Interactive physics simulator
Refraction of Waves
Explore how mechanical waves bend at boundaries. Study water depth transitions, Snell's Law calculations, critical angles, total internal reflection, and 2D ripple tank wavefronts.
Wave Refraction Laboratory
Change the incident angle, water depths, or speeds. Watch how the wave paths bend at the interface.
Live Wave Refraction Telemetry
- Incident Angle (θ₁)
- 30.0°
- Refracted Angle (θ₂)
- 18.2°
- Relative Index (n₁₂)
- 1.60
- Speed Ratio (v₁/v₂)
- 4.0 / 2.5 m/s
- State
- Normal Refraction
Introduction to Wave Refraction
The refraction of waves is the change in the direction of wave propagation when crossing a boundary from one medium to another. Refraction is a fundamental behavior of waves, observed in sound, light, and mechanical ripples alike. It occurs because the wave velocity changes as it transitions into a medium with different physical characteristics.
In water tanks, water wave speed is controlled by depth: waves travel faster in deep water and slower in shallow water. Thus, transitioning depths functions as a boundary between two refractive indices, bending wave trajectories.
The Bending Behavior: Toward vs. Away From Normal
The direction of wave bending is determined by whether the wave speeds up or slows down:
- Bending Toward the Normal (v₁ > v₂): When waves enter a slower medium (such as moving from deep water to shallow water), the wavelength decreases, and the wave path bends **toward the normal** line. The angle of refraction is smaller than the angle of incidence (θ₂ < θ₁).
- Bending Away From the Normal (v₁ < v₂): When waves enter a faster medium (such as moving from shallow water to deep water), the wavelength increases, and the wave path bends **away from the normal** line. The angle of refraction is larger than the angle of incidence (θ₂ > θ₁).
Mathematical Formulation: Snell's Law for Waves
Refraction is governed by Snell's Law, which states that the ratio of the sines of the incident and refracted angles is equal to the ratio of the wave speeds and wavelengths:
Where:
- θ1: The angle of incidence, measured between the incident ray and the normal.
- θ2: The angle of refraction, measured between the refracted ray and the normal.
- v1, v2: The wave velocities in medium 1 and medium 2, respectively.
- λ1, λ2: The wavelengths in medium 1 and medium 2.
- n12: The relative refractive index of medium 2 with respect to medium 1.
Critical Angle and Total Internal Reflection (TIR)
When waves travel from a slower medium to a faster medium, they bend away from the normal. As the angle of incidence θ1 increases, the angle of refraction θ2 increases even faster.
At a specific angle of incidence, the angle of refraction reaches exactly 90°, meaning the refracted wave travels along the interface boundary. This angle is called the **critical angle (θc)**:
If the angle of incidence exceeds this critical value (θ1 > θc), no wave can enter the faster medium. Instead, all of the wave energy is reflected entirely back into the slower medium. This is known as **Total Internal Reflection**.
Solved Examples
A water wave in a ripple tank travels from a deep region where its speed is 30 cm/s to a shallow region where its speed is 18 cm/s. If the wavelength in the deep region is 6.0 cm, calculate: (a) the frequency of the wave, and (b) the wavelength of the wave in the shallow region.
View Step-by-Step Solution
- Identify the given values: speed in deep region v1 = 30 cm/s, wavelength in deep region λ1 = 6.0 cm, and speed in shallow region v2 = 18 cm/s.
- Part (a): Solve for the wave frequency f in the deep water. Using the wave equation v = f · λ:
f = v1 / λ1 = 30 cm/s / 6.0 cm = 5.0 Hz. - Part (b): Refraction conserves frequency. Therefore, the frequency in the shallow water is also f = 5.0 Hz. Using the wave speed and frequency, calculate the new wavelength λ2:
λ2 = v2 / f = 18 cm/s / 5.0 Hz = 3.6 cm.
**Final Answer:** f = 5.0 Hz, λshallow = 3.6 cm
A plane wave in a ripple tank approaches a depth boundary at an angle of incidence of 40° in the deep water. The wave speed in deep water is 0.40 m/s and in shallow water is 0.25 m/s. Calculate: (a) the relative refractive index of the shallow water with respect to the deep water, and (b) the angle of refraction in the shallow water.
View Step-by-Step Solution
- Identify variables: incident angle θ1 = 40°, incident speed v1 = 0.40 m/s, and refracted speed v2 = 0.25 m/s.
