## Kerala Plus One Physics Notes Chapter 15 Waves

Introduction:

The wave is the propagation of disturbance that carries energy from one point to another point, without translatory motion of particles in the medium.

There are three types of wave.

- Mechanical wave :Requires medium for propagation, eg., sound wave, Matter wave, Seismic wave, etc.
- Electromagnetic wave: No medium for propagation, eg., light, X-rays, UV ray etc.
- Matter wave : Wave associated with moving particles (microscopic particle), eg., wave of moving electron, proton etc.

Expression for progressive wave (Displacement relation):

A plane progressive wave propagating along positive direction of ‘x’ is given by.

Direction is given by, v(x,t) = A sin(kx + ωt + φ)

From the above equation we can find the displacements of any part of the string at any time and also the shape of the of the wave at any time.

The wave propagating along negative‘x’ direction is given by y(x, t) = A sin (kx + ωt + φ)

Parameters of wave:

Amplitude : A. It is the maximum displacement of any particle of the medium from its mean position.

Phase and initial phase:

The value (kx + ωt + φ) is called phase and φ is the initial phase. The phase gives the state of motion of wave at position ‘x’ and at time ‘t’Initial phase gives initial state of wave.

Time period (T):

Time for one complete oscillation/vibration is called time period.

Wave length λ:

The linear distance travelled by the wave is one complete oscillation.

Or it can be defined as distance between two consecutive crests or troughs. It is the distance travelled during time period T.

Wave number:

‘k’ is defined as Its unit is radian/m.

Frequency v:

The number of oscillations/vibrations in one second is called frequency. Its unit is s^{-1} of Hz (Hertz).

Angular velocity or angular frequency (ω):

Angular displacement per unit time is called angular velocity or angular frequency.

CAQ.

Question 1.

The displacement y of a particle in a medium can be expressed as

y = 10^{-6 }sin (100t+ 20x + π/4) where t is in second and x in metre. What is the speed of the wave?

Answer:

We compare the given wave equation with the standard wave equation,

y = A sin (ωt + kx + φ)

We get ω =100 rad s^{-1} and k = 20 rad m^{-1}.

Speed of the wave,

Speed of transverse wave on a stretched string:

The wave speed on a string depends on its inertial and elastic inertial and elastic properties. If T is the tension in the string and μ is the mass per unit length of the string the, speed of a transverse wave on the string is

Types of mechanical wave motion:

1. Transverse waves: These are the waves in which individual particles of the medium oscillate perpendicular to the direction of wave propagation (movement).

eg., movement of a guitar string or a violin string or jerking a string on its free end.In the above figure, each part of the string vibrates up and down while the wave travels along the string. So the waves in the string are transverse in nature.

The points, C, C… are called crests and the points T, T… are called troughs.

The distance between two consecutive crests or two consecutive troughs is called

wavelength (λ) of the wave.

The speed of a travelling wave:

Consider a wave travelling in the positive ‘x’ direction. The string element will move up and down (perpendicular) as a function of time, but the wave moves to the right. The displacement of two elements at two instants in a small interval ‘∆t is given in figure.

We can see that the entire wave has moved to the right by a distance ∆x

The ratio ∆x/∆t is the wave speed ‘ V’.

Speed of transverse wave on a stretched string:

The wave speed on a string depends on its intertial and elastic properties.

If T is the tension in the string and p is the mass per unit length of the string the, speed of a transverse wave on the string is

Speed of transverse wave in a solid:

If η is the modulus of rigidity and pis the density of the solid then, the speed of any transverse wave on it is given by

2. Longitudinal waves:

These are waves in which the individual particles of the medium oscillate along the direction of the wave propagation.

Generation of longitudinal waves by tuning I fork:

When prongs of tuning fork moves outward it compresses the surroundings air and a region of increased pressure is formed. This region is called condensation. When the prongs move inward a region of low pressure called ratefraction is formed. Thus condensations and rarefactions are produced.

Speed Of sound wave (Longitudinal wave):

The speed of sound in medium depends on

i. Density of medium (p)

ii. Modulus of elasticity

Case : 1 (In solid),

If solid has Young’s modulus,

Case : 2 (In liquid),

If liquid has Bulk modulus B,

Case: 3 (In gas),

The speed of sound waves in gas was determined Newton. According to Newton, condensations and rarefractions are

isothermal processes. Hence modulus of elasticity is equal to pressure,

This is called Newton’s formula

CAQ

Question 2.

For a steel rod, the Young’s modulus of elasticity is 2.9 × 10^{11} Nm^{-2} and density is

8 × 10^{3} kg m^{-3}. Find the velocity of the longitudinal waves in the steel rod.

Answer:

Correcton in Newton’s formula

Find velocity of sound in air using Newton’s formula. At STP,

(P= 1.013 × 10^{5} Pa, ρ = 1.239 kgm^{-3})

Note:

The velocity of sound at STP is found to be 332 ms^{-1}. This is about 15% less than the experimental speed of sound in air at STP which is 331 ms^{-1}.

