ما هي العلاقة بين سرعة الموجة وترددها؟ وما هو تأثير دوبلر؟ What is the wave equation and Doppler effect?

المعادلة العامة للموجات هي v = fλ (حيث v السرعة بـ m/s، f التردد بالهرتز، λ الطول الموجي بالمتر). وتأثير دوبلر هو التغير الظاهري في تردد الموجة نتيجة الحركة النسبية بين المصدر والمراقب: f_obs = f(v ± v_o)/(v ∓ v_s). [الصيغة الرمزية الموزونة: v = f × λ | f_obs = f × (v ± v_o) / (v ∓ v_s) | T = 1 / f]

Theme C — Wave Behaviour

C.4 Standing Waves and Resonance — IB Physics HL Revision Notes & Topical Questions

5 min read 13 parts Lesson 26 of 37 1 quiz

📚 IB Physics HL Hub / Topical Revision

Curriculum Equation Key Wave Speed & Doppler Effect FormulasFast Exam Solution

v = f × λ  |  f_obs = f × (v ± v_o) / (v ∓ v_s)  |  T = 1 / f

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Lesson summary

C.4 Standing Waves and Resonance — IB PHYSICS HL notes: Describe standing waves, their nodes and antinodes and harmonics on strings and pipes, and explain free and forced oscillations, resonance and damping.Cause — two waves of the same frequency, wavelength and similar amplitude…

All wave types
Fixed positions
Within a loop
Odd number of nodes between
Even number of nodes between
Only two values

Part 1 of 11Formation of standing waves

Formation of standing waves

Cause

Cause two waves of the same frequency, wavelength and similar amplitude travelling in opposite directions along the same line superpose
Usually a travelling wave superposes with its own reflection

Result

Result crests and troughs only move vertically and the pattern does not travel

All wave types

All wave types superposition applies to transverse and longitudinal waves

Energy

Energy a standing wave stores energy and a progressive wave transfers it

Condition

Condition the half-cycles of the wave must be symmetrical, so a shape with a straight leading edge and sinusoidal trailing edge forms no standing wave

Part 2 of 11Nodes and antinodes

Node

Node a point of zero amplitude where the waves are in antiphase and cancel by destructive interference

Antinode

Antinode a point of maximum amplitude where the waves are in phase and add by constructive interference

Fixed positions

Fixed positions nodes do not move and antinodes only oscillate perpendicular to the string

Spacing

Spacing adjacent nodes are λ/2 apart

Part 3 of 11Phase on a standing wave

1Within a loop

Within a loop all points are in phase

2Odd number of nodes between

Odd number of nodes between points are in antiphase

3Even number of nodes between

Even number of nodes between points are in phase

4Only two values

Only two values the phase difference between two points is 0 or π, unlike a travelling wave which can be anything from 0 to 2π

Feature	Travelling wave	Standing wave
Energy transfer	Yes	No net transfer, energy is stored
Amplitude	Same for all points	Zero at nodes to maximum at antinodes
Phase	Changes continuously along the wave	Same between adjacent nodes, antiphase across a node
Wavelength	Crest to crest	Twice the node-to-node distance

Part 4 of 11Boundary conditions

Strings

Strings fixed at both ends, free at both ends, or one end fixed and one free

Pipes

Pipes closed at both ends, open at both ends, or closed at one end

Natural frequencies

Natural frequencies an integer number of half wavelengths fits on the string

Fixed end or closed end

Fixed end or closed end the reflected wave is in antiphase so the end is a node

Free end or open end

Free end or open end the reflected wave is in phase so the end is an antinode

Variables

Variables the allowed patterns depend on the driving frequency and the boundary conditions

String frequency

String frequency it depends on tension and mass per unit length

Part 5 of 11Harmonics on a string fixed at both ends

1First harmonic

First harmonic a single loop with two nodes and one antinode, λ₁ = 2L

2Higher harmonics

Higher harmonics the nth harmonic has (n + 1) nodes and n antinodes

3Wavelength

λ_n = 2L/n for n = 1, 2, 3, …

4Frequency

Frequency f_n = nv/2L, where v is the wave speed on the string

Part 6 of 11Harmonics in pipes

1Open at both ends

Open at both ends the first harmonic has one node and two antinodes
The nth harmonic has n nodes and (n + 1) antinodes
λ_n = 2L/n for all n

