Wave and Motion

 1. How are stationary waves formed?

> Stationary (or standing) wave: when two progressive waves of the same wavelength and amplitude travelling with the same speed through a medium in opposite directions and superimpose upon each other, they give rise to a wave which is called stationary wave.
In stationary wave, there are certain points where the amplitude of vibration is always zero. These points are known as nodes. Midway between these nodes, there are other points when amplitude of vibration is maximum. These points are known as antinodes. The formation a stationary wave along with nodes and antinodes is shown in figure,… 

2. If you are walking on the moon surface, can you hear the cracking sound behind you? Explain.

>There is no atmosphere on the moon because of its weak gravity. For the propagation of sound waves, medium is necessary. This means sound waves cannot propagate through vacuum. So, due of lack of medium (i.e. atmosphere), the propagation of sound waves on moon surface is not possible. So, we cannot hear the cracking sound behind us.

 

3. Do sound waves undergo reflection, refraction and polarization phenomena? Explain.
>Sound waves undergo reflection and refraction phenomenon. Sound waves are reflected from surfaces like walls, ground, big halls etc. During night hearing of sound is clearer than at day time due to the refraction of sound, But sound wave does not undergo polarization phenomenon because it is a longitudinal
wave, and only transverse wave undergoes the phenomenon of polarization.

 

4. Which types of wave propagate in liquid, explain.
>For the propagation of transverse wave, the modulus of rigidity of the medium is responsible and for the propagation of longitudinal wave, the bulk modulus of elasticity is responsible. A solid has both modulus of rigidity and bulk modulus but liquids have only bulk modulus. Hence only longitudinal wave can propagate through the liquid.

 

5.  Longitudinal waves cannot be polarized. Why?
> Polarization is the phenomenon of restriction of wave to vibrate in a single direction. The transverse wave vibrates in all direction and we can cut of other directions restricting vibration in a single direction. The longitudinal waves vibrate in a single direction in the direction of propagation of wave). Due to this reason, longitudinal waves cannot be polarized.

 

6.  Distinguish between light waves and sound waves.
> The main differences between sound waves and light waves are:
Light waves
1. Light waves are electromagnetic waves because these can travel in medium as well as in       vacuum.
2. The speed of light wave is greater i.e.3* 10 m/sec in vacuum or air.
3. They are transverse wave.                                                                                            4.Their wavelength is short.                                                                                                5.Light wave can be polarized
Sound waves
1. Sound waves are mechanical waves because they travel only in medium.
2. The speed of sound wave is smaller i.e. 330m/sec at 0 degree C in air.
3. They are longitudinal wave.
4. Their wavelength is long.                                                                                                        5. Sound wave cannot be polarized.

 

7.  A radio station broadcasts at 800 KHz. If the radio waves (em-waves) travel with a speed of 3x10 m/s, what will be the wavelength of the wave?
>  Frequency (F) = 800 KHz = 800*10^3 Hz
Wavelength = ?
Speed (c) = 3×10^8 m/s

We have,
wavelength = c/f  =3×10^8m/s/800×10³

                           = 375 m.

 

8.  Why echo cannot be heard in a small room?
> The minimum distance between the speaker (source) and the wall (reflector) must be 17m to hear an echo. If a room is small, this requirement is not fulfilled. Hence, we cannot hear an echo in a small room.

 

9.  Frequency is the most fundamental property of a wave. Why?
> The frequency is the most fundamental property of a wave In a wave motion, its velocity and wavelength may change with the medium in which it passes but frequency does not change. Due to this reason, frequency of a wave is taken empirical parameter. 

 

