# IIT JEE / CET – Main / Advanced Physics 2018 & 2019 : Physics free notes – Waves – All the essentials in one page [ PO’PS.Series.2 ]

Wave motion refers to the transfer of energy ( via periodic disturbances ) through space where there is no transfer of matter . When compared to mechanical waves , electromagnetic waves don’t need a medium to travel . The media through which the mechanical wave is transmitted should have elasticity ( for oscillations ) and inertia ( for storing and transporting energy ).

• Transverse wave –  Here , the elements of the disturbed media of the travelling wave , move perpendicular to the direction of the wave’s propagation. A particle at the crest / trough has zero velocity. The distance between two consecutive crests / troughs is equal to the wavelength of the wave . Therefore , the distance between a consecutive pair of crest / trough is half of the wave’s wavelength . The distance of an element / particle from the mean position is called the amplitude.
• Longitudinal wave – Here , the elements of the disturbed media of the travelling wave , move parallel to the direction of the wave’s propagation. Such a wave moves by means of compressions and rarefactions . In compressions , the distance between any two particles is lesser than the normal distance , hence ,it’s more denser here. The opposite is the case with rarefactions , where , consequently , density is lesser.At a particular instant of time , an element / particle in the compressional phase and another one in the rarefactional phase are exactly out of phase with each other.

Wave Features

• Phase – it refers to the position and velocity of a particle oscillating in a wave. Same phase refers to the particles of the oscillating medium which are in the same displacement from their respective mean positions , moving in the same direction. Here the phase difference is given by 2 n π , where n = 1,2,3… . The opposite case is referred to as out of phase , where the phase difference is given by , where n = 1,3,5 …
• Amplitude ( A ) –  it refers to the maximum displacement of an oscillating particle from its mean position on either side.
• Wave speed – it refers to the distance travelled by the wave in unit time .
• Wavelength ( λ ) – refers to the distance between two nearest particles [ along the direction of the wave’s propagation ] which are in the same phase of vibration .  2) Distance travelled by the wave in one time period of oscillation .
• Wave frequency ( f ) – refers to the no of times an oscillating particle is at it’s maximum displacement on one side during a second of it’s motion.  unit  –  1 Hertz  = 1 cycle / sec .
• Time period ( T )  –  refers to the time the ocillating particle needs to complete one cycle .
• Wave Intensity – refers to the energy transmitted by the wave per second . Note that there is no transfer of matter , but the particles by means of their oscillations transfer energy. For simplicity , it is considered that there is no loss of energy during this process .

W A V E   F U N C T I O N                            –

To mathematically describe the propagation of a wave , functions depending on two variables like x and t are needed . For a wave /pulse travelling towards right , considering a stationary frame with the origin at O , the transverse position y for all positions and times is given by

y ( x , t )  =  f ( x – vt ) . If the wave travels towards left , then    y ( x , t )  =  f (  x + vt ) . As denoted , the function y called wave function depends on the variables x and t . Note that the wave function y ( x , t ) represents the y co-ordinate or the transverse position of any element located at position x at any time t .

TRAVELLING  WAVE  /  PLANE   PROGRESSIVE   WAVE    –

Animation : 3D rendition of a plane wave whose x axis is in the direction of motion.

travelling sinusoidal wave

Defined as a wave travelling in a particular direction with constant amplitude .

One of the basic examples of a periodic continuous wave is a sinusoidal wave . It’s called a sinusoidal wave as the curve of the wave is the same as that of the function sin θ plotted against θ . For such a wave , at t = 0, the wave function is expressed as y ( x,0 ) = A sin ax, where A is the amplitude and a is a constant. When x = 0, y (0,0 ) = A sin a ( 0 ) = 0. Hence , the next value for x for which y =0 is x = λ/2 . Therefore,

y [ λ/2 , 0 ]  =  A sin [ a λ/2 ] = 0

The above equation is valid only if , a λ/2 = π  i.e, a =  2 π /λ   –

y [ x,0 ] = A sin [  ( 2 π /λ ) x ]    .  If the wave moves to the right with a speed v , the wave function at a later time is –

y [ x,0 ] = A sin [  ( 2 π /λ ) ( x – vt ) ]

( to continue ……… )