Oscillations and waves (Class 11): SHM, superposition and beats
SHM appears wherever a restoring force grows in proportion to the displacement. Waves then carry that motion through a medium, and two waves meeting add up. Explore both with experiments you can change.
NCERT Class 11 Physics, Chapter 13 (Oscillations) and Chapter 14 (Waves). Every lab below runs in 3D in your browser or on a phone, in English and Hindi.
Try the labs
Floating cylinder oscillating in a liquid: SHM from buoyancy · plan · Chapter 13
To show that a cylinder floating upright in a liquid performs simple harmonic motion when pushed down, and to find its time period.
Superposition of waves: constructive and destructive interference on a string · plan · Chapter 14
To see the principle of superposition: when two waves meet, the displacement is the sum of the displacements, which gives constructive or destructive interference, standing waves and beats.
Damped and forced oscillations: resonance · plan · Chapter 13
To study damped and forced oscillations and understand resonance and the quality factor.
Pendulum in an accelerating frame: effective gravity · plan · Chapter 13
To show that a pendulum in an accelerating frame sees an effective gravity g_eff = g − a and find its time period.
Physical pendulum and torsion pendulum · plan · Chapter 13
To study a physical (compound) pendulum and a torsion pendulum and compare them with the simple pendulum.
SHM as projection of uniform circular motion · plan · Chapter 13
To show that simple harmonic motion is the projection of uniform circular motion on a diameter and to read x, v and a from the reference circle.
Simple pendulum: g from T and large-angle correction · free · Chapter 13
To study the motion of a simple pendulum, measure its time period and find the acceleration due to gravity g, and to see where the small-angle approximation T = 2π√(L/g) stops being valid.
Springs in series and parallel: equivalent k and SHM · plan · Chapter 13
To find the equivalent spring constant of two springs in series and in parallel and verify T = 2π√(m/k_eq).
Spring-mass SHM: energy, springs in series/parallel · plan · Chapter 13
To study simple harmonic motion of a spring–mass system, verify T = 2π√(m/k) and understand energy exchange; to find the effective spring constant of series and parallel combinations.
SHM of a liquid column and a floating cylinder · plan · Chapter 13
To show that a displaced liquid column in a U-tube and a floating cylinder both execute simple harmonic motion and to find their periods.
Beats from two close frequencies · plan · Chapter 14
To study the phenomenon of beats produced by superposition of two sound waves of nearly equal frequency and determine the beat frequency.
Doppler effect · plan · Chapter 14
To understand the Doppler effect for sound and apply the formula for the apparent frequency when the source and/or observer move.
Melde's experiment: standing waves on a string · plan · Chapter 14
To study transverse standing waves on a string with Melde's apparatus and verify v = √(T/μ) and f = (p/2L)√(T/μ).
Resonance tube (open and closed organ pipes) · plan · Chapter 14
To find the speed of sound in air using a resonance tube and to study the harmonics of closed and open pipes.
Resonance tube: speed of sound and end correction · plan · Chapter 14
To determine the speed of sound in air and the end correction of a tube using a resonance tube and a tuning fork.
Sonometer: frequency, tension, length laws · plan · Chapter 14
To verify the laws of transverse vibration of a stretched string using a sonometer and to find the frequency of a tuning fork.
Sound intensity, inverse-square law and decibels · plan · Chapter 14
To understand sound intensity, the inverse-square law and the decibel scale; to add intensities from several sources.
Standing waves on a string and Melde's experiment · plan · Chapter 14
To study standing waves on a stretched string, find the resonant frequencies and verify f = (n/2L)√(T/μ) (Melde's experiment).
Transverse and longitudinal waves on a medium · plan · Chapter 14
To understand transverse and longitudinal mechanical waves, and the relation v = fλ.
Reflection and transmission of a pulse at a boundary · plan · Chapter 14
To see how a pulse is reflected and transmitted at a fixed end, a free end and at the junction of two strings of different linear mass density.
A floating cylinder in simple harmonic motion
A cylinder of length L and density ρc floats in a liquid of density ρl with a length h = L ρc / ρl under the surface. Push it down by x and the extra buoyant force is ρl A g x upward, a restoring force proportional to x. The motion is SHM with ω = √(g / h), which gives T = 2π √(h / g).
The period depends on the immersed length h and not on the area or the amplitude. Raise the damping in the lab to see the amplitude die away.
h = L ρc / ρlT = 2π √(h / g)
Superposition of waves
When two waves overlap, the displacement is the sum of the two. For equal frequencies the resultant amplitude is A = √(A₁² + A₂² + 2 A₁ A₂ cos φ). It is largest, A₁ + A₂, when the waves are in phase (φ = 0) and smallest, |A₁ − A₂|, when they are opposite (φ = 180°).
A = √(A₁² + A₂² + 2A₁A₂ cos φ)constructive: φ = 0, 2π, ...destructive: φ = π, 3π, ...
Standing waves and beats
Two identical waves travelling in opposite directions form a standing wave. The nodes, where the string never moves, are λ / 2 apart, with antinodes in between. If instead the two waves have slightly different frequencies the resultant swells and fades with the beat frequency |f₁ − f₂|.
distance between nodes = λ / 2beat frequency = |f₁ − f₂|
Frequently asked questions
Why is a floating cylinder in SHM?
Pushing it down by x increases the buoyant force by ρl A g x. That net force acts upward and is proportional to x, which is the condition for SHM.
What is the time period of a floating cylinder?
T = 2π √(h / g), where h is the length immersed in equilibrium. It does not depend on the amplitude.
What are constructive and destructive interference?
Constructive interference adds the waves to a larger amplitude, when they are in phase. Destructive interference reduces the amplitude, in the extreme case to zero, when they are in opposite phase.
How far apart are nodes in a standing wave?
λ / 2 apart, and the same distance separates two neighbouring antinodes.
How are beats produced?
When two waves of nearly equal frequency overlap, the resultant amplitude rises and falls |f₁ − f₂| times per second.