soln:-

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6. What are beats? Show that the number of beats produced per secec is equal to the difference in the frequencies of the two sounding bodies.(Mark:- 4)

Beats:
The periodic rise and fall in intensity of sound produced by the superposition of two sound waves of nearly equal frequencies is called beats.
Let the frequencies of two waves be:

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7. What are Lissajous figures? How will you trace graphically the Lissajous figures when time periods are equal and phase difference is π/4?(Mark:- 4)

Lissajous Figures:
The figures obtained by the combination of two mutually perpendicular simple harmonic motions are called Lissajous figures. When the time periods are equal and phase difference is π/4, let the equations of motion be:

x = a sin(ωt)

y = a sin(ωt + π/4)

Expanding:

y = a [sin(ωt)cos(π/4) + cos(ωt)sin(π/4)]

Since:

cos(π/4) = sin(π/4) = 1/√2

Therefore:

y = (a/√2)[sin(ωt) + cos(ωt)]

The curve obtained is an ellipse inclined to the axes. Steps for graphical tracing:

Lissajous Figure Graph

For equal frequencies and phase difference π/4, the Lissajous figure is an inclined ellipse.

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2022

(FYUG PHY-DSC201 Same as CBCS PHҮНСС–202T )

8. What is simple harmonic motion? What is the necessary condition of a particle executing SHM?(Mark:- 2)

Simple Harmonic Motion (SHM)

Simple harmonic motion is a type of periodic motion in which a particle moves to and fro about a fixed mean position under the action of a restoring force directed towards the mean position.

Formula

F ∝ -x

a ∝ -x

Here,
F = Restoring force
a = Acceleration
x = Displacement

Necessary Condition for SHM

The restoring force or acceleration must be directly proportional to the displacement and always directed towards the mean position.

a = -ω²x

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9. Distinguish between longitudinal and transverse waves.(Mark:- 2)

Longitudinal Waves	Transverse Waves
Particles vibrate parallel to the direction of wave propagation.	Particles vibrate perpendicular to the direction of wave propagation.
Consist of compressions and rarefactions.	Consist of crests and troughs.
Can travel through solids, liquids, and gases.	Mainly travel through solids and on liquid surfaces.
Example: Sound waves.	Example: Light waves and water waves.

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10. Show that y = f(ct + x) is a solution of the wave equation . (Mark:- 2)

∂²y / ∂x² = (1 / c²) (∂²y / ∂t²)

where c is the wave velocity. (Mark:- 2)

Let,

y = f(ct + x)

Put,

u = ct + x

y = f(u)

First partial derivative with respect to x

∂y/∂x = f'(u) · ∂u/∂x

∂y/∂x = f'(u)

Second partial derivative with respect to x

∂²y/∂x² = f''(u)

First partial derivative with respect to t

∂y/∂t = f'(u) · ∂u/∂t

∂y/∂t = c f'(u)

Second partial derivative with respect to t

∂²y/∂t² = c² f''(u)

Therefore,

(1 / c²)(∂²y/∂t²) = f''(u)

(1 / c²)(∂²y/∂t²) = ∂²y/∂x²

Hence proved that y = f(ct + x) is a solution of the wave equation.

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11. State and prove the principle of superposition and explain its linearity. (Mark:- 3)

Principle of Superposition

The principle of superposition states that:

When two or more waves travel simultaneously through the same medium, the resultant displacement at any point is equal to the algebraic sum of the individual displacements produced by each wave separately.

If two waves produce displacements y₁ and y₂, then the resultant displacement is

y = y₁ + y₂

For many waves,

y = y₁ + y₂ + y₃ + ... + yₙ

Proof of Principle of Superposition

Consider a stretched string vibrating with small amplitude.

Let:

y₁(x,t) = displacement due to first wave

y₂(x,t) = displacement due to second wave

For a linear medium, each wave satisfies the one-dimensional wave equation:

∂²y₁ ∂x² = 1 c² ∂²y₁ ∂t²

and

∂²y₂ ∂x² = 1 c² ∂²y₂ ∂t²

Now add the two equations:

∂²y₁ ∂x² + ∂²y₂ ∂x² = 1 c² ( ∂²y₁ ∂t² + ∂²y₂ ∂t² )

Using addition of derivatives,

∂²(y₁+y₂) ∂x² = 1 c² ∂²(y₁+y₂) ∂t²

Let

y = y₁ + y₂

Then,

∂²y ∂x² = 1 c² ∂²y ∂t²

Thus the resultant displacement y also satisfies the wave equation.

Hence proved that the resultant disturbance is the sum of individual disturbances. This is called the principle of superposition.

