Superposition Of Waves

Progressive Wave

Properties of Progressive Waves

1. Each particle executes the same type of vibration.

2. Particles at the mean position perform SHM.

3. Have e same amplitude, period, and frequency.

4. The state of vibration of a particle (phase) changes from one period to another.

5. V = 0 at extreme position.

6. The particle attains the maximum velocity after passing through the mean position.

7. Progressive wave 2 types -

   1. Transverse Wave
   2. Longitudinal Wave

Types Of Progressive Waves

1. Transverse Wave

Vibrations of particles are perpendicular to the direction of propagation of the wave. Produces crests and troughs.

Can propagate through solids.

2. Longitudinal Wave

Vibrations of particles produce compressions and rarefaction along the wave’s propagation direction.

Can propagate through solids and fluids.

\

Equation of a Progressive Wave travelling along…

1. Positive x-direction

   y(x,t)=A sin(kx - wt)

2. Negative x-direction

   y(x,t)=A sin(kx + wt)

Equation of a Simple Harmonic Progressive Wave in different forms

y = A sin(wt - kx)

y = A sin [wt - (2π/λ)x]

y = A sin 2π(nt - x/λ)

y  = A sin 2πn (t - x/v)

y = A sin 2π/λ (vt - x)

y = A sin 2π(t/T - x/λ)

Particle velocity and Wave speed

If the simple harmonic progressive wave is given by

y = A sin (wt - kx)

then,

1. Particle velocity,

   dy/dt = wA cos (wt - kx)

2. Wave speed,

   v = w/k

Reflection Of Waves

The phenomenon in which a part of the wave is returned to the original medium with the reduction in its energy and hence in intensity and amplitude is called reflection of waves.

Reflection of a Transverse Waves from…

1. A denser medium

   1. The crest is reflected as a trough
   2. The phase change of 180° or π radians, when transverse waves are reflected from a denser medium

2. A rarer medium

   1. The crest is reflected as the crest and vice-versa with a reduction in amplitude.
   2. No change in phase angle.

Reflection of Longitudinal Waves (Sound Waves) from…

1. A denser medium

   1. To return to normal condition, the particle begins to press in opposite direction, therefore compression gets reflected as compression and vice-versa.
   2. Displacement in reflected is opposite to displacement of the incident wave, therefore phase angle is 180° or π radians.

2. A rarer medium

   1. \

\
\