Theme C — Wave Behaviour

C.1.8 Phase Angles in Simple Harmonic Motion (SHM) — IB Physics HL Revision Notes & Topical Questions

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C.1.8 Phase Angles in Simple Harmonic Motion (SHM) — IB PHYSICS HL notes: Use the phase angle φ in x = x0 sin(ωt + φ) and compare phases of oscillations, x, v and a.In phase — two points on a wave or on different waves are in phase when they are at the same point in their cyclePh…

Phase constant
Displacement
Acceleration
Sine leading
Cosine lagging
sin(ωt − φ)

Part 1 of 9Phase and phase angle

1In phase

In phase two points on a wave or on different waves are in phase when they are at the same point in their cycle

2Phase angle

Phase angle the difference in angular displacement compared with an oscillator that has x = 0 when t = 0

3Range

Range it can vary from 0 to 2π radians, one complete cycle

4Phase constant

φ gives the position in the cycle at t = 0

Part 2 of 9Equations with a phase angle

1Displacement

Displacement x = x₀ sin(ωt + φ)

2Velocity

Velocity v = ωx₀ cos(ωt + φ)

3Acceleration

Acceleration a = −ω²x₀ sin(ωt + φ)

4Units

φ is in radians

Phase angle φ	Start of the motion
0	At equilibrium moving in the positive direction
π/2	At the positive extreme, x = x₀ cos ωt
π	At equilibrium moving in the negative direction
3π/2	At the negative extreme

Part 3 of 9Sine and cosine

Relation

Relation they differ by π/2 radians

Sine leading

Sine leading x₀ sin(ωt + π/2) = x₀ cos ωt

Cosine lagging

Cosine lagging x₀ cos(ωt − π/2) = x₀ sin ωt

Part 4 of 9Shifting graphs

1sin(ωt − φ)

sin(ωt − φ) shifts to the right by φ
- (positive direction)

2sin(ωt + φ)

sin(ωt + φ) shifts to the left by φ

3cos(ωt − φ)

cos(ωt − φ) shifts to the right by φ

4cos(ωt + φ)

cos(ωt + φ) shifts to the left by φ

5One cycle

One cycle a complete oscillation is 2π rad

Part 5 of 9Phase difference

1Definition

Δφ is the difference between the phases of two oscillations of the same frequency

2From a time delay

Δφ = 2πΔt/T = ωΔt

3Values

0 is in phase, π is antiphase and π/2 is a quarter of a cycle apart

4Equivalence

Equivalence differences that differ by 2π are identical

5Leading

Leading the oscillation that reaches each point of the cycle earlier leads

Part 6 of 9Phase of x, v and a

1Velocity

Velocity v = ωx₀ sin(ωt + φ + π/2) so v leads x by π/2

2Acceleration

Acceleration a = ω²x₀ sin(ωt + φ + π) so a is in antiphase with x

3Relative to velocity

Relative to velocity a leads v by π/2

4Graphs

Graphs v peaks a quarter of a period before x and a is the mirror image of x
These relations hold for any initial phase

Part 7 of 9Reading phase from graphs

1Starting value

Starting value sin φ = x(0)/x₀
The gradient at t = 0 chooses between the two possible values of φ

2Comparing traces

Comparing traces measure the horizontal shift between corresponding peaks, divide by the period and multiply by 2π

3Which leads

Which leads the curve with the earlier peak leads

4Quarter period shift

T/4 corresponds to +π/2 lead or 3π/2 lag, which describe the same shift

Part 8 of 9Energy and driven oscillations

Energy and driven oscillations

Kinetic and potential energy

Kinetic and potential energy they oscillate at 2ω and are in antiphase, so when one is at maximum the other is at minimum

Resonance

Resonance the phase of the displacement relative to the driving force changes with driving frequency
It lags by π/2 at the natural frequency and approaches antiphase at high frequency

Power

Power at resonance the driving force is in phase with the velocity

Part 9 of 9More from the specification

Simple Harmonic Motion Defining Equation

An oscillating object executes SHM when its acceleration is directly proportional to its displacement from equilibrium and directed toward it, governed by the defining equation a = -ω²x.

Mass-Spring System Oscillations

In a mass-spring system oscillating with amplitude x₀, the restoring force follows Hooke's law F = -kx,
resulting in an angular frequency ω = √(k/m) and time period equation T = 2π√(m/k).

Simple Pendulum in a Gravitational Field

A simple pendulum system undergoing small-angle oscillations experiences a restoring gravitational component giving the period equation T = 2π√(L/g), independent of the oscillating mass.

Phase Angle and Kinematic Equations

The displacement of an oscillating object is described by harmonic equations such as x = x₀ sin(ωt + φ),
where φ is the phase angle;
- two systems are in phase when their phase difference is equal to zero or an integer multiple of 2π.

Simple Harmonic Motion Defining Equation

An oscillating object executes SHM when its acceleration is directly proportional to its displacement from equilibrium and directed toward it, governed by the defining equation a = -ω²x.

Mass-Spring System Oscillations

In a mass-spring system oscillating with amplitude x₀, the restoring force follows Hooke's law F = -kx,
resulting in an angular frequency ω = √(k/m) and time period equation T = 2π√(m/k).

Simple Pendulum in a Gravitational Field

A simple pendulum system undergoing small-angle oscillations experiences a restoring gravitational component giving the period equation T = 2π√(L/g), independent of the oscillating mass.

Phase Angle and Kinematic Equations

The displacement of an oscillating object is described by harmonic equations such as x = x₀ sin(ωt + φ),
where φ is the phase angle;
- two systems are in phase when their phase difference is equal to zero or an integer multiple of 2π.

Displacement-time graph

A graph of displacement against time whose gradient is velocity.

Velocity-time graph

A graph of velocity against time whose gradient is acceleration and area is displacement.

Amplitude

Maximum displacement from equilibrium.

Pendulum

A weight swinging on a string, exchanging GPE and KE.

Simple pendulum

A small bob on a light string of length L

Restoring force

Force directed toward equilibrium.

Gravitational potential

Work done per unit mass in bringing a test mass from infinity to the point.

Key terms

Term	Meaning
Phase	The argument ωt + φ that locates the position in the cycle.
Phase angle φ	The initial phase at t = 0.
Phase difference Δφ	The difference in phase between two oscillations of equal frequency.
In phase	Phase difference 0 (or 2π).
Antiphase	Phase difference π radians.
Leads	Reaches each point of the cycle earlier.

Avoid these mistakes

Saying acceleration leads displacement by π/2 — it is in antiphase (π).

Mixing up leading and lagging — a leading curve peaks earlier.

Using degrees for φ with a radian equation.

Comparing the phase of oscillations of different frequencies — phase difference needs equal frequencies.

Giving a phase difference larger than 2π instead of reducing it.

Last updated: 15 October 2026

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