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.3.4 Refraction of Waves — IB Physics HL Revision Notes & Topical Questions

5 min read 11 parts Lesson 20 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.3.4 Refraction of Waves — IB PHYSICS SL notes: Explain refraction in terms of changing wave speed and wavelength, and apply refractive index, Snell's law and the critical angle.Refraction — the change in direction of a wave when it passes from one medium into another or through…

Pivoting wavefront
Car analogy
Normal incidence
What the symbols mean
Density link
Calculator

Part 1 of 9Why waves change direction

Refraction

Refraction the change in direction of a wave when it passes from one medium into another or through a region where its speed changes

Cause

Cause the change in speed

Frequency

Frequency it is fixed by the source, so wavelength changes in proportion to speed, λ = v/f

Pivoting wavefront

Pivoting wavefront the part of a wavefront that reaches a slower medium first slows first
The rest still travels at the original speed and covers more distance in the same time.
The wavefront pivots, so the ray turns towards the normal.

Car analogy

Car analogy a car driving at an angle onto mud turns because the wheel entering the mud slows first

Part 2 of 9Wavefront model

Spacing

Spacing wavefronts in the slower medium are closer together

Angles

Angles the angle between wavefront and boundary equals the angle between ray and normal

Ratio

Ratio sin θ₁/sin θ₂ = v₁/v₂ = λ₁/λ₂
θ₁ and θ₂ are the angles of the ray to the normal in each medium.

Normal incidence

Normal incidence the wave does not change direction but its wavelength still changes

Drawing

Drawing wavefronts are perpendicular to the rays and the refracted wavefronts in a slower medium are closer together

Part 3 of 9Refractive index

Refractive index

Refractive index n = c/v, a measure of how optically dense a medium is
What the symbols mean: n = refractive index (no unit) · c = speed of light in a vacuum (3.00 × 10⁸ m s⁻¹) · v = speed of light in the material (m s⁻¹)
c = 3.00 × 10⁸ m s⁻¹ is the speed of light in a vacuum and v is the speed in the medium.
It is a ratio and has no units.

Air

Air n is taken as 1

Density link

Density link the higher the n, the more optically dense the medium and the slower light travels

Check

Check a calculated value below 1 means an error has been made

Part 4 of 9Snell's law

Equation

Equation n₁ sin θ₁ = n₂ sin θ₂
What the symbols mean: n = refractive index (no unit) · i or θ₁ = angle of incidence · r or θ₂ = angle of refraction
- (both measured from the normal)
Equivalent form: n₂/n₁ = sin θ₁/sin θ₂ = v₁/v₂.
θ₁ is the angle of incidence and θ₂ the angle of refraction.

Angles

Angles measure them from the normal; if the angle to the boundary is given, use 90° minus that angle

Calculator

Calculator keep it in degree mode

Slower medium

Slower medium the ray bends towards the normal and the angle of refraction is smaller

Faster medium

Faster medium the ray bends away from the normal and the angle of refraction is larger

Part 5 of 9Critical angle and total internal reflection

Approach

Approach as the angle of incidence rises, the angle of refraction rises towards 90°

Critical angle

Critical angle the angle of incidence for which the angle of refraction is exactly 90°, with the ray refracted along the boundary
sin θ_c = n₂/n₁
The larger the refractive index of a material, the smaller its critical angle.

Below the critical angle

Below the critical angle the ray is refracted and exits

Above the critical angle

Above the critical angle the ray undergoes total internal reflection, following the law of reflection

Two conditions

Two conditions n₂ < n₁, so the light starts in the denser medium, and the angle of incidence exceeds θ_c

Formula

Formula the critical angle equation is not given in the data booklet

Part 6 of 9Water waves and shallow water

Speed and depth

Speed and depth surface waves travel slower in shallow water

Ripple tank

Ripple tank a submerged plate makes a shallow region
Plane waves crossing its edge at an angle turn towards the normal and the wavefronts bunch up.

Beaches

Beaches the part of a wave reaching shallow water first slows, so waves arrive nearly parallel to the shore

Swell on a slope

Swell on a slope decreasing depth lowers the speed, shortens the wavelength and raises the wave height

Headlands and bays

Headlands and bays rays converge on a headland and concentrate energy, causing erosion, while in bays they spread out and the water is calmer

Tsunami

Tsunami their speed depends on the depth of the water

Part 7 of 9Other waves

Sound

Sound it refracts when its speed changes with temperature
On a still evening the cooler air at ground level makes sound slower there, so rays bend downward and travel farther.
By day the effect reverses and sound bends upward.

Seismic waves

Seismic waves they refract at layers of different density, so arrival times reveal the structure of the Earth

Light

Light it refracts in lenses, prisms and the atmosphere
Stars twinkle and the setting Sun looks slightly flattened for this reason.

Continuous change of n

Continuous change of n hot air above a road bends rays smoothly, producing a mirage

Part 8 of 9Reflection and energy at a boundary

1Reflected wave

Reflected wave some wave is always reflected, with intensity depending on how different the media are

2Conservation

Conservation incident energy equals reflected plus transmitted plus absorbed energy

3Contrast with diffraction

Contrast with diffraction diffraction does not alter the wavelength, whereas refraction does

4Final check

Final check make sure the direction of bending agrees with whether the wave slowed or sped up

Part 9 of 9More from the specification

Young's Double-Slit Experiment

Coherent monochromatic wave fronts diffracting through two narrow slit apertures in a barrier superpose to create an interference pattern of bright and dark fringes on a screen.

Superposition Principle and Phase Difference

According to the principle of superposition, total wave displacement equals the vector sum of individual displacements depending on their relative phase.

Constructive Interference Conditions

Constructive interference occurs where coherent waves arrive with a path difference equal to an integer number of wavelengths (nλ),
producing intensity maxima.

Destructive Interference Conditions

Destructive interference occurs where waves arrive in antiphase with a path difference of (n + ½)λ, cancelling wave amplitudes to produce dark minima.

Interference

Effect of the overlap of waves.

Constructive

Waves in phase giving larger amplitude.

Destructive

Waves in antiphase giving reduced amplitude.

Superposition

Resultant displacement is the sum of the displacements of the individual waves.

Coherent

Constant phase difference and same frequency.

Transmission

Sending electricity along power cables from the power station to consumers.

Destructive interference

Waves meet in antiphase giving reduced amplitude.

Constructive interference

Waves meet in phase giving increased amplitude.

Key terms

Term	Meaning
Refraction	Change of wave direction caused by a change of speed.
Angle of refraction	Angle between the refracted ray and the normal.
Refractive index	n = c/v for light.
Critical angle	Angle of incidence in the denser medium for which the refracted angle is 90°.
Total internal reflection	Complete reflection when light in a denser medium meets the boundary above the critical angle.

Avoid these mistakes

Claiming frequency changes when a wave refracts.

Bending towards the normal when the wave speeds up — a faster medium bends the ray away from the normal.

Drawing wavefronts at the same spacing on both sides of the boundary.

Using the angle between the wavefront and the normal instead of between the ray and the normal.

Assuming refraction applies only to light — it applies to all waves.

Last updated: 15 October 2026