Linearity of Superposition

The principle of superposition is valid only for linear systems.

A system is said to be linear if:

• The restoring force is directly proportional to displacement.
• The governing differential equation is linear.

For a linear equation:

L(y₁) = 0

and

L(y₂) = 0

then,

L(y₁ + y₂) = 0

Therefore, the sum of two solutions is also a solution.

This property is called linearity.

Explanation of Linearity

In linear media:

• Waves pass through each other without permanent change.
• Individual wave shapes remain unchanged after interaction.
• Resultant displacement is obtained by simple algebraic addition.

Examples:

• Sound waves of small amplitude
• Light waves
• Vibrations of stretched strings

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12. Obtain the expression for pressure at different points in longitudinal waves.(Mark:- 3)

Pressure at Different Points in Longitudinal Waves

The displacement equation for a plane longitudinal wave is

y = a sin ( 2π λ (vt - x) ) (1)

Let P be the instantaneous change in pressure at a point x, then

P = -K dy dx (2)

where K is the bulk modulus.

Differentiate equation (1) with respect to x,

dy dx = - 2πa λ cos ( 2π λ (vt - x) ) (3)

Using equation (3) in equation (2),

P = K 2πa λ cos ( 2π λ (vt - x) ) (4)

The velocity of a longitudinal wave is

v = √ K ρ

Therefore,

K = ρv² (5)

Using equation (5) in equation (4),

P = ρv² 2πa λ cos ( 2π λ (vt - x) )

P = 2πρv²a λ cos ( 2π λ (vt - x) )

The quantity

2πρv²a λ = P_m

is called pressure amplitude.

Hence, the pressure equation of a longitudinal wave is

P = P_m cos ( 2π λ (vt - x) )

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13. A particle is subjected simultaneously to two SHMs of the same period but of different amplitudes and phases in perpendicular direction. Find the expression for the resultant motion. For what condition, the path may be straight line, ellipse and circle?(Mark:- 3+3=6)

Resultant Motion of Two Perpendicular SHMs

Consider a particle executing two simple harmonic motions simultaneously along two mutually perpendicular directions X and Y having the same time period but different amplitudes and phase difference.

Let the displacement along X-axis be

x = a sin ωt (1)

Let the displacement along Y-axis be

y = b sin (ωt + φ) (2)

where,

• a and b are amplitudes
• ω is angular frequency
• φ is phase difference

Derivation of Resultant Motion

Expanding equation (2),

y = b [sin ωt cos φ + cos ωt sin φ]

Dividing equation (1) by a,

x a = sin ωt (3)

Therefore,

cos ωt = √ ( 1 - x² a² ) (4)

Using equations (3) and (4) in equation (2),

y = b [ x a cos φ + sin φ √ ( 1 - x² a² ) ]

Rearranging,

y - bx cos φ a = b sin φ √ ( 1 - x² a² ) (5)

Squaring both sides,

( y - bx cos φ a ) ² = b² sin²φ ( 1 - x² a² )

Expanding,

y² - 2bxy cos φ a + b²x² cos²φ a² = b² sin²φ - b²x² sin²φ a²

Bringing all terms to one side,

y² - 2bxy cos φ a + b²x² cos²φ a² + b²x² sin²φ a² = b² sin²φ

Using identity,

sin²φ + cos²φ = 1

Therefore,

y² - 2bxy cos φ a + b²x² a² = b² sin²φ

Dividing throughout by b²,

x² a² + y² b² - 2xy cos φ ab = sin²φ

This is the general equation of the resultant motion.

Condition for Straight Line

If phase difference

φ = 0

then

cos φ = 1

sin φ = 0

Substituting in the general equation,

x² a² + y² b² - 2xy ab = 0

( x a - y b ) ² = 0

y = b a x

Hence the resultant path is a straight line.

Condition for Ellipse

For any value of phase difference other than 0 or π,

φ ≠ 0 , π

the resultant path is an ellipse.

Condition for Circle

The ellipse becomes a circle when

a = b

and

φ = π 2

Then,

cos φ = 0

sin φ = 1

Therefore the general equation becomes

x² a² + y² a² = 1

x² + y² = a²

Hence the resultant path is a circle.

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2023

(FYUG PHY-DSC201 Same as CBCS PHҮНСС–202T )

14. Define beats. What is the condition for frequency difference between two sound waves to produce beats? (Mark:- 2)

1. Beats

Beats are periodic variations in the intensity of sound produced when two sound waves of nearly equal frequencies interfere with each other.

Condition for production of beats:

|f₁ - f₂| ≤ 10 Hz

i.e., the frequency difference between the two sound waves must be small.

