of the particle.

If two particles of the medium, at any instant are at the same distance in the same direction from their equilibrium positions and are moving in the same direction then they are said to be in same phase e.g. In the following figure particles 1, 3 and 5 are in same phase and point 6, 7 are also in same phase.

(15) Intensity of wave : The wave intensity is defined as the average amount of energy flow in medium per unit time and per unit of it's cross-sectional area. It's unit is \[W/{{m}^{2}}\]

Hence intensity \[(l)=\frac{\text{Energy }}{\text{Area }\times \text{ Time}}=\frac{\text{Power}}{\text{Area}}

=2{{\pi }^{2}}{{n}^{2}}{{a}^{2}}\rho v\]

\[\Rightarrow \] \[I\propto {{a}^{2}}\] (when v, \[\rho =\]constant)

where a = Amplitude, n =Frequency, v = Wave velocity, \[\rho =\] Density of medium.

At a distance r from a point source of power P the intensity is given by \[I=\frac{P}{4\pi {{r}^{2}}}\,\Rightarrow \,\,I\propto \frac{1}{{{r}^{2}}}\]

The human ear can hear sound of intensity up to \[{{10}^{-12}}\,W/{{m}^{2}}\]. This is called threshold of intensity. The upper limit of intensity of sound which can be tolerated by human ear is \[1\,W/{{m}^{2}}\]. This is called threshold of pain.

(16) Energy density : The energy associated with unit volume of the medium is defined as energy density

Energy density \[=\frac{\text{Energy}}{\text{Volume}}=\frac{\text{Intensity}}{\text{Velocity}}=\frac{2{{\pi }^{2}}{{n}^{2}}{{a}^{2}}\rho v}{v}=2{{\pi }^{2}}{{n}^{2}}{{a}^{2}}\rho \]