CAQ

Question 3.

In which medium, do the sound waves travel faster, solids, liquids or gases? Give reason.

Answer:

Sound waves travel in solids with highest speed. This is because the coefficient of elasticity of solids is much greater than coefficient of elasticity of liquids and gases.

Laplace’s Correction:

Laplace corrected Newton’s formula taking condensation and rarefraction as adiabatic process. The modulus of elasticity is now

‘γp’; where is specific heat capacity at constant pressure and C_{v} is specific heat capacity at constant volume.

For air γ = 7/5, so speed of sound in air at STP will be

This value is very close tot eh experimental value.

or

The overlapping waves algebraically add to produce a resultant wave.

Question 4.

We always see lightning before we hear thundering. Why?

Answer:

The speed of light (3 × 10^{8} ms^{-1}) is much larger than the speed of sound (~ 340 ms^{-1}). Consequently, the flash of light reaches us much earlier than the sound of thunder.

The principle of superposition of waves:

It states that when two or more waves pass through a media the net displacement of particle at any time is the algebraic sum of displacements due to each wave.

or

The overlapping waves algebraically add to produce a resultant wave.

Standard Waves:

When two waves of same amplitude and frequency travelling in opposite direction super impose the resulting wave pattern does not move to either sides. This pattern is called standing wave.

In the case of standing wave, at any point ‘x’ at any time ‘ t, there are always two waves, one move to the left and another to the right.

The wave travelling in positive direction of x axis, y_{1}(x,t) = a sin (kx-t)

The wave travelling in negative direction of x axis, y_{2}(x,t) = a sin (kx-t)

According to superposition principle, the combined wave is

y(x,t) =y_{1}(x,t) +y_{2}(x,t)

y(x,t) = a sin (kx -ωt) + a sin (kx + ωt)

But, sin (A+B) + sin (A-B)

=

Hence we get, combined wave as y(x,t) = 2a sin (kx) cos ωt

This wave has an amplitude of ‘2a sin kx’, and it is not a travelling wave.

Nodes & Antinodes:

The position of maximum amplitude in a standing wave is termed as antinode and position of minimum amplitude (zero) is termed as node.

Node:

The amplitde of standing wave is ‘2a sin kx’. It becomes zero when kx = 0, π, 2π.. etc.

Antinode:

The amplitude has maximum value 2a when (2a sin kx = 2a).

Note: Distance between any two nodes or antinodes is half its wavelength.

Standing waves in stretched string & Modes of vibration of string:

A string of length L is fixed at two ends. The position of one end is chosen as x = 0, then the position of other end will be x = L. At x = 0, there will be node. To occur node at x = L, it must satisfy.

The frequency of vibrations of stretched string of length L is ν=nv/2L ;n=1,2,3….etc.

This set of frequencies at which the string can vibrates are called natural frequencies or modes of vibration or harmonics. The above equation shows that the nodes of vibration (natural frequencies) of string are integral multiple of lowest frequency,

Fundamental mode (or) First harmonic:

If the string is plucked in the middle and released, then it vibrates in one segment with nodes at its ends and an antinode in the middle.

This is the lowest frequency with which string vibrates.

Second harmonic:

If the string is pressed in the middle and plucked at one-fourth of its length, then the string vibrates in two segments.

Third harmonic:

If the striping is pressed at one-third of its length from one end and plucked at one-sixth its length, it will vibrate in three segments.

Thus collection of all possible mode is called harmonic series and n is called harmonic number.

Beats:

When two sound waves of nearly same frequency and amplitude travelling in same

direction super imposed and periodic variation of sound intensity (wavering of sound or waxing and waning of sound) is produced, this is called beats.

The Dopping Effect:

While standing on the road; we have.heard a car horn as the car speeds by. Have you

noticed the pitch? When the car approaches, the pitch is high and when the car leaves, the pitch is low. is it felt by the car driver? No. This apparent change in frequency when the source of sound (or listener) moves is called Doppler Effect.

The apparent frequency when the source or listener (or both) move is given in the general formula as,

V = speed of sound

V_{0} = speed of observer (listener)

V_{s} = speed of source

Now, when the source and observer (listener) approach, the apparent frequency increases, (when they recede, the apparent frequency, decreases) .This is the only rule that we need to remember. So,

- Source moves towards (observer stationary)

- Sources moves away(observer stationary)

- Observer moves towards (source stationary)

- Observer moves away (source stationary)

- Both moves towards,

- Both moves away,

See with this pattern we have dealt with all cases of Doppler effect.

CAQ

Question 5.

The sirens of two fire engines have a frequency of 600 Hz each. A man hears the sirens from the two engines, one approaching him with a speed of 36 km h^{-1} and the other going away from him at a speed of 54 km h^{-1}. What is the difference in frequency of two sirens heard by the man? Take the speed of sound to be 340 ms^{-1}.

Answer:

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