Answer of Wave and motion

Question And answer of Wave and motion

 A . Short Answer and questions

1.

In a standing wave no energy is transferred at node becausedisplacement node is point of zero velocity, thus zero energy.

 

2.

If we drop stone in a pond, the energy carried by the stone disturbs the water molecule near to it. Water molecules near to stone start vibrating up and down and transfer energy to water molecules next to each other.These molecules also start vibrating. This shows that kinetic energy has been transferred from one molecule to the other. In this way, energy is transported by wave. It is the transfer of the energy not the matter itself.

 

3.

When two progressive waves of the same wavelength and same amplitude travelling with the same speed in opposite direction to each other in a medium meet they give rise to a wave called stationary wave.

A stationary wave has nodes where the amplitude is zero and antinodes where the amplitude is maximum. At nodes the particles are completely at rest but at antinodes the particle are vibrating with maximum amplitude.

 

4.

The stationery wave are called so because the wave seems not to be moving as there is no net flow of energy along the wave.

 

5.

Force vibration	Free vibration
When the body vibrates with applying external force then it is called force vibration.	When the body vibrate with its own natural frequency without the help ofexternal force then it is free vibration.
the vibration of a pendulum is free vibration which needs no external force to vibrate.	The vibration of a machine like a drill is forced vibration which needs an external force to vibrate.

 

 

6.

It is because when they are in March in steps the frequency of marching coincides with the natural frequency of bridge due to which the resonance in the bridge occurs and the bridge begins to vibrate with maximum frequency whichmay cause the destruction of bridge.

 

7.

The relevant properties of the medium are follows:

1. The medium should possess the property of elasticity.

2. The medium should possess the property of inertia.

3. The medium should have minimum friction.

 

8.

Force vibration	Free vibration
1. When the body vibrates with applying external force then it is called force vibration.	When the body vibrate with its own natural frequency without the help of external force then it is free vibration.
2. The vibration of a pendulum is free vibration which needs no external force to vibrate.	The vibration of a machine like a drill is forced vibration which needs an external force to vibrate.

 

B.

Long answers questions

 

1.

The characteristics of wave motion are:

a. Particles vibrate about their mean position when disturbance is produced in medium

b. There is transference of energy to the nearest particle as disturbance occurs, which dependson the nature of the medium.

c. Total displacement of the particle resultant over the one period is zero.

 d. Consecutive particles have certain phase difference as vibration occurs.

e. Vibrating particleposses both kinetic and potential energy.

f. Wave velocity and particle velocity of the medium are different.

 

Waves motion are classified into

Transverse waves: the waves in which particles of the medium vibrate at right angles to the direction of wave motion is called transverse wave. The wave is propagated in the form of crests and troughs. This type of wave motion is possible in solids and on liquid surfaces. These waves can undergo polarization

 

Longitudinal waves: The waves in whichparticles of the medium vibrate parallel to the direction of wave motion is called longitudinal wave. The wave is propagated in the form of compressions and rarefactions. This type of wave motion is possible in any medium (solid, liquid or gas). These waves do not undergo polarization.

 

2.

Transverse waves	Longitudinal waves
The particles of the medium vibrate at right angles to the direction of wave motion.	The particles of the medium vibrate parallel to the direction of wave motion.
The wave is propagated in the form of crests and troughs.	The wave is propagated in the form of compressions and rarefactions.
This type of wave motion is possible in solids and on liquid surfaces.	This type of wave motion is possible in any medium (solid, liquid or gas).
These waves can undergo polarization	These waves do not undergo polarization.

 

 

3.

Progressive wave	Stationery waves
a. Wave transmits the energy b. All particles vibrate with the same amplitude and frequency. c. Pressure and density variation occurs in succession and it issame for all the particlein the medium d. Waves have definite velocity in the medium. e. Neighboring points are notin phase	a. Waves do not transmit energy b. In these waves, amplitude of vibration is maximum at antinodes and minimum at nodes. c. Variation in pressure and density is maximum at nodes and minimum at antinodes.   d. Wavesremain stationery between the boundaries e. All the points between two successive nodes vibratein phase with each other.

 

Characteristics of stationery waves

a. Waves do not transmit energy.

b. In these waves, amplitude of vibration is maximum at antinodes and minimum at nodes.

c. Variation in pressure and density is maximum at nodes and minimum at antinodes.

d. Wavesremain stationery between the boundaries

e. All the points between two successive nodes vibrate in phase with each other.

