Oscillations And Waves

Oscillations and waves are fundamental concepts in physics that explain the movement and transfer of energy. Oscillations refer to repetitive motion back and forth around a central point, such as a pendulum or a vibrating string. Waves, however, are disturbances that carry energy across space and time.

Categories of Motion

• Periodic Motion: It repeats at consistent time intervals (e.g., the motion of planets, a swinging pendulum).
• Oscillatory Motion: Movement back and forth about a fixed point (e.g., vibration of a string or mass on a spring).

Simple Harmonic Motion (SHM) and Its Conditions

SHM is the simplest form of vibratory motion where a particle oscillates about an equilibrium position.

Conditions for SHM:

The restoring force FF is proportional to displacement xx and acts towards the equilibrium position:

$F\propto -x\text{or}a\propto -x$

$F=-kx,k\to \text{Force constant of the S.H.M.}$

1.0Types of SHM

Linear SHM: Particles move in a straight line about the equilibrium (e.g., mass-spring system).

Angular SHM: Particle oscillates about an axis (e.g., simple pendulum).

Equations of SHM

The differential equation for SHM is:

$\frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=0\text{where}\omega =\sqrt{\frac{k}{m}}$

Its solution is

$x=A\sin (\omega t+\varphi )$

A =Amplitude  

$\omega$ =Angular Frequency  

$\varphi$ =Initial Phase

Terms Associated with SHM

• Amplitude (A): The greatest displacement from the equilibrium position.
• Period (T): Time to complete one oscillation, $T=\frac{2\pi }{\omega }$
• Frequency (f): Number of oscillations per second, $f=\frac{1}{T}\text{measured in Hertz (Hz)}$

Kinematics of SHM

• Velocity (v):

$v=\frac{dx}{dt}=A\omega \cos (\omega t+\varphi )$

At x = 0, $v_{\max }=\mp \omega A$

• Acceleration (a):

$a=-{\omega }^{2}x$

At extreme positions, $a_{\max }=-{\omega }^{2}A$

Energy in SHM

• Kinetic Energy (K):

$K=\frac{1}{2}m{v}^2=\frac{1}{2}m{\omega }^2(A^2-x^2)=\frac{1}{2}k(A^2-x^2)$

At $x=0,K.E_{\max }=\frac{1}{2}k{A}^2$

• Potential Energy (U):

$U=\frac{1}{2}k{x}^2$

$x=\pm A,P.E_{\max }=\frac{1}{2}k{A}^2$

• Total Energy (E):

$E=\frac{1}{2}k{A}^2$, The total energy remains constant throughout the oscillation.

2.0Spring-Mass System

For a spring-mass system, the force equation is:

F= -kx

The time period T is given by:

$T=2\pi \sqrt{\frac{m}{k}}$

Combination of Springs

• Series Combination: Effective spring constant:

$\frac{1}{k_{eq}}=\frac{1}{k_1}+\frac{1}{k_2}$

The time period for combined springs:

$T=2\pi \sqrt{\frac{m}{k_{eq}}}$

Parallel Combination: Effective spring constant:

$k_{\text{eq}}=k_1+k_2$

$T=2\pi \sqrt{\frac{m}{k_{\text{eq}}}}$

Simple Pendulum

The time period for a simple pendulum is:

$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{l}{g}}=2\pi \sqrt{\frac{\text{length of simple pendulum}}{\text{acceleration}}}$

Where l represents the length and g is the acceleration due to gravity.For small angles, the motion is SHM, and the time period depends only on the length of the pendulum.

Physical Pendulums

A compound pendulum is a rigid body oscillating about a fixed axis.The time period is:

$T=2\pi \sqrt{\frac{I_{COM}+m{l}^2}{mgl}}$

$I_{COM}=m{k}^2$ ∴ k is radius of gyration about axis passing from centre of mass

3.0Introduction to Superposition

When two SHM motions of the same frequency combine, the resultant motion is also SHM. The displacement xx is the sum of the individual displacements:

$x_{\text{res}}=\left(x_1+x_2\right)$

For SHMs in phase, the amplitudes add up; if out of phase, they subtract.

Damped Oscillations

In damped oscillations, resistive forces (like friction or air resistance) cause the amplitude to decrease over time. 

The equation of motion is:

$m\frac{d^{2}x}{d{t}^2}=mg-b\frac{dx}{dt}-kx$

The solution shows that the amplitude decays exponentially.

$x(t)=A_0{e}^{\frac{-bt}{2m}}\cos (\omega t+\varphi )$ Where b  is the damping coefficient.