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15. What are the differences between a longitudinal and a transverse wave?(Mark:- 2)

Longitudinal Wave	Transverse Wave
Particles vibrate parallel to the direction of propagation.	Particles vibrate perpendicular to the direction of propagation.
Consists of compressions and rarefactions.	Consists of crests and troughs.
Example: Sound waves.	Example: Light waves.

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16. What is the relation between .group velocity and phase velocity? (Mark:- 2)

The relation between group velocity and phase velocity is

v_g = v_p + f dv_p df

where,

• v_g = group velocity
• v_p = phase velocity
• f = frequency

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17. Obtain the expression of resultant for superposition of two collinear oscillations with equal frequencies. ? (Mark:- 3)

Let the two simple harmonic motions be

x₁ = a sin ωt

x₂ = b sin (ωt + φ)

According to the principle of superposition,

x = x₁ + x₂

x = a sin ωt + b sin (ωt + φ)

Expanding the second term,

x = a sin ωt + b(sin ωt cos φ + cos ωt sin φ)

x = (a + b cos φ) sin ωt + b sin φ cos ωt

Let the resultant SHM be

x = A sin(ωt + θ)

Expanding,

x = A sin ωt cos θ + A cos ωt sin θ

Comparing coefficients,

A cos θ = a + b cos φ

A sin θ = b sin φ

Squaring and adding,

A² = (a + b cos φ)² + b² sin² φ

A² = a² + b² + 2ab cos φ

A = √(a² + b² + 2ab cos φ)

Also,

tan θ = (b sin φ) / (a + b cos φ)

Therefore, the resultant oscillation is

x = A sin(ωt + θ)

where

A = √(a² + b² + 2ab cos φ)

is the amplitude of the resultant SHM.

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18. Resultant of two SHMs at right angle to cach other and having equal time periods and unequal amplitudes is given by

x² a² + y² b² cos²φ - 2xy ab = sin²φ

Draw diagrams corresponding to φ = 45° and 90° (Mark:- 3)

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19. Obtain the differential equation of wave motion. (Mark:- 3)

Intensity of a Wave

The intensity of a wave is defined as the amount of energy transmitted per second through unit area perpendicular to the direction of propagation of the wave.

SI unit of intensity is watt per square metre (Wm⁻²).

Total Energy of a Transversely Vibrating String

Consider a transverse wave travelling along a stretched string.

Let the displacement equation be

where,

• a = amplitude
• ω = angular frequency
• k = wave number

The kinetic energy of a vibrating particle is

Since

Therefore,

The potential energy is

Hence total energy is

Using

Therefore,

This is the expression for total energy of a transversely vibrating string.

The principle of superposition states that when two or more waves travel simultaneously through the same medium, the resultant displacement at any point is equal to the algebraic sum of the individual displacements produced by each wave.

where y₁ and y₂ are the displacements due to individual waves.

• To determine the phase difference between two SHMs.
• To compare the frequencies of two vibrating sources.

Given,

The one-dimensional wave equation is

Differentiating partially with respect to x,

Differentiating partially with respect to t,

Therefore,

Since

therefore,

is not a solution of the one-dimensional wave equation.

Resultant Amplitude of Two SHMs Having Equal Frequencies and Travelling in the Same Direction

Consider two simple harmonic motions having the same frequency and travelling in the same direction.

Let the first SHM be

Let the second SHM be

where,

• a₁ and a₂ are amplitudes
• ω is angular frequency
• φ is phase difference

The resultant displacement is

Substituting equations (1) and (2) in equation (3),

Expanding the second term,

Rearranging,

Let the resultant SHM be

Expanding equation (5),

Comparing equations (4) and (6),

Squaring and adding equations (7) and (8),

Using identity,

Therefore,

Hence resultant amplitude is

Special Cases

When phase difference

then

Maximum amplitude is obtained.

When phase difference

then

Minimum amplitude is obtained.

Resultant Motion of Two Perpendicular SHMs

Consider a particle subjected simultaneously to two simple harmonic motions along two mutually perpendicular directions X and Y having the same time period but different amplitudes and phase difference.

Let the displacement along X-axis be

Let the displacement along Y-axis be

where,

• a and b are amplitudes
• ω is angular frequency
• φ is phase difference

Derivation of Resultant Motion

Expanding equation (2),

From equation (1),

Squaring equation (4),

Using identity,

Therefore,

Taking square root,

Substituting equations (4) and (6) in equation (3),

Rearranging,

Squaring both sides,

Expanding,

Bringing all terms to one side,

Using identity,

Therefore,

Dividing throughout by b²,

This is the required expression for the resultant motion of two perpendicular SHMs.

The resultant path is called a Lissajous figure.