 

Characteristics of progressive waves

a. Wave transmits the energy.

b. All particles vibrate with the same amplitude and frequency.

c. Pressure and density variation occurs in succession and it issame for the entire particlein the medium

d. Waves have definite velocity in the medium.

e. Neighboring points are notin phase

 

 

4.

Phase of the wave defines the position of the wave at any point with reference to the particular point.

It is measured in degrees. Path difference is defined as the phase angle or time interval by which given wave leads or lags another.

Relation between the path difference and phase difference

Consider two crestsc1${\mathrm{c}}_1$ and c2${\mathrm{c}}_2$.pathdifference between them isλ with the time difference T

As wave reaches to c2${\mathrm{c}}_2$fromc1$\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}{\mathrm{c}}_1$ it completes one vibration so the phase difference is 2π.

Also consider two any point P& Q in the medium as shown in figure.

 

Figure

Let Q beat a distance of x2, and P beat a distance of x1.

Then path difference is given byx= x2 – x1

x isthe path difference ofdenoted by ∆ɸ

From the figure, λ corresponds to the phase difference of 2π
x corresponds to phase difference of2πxλ$\frac{2\pi \mathrm{x}}{\lambda }$

Therefore, ∆ɸ =2πxλ$\frac{2\pi x}{\lambda }$

This is the relation between path difference and phase difference.

 

5.

When two progressive waves of the same wavelength and same amplitude travelling with a same speed in opposite direction to each other in a medium are superposed they give rise to wave called stationary wave or standing wave. These waves are called stationery because they seem to be remained stationery and there is no net transfer of the energy.

The superposition of two waves results in the points such that there is no displacement a (i.e amplitude of vibration) a point called nodes and maximum displacement occurs at the point called antinodes.

Property of stationery waves are:

a. Waves do not transmit energy.

b. In these waves, amplitude of vibration is maximum at antinodes and minimum at nodes.

c. Variation in pressure and density is maximum at nodes and minimum at antinodes.

d. Waves remain stationery between the boundaries.

e. All points between two successive notes vibrate simultaneously.

f. The distant between two adjacent nodes or antinodes is λ/2 and the distance between adjacent nodes and antinodes is λ/4

g.The wave profile doesn’t move in the direction of propagation.

 

6.

When the body vibrate with its own natural frequency without the help of external force then it is free vibration. When the body vibrate with its own natural frequency without the help of external force then it is free vibration. Thevibration of a machine like a drill is forced vibration which needs an external force to vibrate.

When the body vibrates with applying external force then it is called force vibration.The vibration of a pendulum is free vibration which needs no external force to vibrate.

If the amplitude of vibration goes on decreasing with time then the vibration is said to be damped vibrations. It is because of the presence of friction which dissipates the energy of the oscillator.

 

7.

Resonance is a phenomenon that occurs when a given system is driven by another vibrating system or external force to oscillate with greater amplitude at a specific frequency. Frequencies at which the response amplitude is a relative maximum are known as resonance frequencies. At resonant frequencies, small periodic driving forces have the ability to produce large amplitude oscillations.

Examples;Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale, such as electrons in atoms.Many sounds we hear, such as when hard objects of metal, glass, or wood are struck, are caused by brief resonant vibrations in the object.

 

8.

When two wave trains of same frequency and amplitude travel with the same velocity along the same straight line in opposite directions, they superimpose and produce a new type of wave called stationary wave or standing wave.

The name stationary for such type of waves is justified because there is no flow of energy along the wave. Let the incident wave propagating along Y- axis be

y1= a sin ( ωt - kx ) …(1)

and the wave reflected from the boundary traveling along negative X-axis is

y2=a sin ( ω t + kx )….(2)

When two wave superimposed then the stationary wave is produce then the resultant wave is given by y=y1 +y2=a sin(ωt+kx)+asin(ωt – kx)

or, y=2asin(ωt−kx+ωt+kx2)cos(ωt−kx−ωt−kx2)=2a$2\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\omega t-kx+\omega t+kx}{2}\right)\cos \left(\frac{\omega t-kx-\omega t-kx}{2}\right)=2a$sinωt. Cos(-kx)=2a coskx. Sinωt

Or, y=2a cos2πλ$\frac{2\pi }{\lambda }$x sinωt……..3

Eq.3represents the equation of stationary wave.

Here 2acoskx is amplitude and sinωt give the nature of the amplitude of the oscilliaration.

Special cases

Nodes

 At nodes amplitude is minimum.