Types of Oscillations

• Undamped: No energy loss; oscillation amplitude remains constant.
• Overdamped:The system returns to equilibrium directly (no oscillation).
• Critically damped: The system is critically damped, returning to equilibrium as rapidly as possible without oscillating.
• Underdamped: Oscillations gradually decay.

Forced Oscillations

When an external force drives an oscillating system, the system may enter a state of resonance if the driving frequency matches the system’s natural frequency. The amplitude reaches a maximum under these conditions.

The equation of motion is:

$F_{\text{net}}=F_0\cos \left({\omega }_{d}t+\varphi \right)-b\frac{dx}{dt}-kx+mg=m\frac{d^{2}x}{d{t}^2}$

Where ${\omega }_d$ is the driving frequency and $F_0$ is the amplitude of the driving force.

Maximum oscillation amplitude occurs at resonance (when driving and natural frequencies match).

4.0Wave

A wave is a propagating disturbance that carries energy through a medium, but the medium itself does not travel with the wave. Examples include water waves and sound waves.

Classification of Waves

Waves are classified into:

• Transverse Waves: Particles move perpendicular to wave direction. Example: Light waves.
• Longitudinal Waves: Particles move parallel to wave direction. Example: Sound waves.

Difference Between Transverse and Longitudinal Waves

Transverse waves	Longitudinal waves
Particle motion is transverse to the wave's direction.	Particle vibration is parallel to wave propagation.
It travels in the form of crests (C) and troughs (T).	It travels in the form of compression (C) and rarefaction (R).
Transverse waves can travel through solids and can be generated on the surface of liquids. However, they cannot propagate through liquids or gases.	These waves can travel through solids, liquids, and gases because their propagation requires volume elasticity.

Wave Function

The wave function represents the displacement of particles in a medium. For a wave on a string, it’s the displacement; for sound, it’s pressure or density fluctuations.

• General wave equation:

$y(x,t)=f(x\mp vt)$ where v is the wave speed.

• Phase velocity:

$v=-\frac{\text{ Coefficient of }t}{\text{ Coefficient of }x}$

Wave on a String

For a string under tension, the wave equation is:

$y(x,t)=A\mathrm{Sin}(kx-\omega t)$

• Wave speed:

$v=\sqrt{\frac{\tau }{\mu }}\mathrm{T}\text{ is tension and }\mu$ is the mass per unit length.

Energy Density in Traveling Waves

• Kinetic Energy:

$dk=\frac{1}{2}(\mu dx){\left(-\omega y_m\right)}^2{\mathrm{Cos}}^2(kx-\omega t)dx$

• Potential Energy: Equivalent to kinetic energy in terms of displacement.

Power Transmitted by a Sine Wave

Average power transmitted along a string is:

$P_{avg}=2{\pi }^2\mu v{A}^2{f}^2\Rightarrow P_{\text{avg }}=\frac{1}{2}\mu v{A}^2{\omega }^2$

Interference and Principle of Superposition

When two waves meet, their displacements combine:

• Constructive interference: Amplitudes add.
• Destructive interference: Amplitudes subtract.

Resultant wave: $y=y_1+y_2$

Reflection and Transmission of Waves

Reflection at a Fixed End: Wave inverts.

Reflection at a Free End: Wave doesn’t invert.

For reflection and transmission:

$A_r=\left(\frac{2\sqrt{{\mu }_i}}{\sqrt{{\mu }_i}+\sqrt{{\mu }_t}}\right)A_i\text{ and }{A}_t=\left(\frac{\sqrt{{\mu }_i}-\sqrt{{\mu }_t}}{\sqrt{{\mu }_i}+\sqrt{{\mu }_t}}\right)A_i$

Standing Waves

Standing waves, characterized by nodes and antinodes, result from the interference of identical waves moving in opposite directions.

Equation:

$y(x,t)=(2A\mathrm{Cos}kx)\mathrm{Sin}\omega t$

Vibration of a String

For a string fixed at both ends, standing waves form. The fundamental frequency is: $f_0=\frac{v}{2L}$

Laws of Transverse Vibrations of a String

• Length Law: $f\propto \frac{1}{l}$
• Tension Law: $f\propto \sqrt{T}$
• Mass Law: $f\propto \frac{1}{\sqrt{\mu }}$

Energy Density of Standing Waves 

In a standing wave, energy is the total of the kinetic and potential energy densities.

Power:

$P=-\frac{1}{4}T{A}^{2}k\omega \mathrm{Sin}(2kx)\mathrm{Sin}(2\omega t)$

5.0Sound Waves

Sound waves are longitudinal waves requiring a medium (solid, liquid, or gas) to propagate. It consists of compressions and rarefactions.