 10. Use the principle of superposition of two waves to find the position of nodes and antinodes in a standing wave.
> Stationary (or standing) wave: Whenever two progressive waves of the same wavelength and amplitude travel in opposite directions with the same speed in a medium and undergoes superposition; a resultant wave is formed such
wave is called stationary wave.
These waves are called stationary waves because there is no flow of energy along the wave. When a stationary wave is formed due to the superposition of two waves, the points of maximum and zero amplitude are resulted alternatively in the space. The points where the amplitude of vibration is maximum are called anti-nodes and those where the amplitude is zero are called nodes. The distance between two consecutive nodes or antinodes is equal to half of the wavelength i.e. ½ lemda  where lemda is wavelength of a wave. Also, the distance between adjacent node and antinode is equal to one quarter of wavelength i.e.1/4lemda. >Stationary wave equation: Stationary wave equation can be obtained by opposite adding vectorically the displacements of two waves of equal amplitude, frequency (or period) and wavelength travelling in opposite directions.
Let y1, be the displacement of the wave travelling to the positive x-direction,
yı = a sin (wt - kx)---------🡪(i)
And, y2 be the displacement of the wave travelling to the negative x-direction
y2 = a sin (wt + kx)-------🡪(ii)
By using the principle of superposition of waves, the resultant displacement y is given by
y = y1+y2
= a sin (wt-kx) + a sin(wt + kx)
= a[sin (wt-kx) + sin (wt + kx)
= a [sin wt cos kx - cos wt sin kx + sinwt cos kx + cos wt sinkx]
=2 a sinwt coskx
=2a sin 2π/T × t cos 2π/lemda × X -----------🡪(iii)

where, A = 2a cos 2πx/lemda be the amplitude of resultant wave.

Equation (iii) is the equation of stationary wave equation.

Case 1: For X = 0, lemda/2, 2lemda/2, 3lemda/2
then, A= 2a is maximum amplitude
Thus, these points are antinodes.
Case 2: For X = lemda/4, 3lemda/4, 5lemda/4 …….
then, A=0 is minimum or zero amplitude.
Thus, these points are nodes.
Therefore Distance between two consecutive nodes and antinodes is lemda/2.

11. Define progressive wave. Derive progressive wave equation.
> Progressive wave: A wave that travels from one region of medium to another region carrying energy is called the progressive wave. Both transverse and longitudinal waves are progressive waves. The motion of progressive wave is given below………..fig..???
Equation of progressive wave : Let us consider a wave is travelling from left to right as shown in figure the displacement of the vibrating particle in the medium is given by, y = a sin wt ----🡪(i)

where a is amplitude, t is time and w-2πf and f is frequency of vibration. If φ  be the phase angle of the particle P at distance X from O,then the displacement equation given by,
y = a sin(wt-φ)---🡪 (ii)
Since, for a path diff. lemda, phase diff. is 2π.
And for a path diff. x, phase diff. is 2π/lemda *X.
i.e. φ = 2π/lemda × X                                              

y = a sin (wt - 2π/lemda ×  X )

y = a sin 2π (t/T – x/lemda )                           
y = a sin (2π/T× t - 2π/lemda× x)

y = a sin (wt – kx) ---🡪 (i)

[ Therefore K = 2π/lemda, a wave number or wave vector ]
If the wave is travelling from right to left, then the displacement of the particle is given by,
y = a sin 2π (t/T + x/lemda )--------🡪(ii)

These equations (1) and (ii) are the plane progressive wave equations.
equation in terms of its wave vector and displacement.

12.  How is a progressive wave different from a stationary wave? Derive progressive wave equation.
> Differences between progressive wave and stationary wave are
Progressive waves
i. The disturbance travels in forward direction.
ii. The amplitude of vibration of each particle is same.
iii. Energy is transferred forward along the waves.
iv. No particles in the medium are permanently at rest but momentarily at rest at the extreme positions.
Stationary waves
i. The disturbances are confined to a particular region.
ii. The amplitude is zero at nodes and maximum at antinodes.
iii. There is no transfer of energy in the medium.
iv. Particles at the nodes are permanently at rest.

13.  A wave has the equation (x in metres and in seconds) y = 0.02 sin (30 t - 4x)
Find i. frequency, speed and wave length.

ii. The equation of wave with double the amplitude but travelling in the opposite direction.
Solution
Given,
The given equation is
y = 0.02 sin (30t-4x)
Comparing this equation with the standard wave equation,
y=a sin (wt-kx), where k = 2π/lemda,
we have
w = 30
Or, 2πf = 30
Or, f = 30/2π = 15/π
Frequency, F= 15/π = 4.77 Hz
                              k= 4
Or, 2π/lemda = 4
or, lemda = 2π/4 = π/2
Wavelength, = 1.571m
and Speed, v = ?
We know that
Speed, v= lemda ×f = π/2 = 7.5 ms-1
Hence, frequency, f = 4.77 H
Speed, v - 7.5 ms-1
and wavelength,  = 1.571 m
The equation of the wave moving in opposite direction and double the amplitude is y = 0.04
sin (30t + 4x)