Simple Harmonic Motion and Waves: Notes, Formulas and Key Concepts for NEET

 Simple Harmonic Motion (SHM) and Waves can be challenging to revise for NEET Physics because they involve several equations, graphs and special cases. Questions may test your understanding of displacement, velocity, acceleration, energy changes, time periods, wave speed and standing-wave patterns. Memorising formulas without knowing when to apply them can also make numerical problems difficult to solve.

To prepare effectively, you need to connect each equation with its underlying concept and identify the conditions in which it applies. The following notes explain the key concepts through concise pointers, formula tables and comparisons, helping you revise important topics and practise NEET-level questions.

Simple Harmonic Motion (SHM): Definition and Basic Equations

Simple Harmonic Motion is a type of periodic motion in which the restoring force or acceleration is directly proportional to displacement from the mean position and acts in the opposite direction.

Periodic, Oscillatory and Simple Harmonic Motion

Conditions for SHM:

• The body moves about a stable mean position.

• The restoring force acts towards the mean position.

• The restoring force is proportional to displacement: (F\propto -x).

• Damping is neglected in ideal SHM.

Important SHM Equations

For a particle oscillating with amplitude (A), angular frequency (\omega) and phase constant (\phi):

The displacement equation may also be written using cosine. The choice depends on the initial position and phase of the particle.

SHM at Different Positions

Remember:

• The particle does not remain at rest at the mean position; it crosses it with maximum speed.

• At an extreme position, the particle momentarily stops before reversing direction.

• Acceleration and restoring force always point towards the mean position.

Energy in SHM and Important Graphs

In ideal SHM, the total mechanical energy remains constant. Kinetic energy and potential energy keep converting into each other during oscillation.

Energy Equations

Here, (m) is mass, (k) is the spring constant, (A) is amplitude and (x) is displacement from the mean position.

How SHM Graphs Behave

Time Period of Common Oscillating Systems

For a physical pendulum, (I) is the moment of inertia about the pivot and (d) is the distance between the pivot and the centre of mass.

Springs in Series and Parallel

NEET tip: For a spring-block system, (T\propto\sqrt{m}) and (T\propto\frac{1}{\sqrt{k}}). Increasing the mass increases the time period, while increasing the spring constant decreases it.

Waves: Types, Equations and Wave Speed

A wave transfers energy from one point to another without the overall transfer of matter. The particles of the medium may oscillate as the disturbance travels through it.

Types of Waves

Important Wave Equations

In the wave equation, (A) is amplitude, (\omega) is angular frequency, (k) is wave number and (\phi) is the initial phase.

Wave Speed on a Stretched String

The speed of a transverse wave on a stretched string depends on the tension and linear mass density.

[
v=\sqrt{\frac{F_T}{\mu}}
]

Here, (F_T) is the tension in the string and (\mu) is the mass per unit length.

Key points:

• Greater tension increases wave speed.

• Greater linear mass density decreases wave speed.

• For a given medium under fixed conditions, wave speed is determined by the medium, while frequency is determined by the source.

• When a wave enters another medium, its frequency remains unchanged, but its speed and wavelength may change.

Average Power Transmitted by a Sinusoidal Wave

For a sinusoidal transverse wave on a string:

[
P_{\text{avg}}=\frac12\mu\omega^2A^2v
]

The average power depends on the square of the amplitude and angular frequency, along with the medium's linear mass density and wave speed.

Standing Waves, Strings and Organ Pipes

Standing waves form when two waves of the same frequency and amplitude travel in opposite directions and superpose. The resulting pattern contains fixed points called nodes and points of maximum oscillation called antinodes.

Nodes and Antinodes

Standing Wave on a String Fixed at Both Ends

Both ends of the string are nodes. The allowed frequencies are:

[
f_n=\frac{nv}{2L},\qquad n=1,2,3,\ldots
]

Here, (L) is the vibrating length of the string and (v) is the wave speed.

Standing Waves in Organ Pipes

Organ pipes are classified according to their end conditions. An open end is a displacement antinode, while a closed end is a displacement node.

Important differences:

• An open pipe supports all harmonics: (f_1,2f_1,3f_1,\ldots).

• A pipe closed at one end supports only odd harmonics: (f_1,3f_1,5f_1,\ldots).

• For the same length and wave speed, the fundamental frequency of an open pipe is twice that of a closed pipe.

End Correction and Resonance

For a pipe with an open end, the displacement antinode lies slightly outside the physical opening. This effect is accounted for using end correction.

Here, (r) is the radius of the pipe. These end-correction relations are approximate and apply to common circular-pipe cases.

Sound Waves: Speed and Characteristics

Sound is a mechanical wave, so it requires a medium to travel. Its speed depends on the properties of that medium and the conditions under which the sound propagates.

Speed of Sound

Here, (B) is the bulk modulus, (\rho) is density, (\gamma) is the ratio of specific heats, (P) is pressure, (R) is the universal gas constant, (T) is absolute temperature and (M) is molar mass.

Remember:

• In a given gas, sound speed increases with absolute temperature.

• Sound generally travels faster in solids than in gases, though the exact speed depends on the material and wave type.

• Frequency determines pitch, while amplitude and intensity are related to perceived loudness.

Characteristics of Sound

For a point source radiating uniformly in a free field, intensity decreases with the square of the distance:

[
I\propto\frac{1}{r^2}
]

This inverse-square relationship assumes negligible absorption and reflection.

Beats and Interference of Sound

Beats occur when two sound waves of slightly different frequencies superpose. The resulting sound alternates between louder and softer because of periodic changes in amplitude.

Beats: Important Formulas

Tuning fork questions:

• If beats are heard when a tuning fork is sounded with a known frequency, the unknown frequency may differ from the known frequency by the beat frequency.

• If wax is added to a tuning fork, its mass increases and its frequency generally decreases.

• If a tuning fork is filed, its mass decreases and its frequency generally increases.

• Use the direction of the frequency change, along with the beat count, to identify the unknown frequency.

Interference of Sound

Interference occurs when two or more waves superpose. Depending on their phase difference, they may reinforce or weaken each other.

Here, (n=0,1,2,\ldots). The simple conditions assume coherent waves with the same frequency.

Resultant Amplitude and Intensity

For two waves with amplitudes (A_1) and (A_2) and phase difference (\Delta\phi):

[
A_R=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\Delta\phi}
]

For coherent waves in the same medium, intensity is proportional to the square of amplitude. Therefore:

[
I\propto A_R^2
]

For equal amplitudes (A):

Complete cancellation occurs only when the interfering waves have equal amplitudes and are exactly out of phase.

Important SHM and Waves Formulas for NEET Revision

Use this combined table for a final formula check before solving numerical questions.

Revise Simple Harmonic Motion (SHM) and Waves for NEET by focusing on motion, energy changes, time periods, wave speed, standing waves, organ pipes, beats and interference. Review the key equations, understand when each formula applies, and practise numerical questions to strengthen your conceptual understanding and improve problem-solving accuracy.