2a cos2πλ$\frac{2\pi }{\lambda }$x = minimum= 0

Therefore x=(n+12)2π$\left(n+\frac{1}{2}\right)2\pi$

Or, x=(n+12)λ2$\frac{\left(n+\frac{1}{2}\right)\lambda }{2}$where n is the interger. The point x is always at rest called node.

∴ Displacement at nodes is 0.

 Antinodes

If 2a coskx =±1$\pm 1$ then, kx=nπ

Or, x=nλ/2

Here at the point x maximum displacement occur called antinodes.

9.

When two wave trains of same frequency and amplitude travel with the same velocity along the same straight line in opposite directions, they superimpose and produce a new type of wave called stationary wave or standing wave.

The name stationary for such type of waves is justified because there is no flow of energy along the wave. Let the incident wave propagating along Y- axis be

y1= a sin ( ωt - kx ) …(1)

And the wave reflected from the boundary traveling along negative X-axis is

y2=a sin ( ω t + kx )….(2)

When two wave superimposed then the stationary wave is produce then the resultant wave is given by y=y1 +y2=a sin (ωt+kx)+asin(ωt-kx)

or, y=2asin(ωt−kx+ωt+kx2)cos(ωt−kx−ωt−kx2)=2a$2\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\omega t-kx+\omega t+kx}{2}\right)\cos \left(\frac{\omega t-kx-\omega t-kx}{2}\right)=2a$sinωt. Cos(-kx)=2a coskx. Sinωt

or, y=2a cos2πλ$\frac{2\pi }{\lambda }$x sinωt……..3

eq.3 represent the equation of stationary wave.

Here 2acoskx is amplitude and sinωt give the nature of the amplitude of the oscilliaration.

Special cases

If 2a coskx=0 then coskx=0, therefore x=(n+12)π$\left(n+\frac{1}{2}\right)\pi$

Or, x=(n+12)λ2$\frac{\left(n+\frac{1}{2}\right)\lambda }{2}$where n is the interger. The point x is always at rest called node.

If 2a coskx =±1$\pm 1$ then, kx=nπ

Or, x=nλ/2

Here at the point x maximum displacement occurs which is called antinodes.

 

10.

Progressive waves	Stationery waves
a. Wave transmits the energy. b. All particles vibrate with the same amplitude and frequency. c. Pressure and density variation occurs in succession and it issame for all the particlein the medium d. Waves have definite velocity in the medium. e. Neighboring points are notin phase	a. Waves do not transmit energy. b. In these waves, amplitude of vibration is maximum at antinodes and minimum at nodes. c. Variation in pressure and density is maximum at nodes and minimum at antinodes.   d. Wavesremain stationery between the boundaries e. All the points between two successive nodes vibratein phase with each other

 

Consider a particle O at origin in the medium. The displacement at any instant of time is given by y=Asinwt…………..1

Where A is the amplitude, ω is the angular frequency of the wave. Consider a particle P at a distance x from the particle O on its right. Let the wave travel with a velocity v from left to right. Since it takes some time for the disturbance to reach P, its displacement can be written as

y=A sin (ωt - ɸ)……….2

Where ɸ is the phase difference between the particles O and P

 Hence a path difference of x corresponds to a phase difference of 2πx/λ=ɸ

Substituting the value of the ɸ in equation 2

We get,

y=A sin (ωt -2πx/λ)…………….3

Since we have ω=2π/T=2πf

y=Asin{2πtT−2πxλ}$\mathrm{y}=\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}\left\{\frac{2\pi \mathrm{t}}{\mathrm{T}}-\frac{2\pi \mathrm{x}}{\lambda }\right\}$

y=Asin2πλ{vt−x}$\mathrm{y}=\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}\frac{2\pi }{\lambda }\left\{\mathrm{v}\mathrm{t}-\mathrm{x}\right\}$ ………4

Similarly, for a particle at a distance x to the left of O, the equation for the displacement isgiven byy=Asin2πλ{vt−x}$\mathrm{y}=\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}\frac{2\pi }{\lambda }\left\{\mathrm{v}\mathrm{t}-\mathrm{x}\right\}$

If the wave travels from right to left in negative x-axis then the equation for the displacement isgiven by

y=Asin2πλ{vt+x}$\mathrm{y}=\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}\frac{2\pi }{\lambda }\left\{\mathrm{v}\mathrm{t}+\mathrm{x}\right\}$ ……5

This is the expression of the progressive wave equation.