Wave Motion

Elements of a Wave
Amplitude

The maximum displacement of a particle from mean position is called as Amplitude of a transverse Wave. The maximum pressure exerted on medium is called as Amplitude of a longitudinal Wave.

Period

The time taken for the execution of one complete wave is called as Time Period.

Frequency

The number of waves executed per second is called as Frequency of he wave.

Wavelength

The distance traversed by one complete wave is called as wavelength.

Wave Number

The number of waves passing through 1 metre of medium is called as wave number. It is the reciprocal of wavelength (in metres)

Velocity

The velocity of a wave is the displacement of the ave per unit time.

Phase

Periodicity

Path Difference and Phase Difference

Phase Difference = ( 2π / λ ) Path Difference

Where (2π / λ ) is the propagation constant

Longitudinal
Velocity = v = sqrt [T/μ] where μ is the linear charge density

Simple Harmonic Progressive Waves
A Simple Harmonic Motion is periodic in space as well as in time. This property is called double periodicity.

Displacement :

y = a sin2π [t/T - x/λ]

Reflection of Waves
For transverse waves, the wave is reflected back in the same phase.

For longitudinal Waves, the wave is reflected back in the same phase ; but if at the other end of boundary , there exists a rarer medium , the reverse condition will be valid.

Superposition of Waves
When two or more waves travelling through a medium arrive at a point of the medium simultaneously, each wave produces its own displacement , independently of others. Hence, the resultant displacement at that point is equal to the vector sum of the displacements due to all the waves.

Beats
The alternate waxing and waning of sound after definite intervals of time, due to superposition of two waves of nearly equal frequencies , is called production of beats. The number of times the sound waxes or wanes in one second is called as frequency of waves.

y = R sin 2π nt ; where n is the average frequency of superposing waves, t is the time period of beat.

N = n1 - n2

Imax α (a1+a2)2

Imin α (a1-a2)2

Doppler Effect
na = [v ± vo / v ∓ vs]

If the source and observer are approaching each other, numerator has addition.

If the source and observer are moving apart from each other, numerator has subtraction.

Stationary Waves
When 2 identical progressive waves both travelling in opposite directions, interfere with each other , by superposition of waves resultant wave obtained in the form of loops is called a Stationary Wave.

Resonance
If the resonance frequencies of a body are in the ratio xa : xb : xc , then x is the frequency of the body.