2Open at one end

Open at one end the lowest harmonic is a half loop with one node and one antinode
Only odd harmonics exist, λ_n = 4L/n with n = 1, 3, 5, …

3Frequencies

Frequencies f_n = v/λ_n in both cases

4Next harmonic after the first in a closed pipe

Next harmonic after the first in a closed pipe the third harmonic and not the second

5Recall

Recall these wavelength expressions are not given in the data booklet

Part 7 of 11Free and forced oscillations

Free oscillation

Free oscillation only internal forces act and there is no energy input
The system oscillates at its natural frequency

Natural frequency f0

Natural frequency f0 the frequency of oscillation when the system is allowed to oscillate freely

Forced oscillation

Forced oscillation produced by a periodic external driving force
The system oscillates at the driving frequency

Examples

Examples a struck tuning fork is free, a glass shattered by sound and a clarinet are forced

Part 8 of 11Resonance

Condition

Condition driving frequency f = natural frequency f₀

Effect

Effect the amplitude of the oscillations is a maximum

Energy

Energy the energy transfer from driver to system is most efficient

Away from resonance

Away from resonance if the driving frequency is slightly higher or lower, the amplitude is smaller

Example

Example a child pushed on a swing at the swing's natural frequency swings highest

Uses and hazards

Uses and hazards useful in musical instruments and microwave ovens, harmful in bridges and buildings

Part 9 of 11Damping

Definition

Definition the reduction in energy and amplitude of oscillations due to resistive forces such as friction and air resistance

Frequency

Frequency the frequency of damped oscillations does not change as the amplitude falls

Light damping

Light damping the amplitude decays exponentially over many cycles with a constant period
A pendulum slowly stops

Critical damping

Critical damping the system returns to equilibrium in the shortest time without oscillating
Examplecar suspension

Heavy damping

Heavy damping the system returns slowly without oscillating
Exampledoor dampers

Part 10 of 11Resonance curve and damping

1Curve

Curve amplitude against driving frequency peaks at the natural frequency

2More damping

More damping the peak is lower, the curve is broader and the peak moves slightly to a lower frequency

3Natural frequency

Natural frequency it stays the same

4Resistive and restoring forces

Resistive and restoring forces a resistive force opposes motion and causes damping while a restoring force returns the system to equilibrium

Part 11 of 11More from the specification

Standing waves

Standing waves formed by superposition of two identical waves travelling in opposite directions, giving stationary waves with nodes and antinodes

Nodes and antinodes

Nodes and antinodes nodes have zero amplitude and antinodes maximum, separated by half a wavelength

Harmonics

Harmonics in a string fixed at both ends, the fundamental has length equal to half wavelength and higher modes are whole multiples

Graph and scale

Graph and scale a displacement-position graph with a scale shows the envelope of the stationary wave, with no net energy transfer

Stationary

Not moving.

Stationary wave

Wave formed by two progressive waves of the same frequency travelling in opposite directions.

Key terms

Term	Meaning
Standing wave	A wave pattern formed by the superposition of two waves travelling in opposite directions, with fixed nodes and antinodes.
Node	A point of zero amplitude in a standing wave.
Antinode	A point of maximum amplitude in a standing wave.
Harmonic	A mode of vibration whose frequency is a whole-number multiple of the first harmonic.
Natural frequency	The frequency at which a system oscillates when displaced and released.
Resonance	The large increase in amplitude when the driving frequency equals the natural frequency.
Damping	The loss of energy and amplitude from an oscillating system due to resistive forces.
Critical damping	Damping that returns a system to equilibrium in the shortest time without oscillation.

Avoid these mistakes

Saying a standing wave transfers energy along the medium — there is no net energy transfer.

Using the open-pipe formula for a closed pipe — a pipe closed at one end has only odd harmonics with λ = 4L/n.

Saying points on opposite sides of a node are in phase — they are in antiphase.

Confusing resistive force with restoring force.

Saying damping changes the natural frequency — it lowers the amplitude and broadens the peak only.

Last updated: 15 October 2026