Displacement and Pressure Waves in Sound

Displacement Wave: Describes the motion of particles.

Pressure Wave: Describes changes in pressure due to compressions and rarefactions.

Relationship:

$P_0=\frac{B\omega }{v}{S}_0=Bk{S}_0$

Speed of Sound in Various Mediums

• In solids: $v=\sqrt{\frac{k+\frac{4}{3}\eta }{\rho }}$ ,k== Bulk modulus, $(\eta )$== Modulus of rigidity, $(\rho )$=Density
• Solid (long bar), $v=\sqrt{\frac{Y}{\rho }}$ Y== Young's modulus of rod material of rod.
• Velocity of sound waves in a  medium (liquid or gas), $V=\sqrt{\frac{B}{\rho }}$ where 

$B=-V\frac{dP}{dV}$

Factors Affecting Speed of Sound

• Temperature: $v\propto \sqrt{T}$
• Humidity: Higher humidity increases sound speed due to reduced air density.
• Wind: Wind speed can modify the velocity of sound in the wind direction.

Intensity of Sound Waves

Intensity is the energy transmitted per unit time and per unit area, and it is proportional to the square of the pressure amplitude.

Average Intensity $=\frac{\text{ Average Power }}{\text{ Area }}$

$<I>=\frac{P_0^2}{2\rho v}$

The decibel scale (dB) is used to measure sound intensity:

$\beta =10\log \left(\frac{I}{I_0}\right)dB$

Pitch and Frequency

• Pitch: A psychological perception related to frequency.
• Frequency: The number of oscillations per second, measured in Hz.

Loudness and Intensity

Loudness is a subjective perception, while intensity is an objective measure of sound energy. Loudness increases logarithmically with intensity.

Interference of Sound Waves

• Constructive interference: Occurs when waves are in phase.
• Destructive interference: Occurs when waves are out of phase.

Beats

Slightly different frequency waves interfering produce beats: periodic intensity variations.

Beat Frequency = $|f_1-f_2|$

Longitudinal Standing Waves: Two longitudinal waves of the same frequency traveling in opposite directions create standing waves, with nodes and antinodes for pressure variation.

Vibration of Air Columns (Organ Pipes)

Closed Organ Pipe: Only odd harmonics.

Fundamental frequency:

$f_0=\frac{v}{{\lambda }_0}=\frac{v}{4l}\left({\lambda }_0=4l\right)$

$n^{th}\text{ overtone }{f}_n=(2n+1)f_0$

Open Organ Pipe:  Odd and even harmonics are present

Fundamental frequency: $f_0=\frac{v}{{\lambda }_0}=\frac{v}{2l}\left({\lambda }_0=2l\right)$

$n^{\text{th }}\text{ overtone }{f}_n=(n+1)f_0$

End Correction

The end correction compensates for the displacement antinode at the open end of a pipe:

• Closed pipe:( l+0.6r )
• Open pipe:( l+1.2r )

Apparatus for Determining Speed of Sound

• Quincke’s Tube: Interference of sound waves in tubes.
• Kundt’s Tube: Stationary waves form in a tube.
• Resonance Tube: Measures sound speed via resonance.

Doppler Effect

When there is relative motion between a sound or light wave source and an observer along the line connecting them, the frequency observed differs from the source's frequency. This phenomenon is known as the Doppler Effect.

• If the observer and the source are moving toward each other, the observed frequency increases, becoming higher than the source's frequency.
• If they are moving away from each other, the observed frequency decreases, becoming lower than the source's frequency.

v= velocity of sound w.r.t. ground, c=velocity of sound with respect to medium,

$v_m$= velocity of medium,

$v_o$ =velocity of observer,

$v_s$ =velocity of source

$f^{\prime }=f\left(\frac{v^{\mp }{v}_o}{v\mp v_x}\right)$

6.0Solved Examples

Q-1.A body of mass m is attached to a spring with spring constant k. If the amplitude of oscillation is A, what is the total mechanical energy in the system?

Solution:

The total mechanical energy in SHM is given by $E=\frac{1}{2}k{A}^2$ .This is independent of the mass and depends only on the spring constant and amplitude.

Q-2.In simple harmonic motion with angular frequency ω, how are ω and the period (T) related

Solution:

The angular frequency ω and the time period T are related by the equation, $\omega =\frac{2\pi }{T}$ . So, the time period T is the reciprocal of the frequency and is proportional to the angular frequency.

Q-3.How does wave energy change when amplitude doubles (constant tension)?

Solution:

Wave energy is proportional to the square of the amplitude $E\propto A^2$ ; thus, doubling the amplitude increases energy fourfold,the energy will increase by a factor of 4.