- Part (a): The relative index of refraction n12 represents the ratio of speeds between the two regions:
n12 = v1 / v2 = 0.40 m/s / 0.25 m/s = 1.60. - Part (b): Use Snell's Law for waves: sin(θ1) / sin(θ2) = v1 / v2 = n12.
- Solve for sin(θ2):
sin(θ2) = sin(θ1) / n12 = sin(40°) / 1.60 ≈ 0.6428 / 1.60 ≈ 0.4017. - Compute the inverse sine: θ2 = arcsin(0.4017) ≈ 23.7°.
- The wave bends toward the normal line since it slows down in the shallower water.
**Final Answer:** n12 = 1.60, θrefracted ≈ 23.7°
A water wave travels from a shallow region where its speed is 0.15 m/s to a deep region where its speed is 0.30 m/s. (a) Calculate the critical angle for total internal reflection. (b) Explain what happens if a wavefront approaches the boundary at an angle of incidence of 45°.
View Step-by-Step Solution
- Identify variables: incident speed v1 = 0.15 m/s, and refracted speed v2 = 0.30 m/s.
- Part (a): Total internal reflection can occur when waves transition from a slower medium to a faster medium (v1 < v2). The critical angle θc occurs when the angle of refraction is 90°:
sin(θc) = v1 / v2 = 0.15 / 0.30 = 0.50. - Calculate the angle: θc = arcsin(0.50) = 30°.
- Part (b): Since the angle of incidence θ1 = 45° is greater than the critical angle θc = 30°, refraction is mathematically impossible (sin(θ2) would equal sin(45°) · (0.30 / 0.15) = 0.7071 · 2 = 1.414, which exceeds 1). The wave experiences total internal reflection, reflecting back into the shallow region at 45°.
**Final Answer:** θc = 30°, Total Internal Reflection occurs at 45°
Common Misconceptions & Pitfalls
- Misconception: The frequency of a wave changes when it enters a new medium.
**Reality:** No. Frequency is determined by the wave generator source. As wavefronts pass through the boundary, the rate at which they cross the line remains constant. Only wave speed and wavelength adjust. - Misconception: Total internal reflection can happen in any depth transition.
**Reality:** No. Total internal reflection only occurs when waves travel from a **slower** medium to a **faster** medium (e.g. from shallow to deep water). Going from fast to slow water (deep to shallow) always allows refraction. - Misconception: Wavefronts and wave rays are the same thing.
**Reality:** They are related but perpendicular. A wavefront represents the crest or trough of a wave (a line of constant phase). A ray is a vector pointing in the direction of wave travel, which is perpendicular to the wavefronts.
Practice Questions
Question 1
Explain why the frequency of a wave remains constant when it undergoes refraction, while its speed and wavelength change.
Show Explanation
The frequency of a wave is determined solely by the source generator (such as the oscillating paddle in a ripple tank). As the wave crosses a boundary, each wave crest arriving at the interface drives the oscillation in the second medium, generating exactly one new crest. Since crests cannot accumulate or disappear at the boundary, the number of cycles passing per second (frequency) remains identical. Because the wave speed changes due to the physical properties of the new medium, the distance between crests (wavelength) must adjust proportionally (λ = v/f) to maintain this constant rate.
Question 2
How does the depth of water in a ripple tank affect the speed of water waves, and how does this lead to refraction?
Show Explanation
Water waves in a ripple tank are surface gravity waves whose propagation speed depends on the water depth (v ≈ √(gh) in shallow water, where g is gravity and h is depth). Therefore, waves travel faster in deep water and slower in shallow water. When wavefronts approach a diagonal boundary at an angle, one side of each wavefront crosses into the shallow water first and slows down, while the other side remains in deep water and continues at full speed. This difference in velocity across the width of the wavefront causes the wavefront to tilt, pivoting the direction of propagation (the ray) toward the normal.
Question 3
Under what conditions does total internal reflection occur for mechanical waves at a boundary?
Show Explanation
Total internal reflection (TIR) occurs when two conditions are met: (1) The wave must travel from a slower medium (where wave speed is lower, like shallow water) toward a faster medium (where wave speed is higher, like deep water). (2) The angle of incidence θ₁ must exceed the critical angle θ_c = arcsin(v₁ / v₂). If both conditions are satisfied, the wave cannot enter the faster medium because the calculated angle of refraction exceeds 90°, and all incident wave energy reflects back into the slower medium.
Frequently Asked Questions
- What is wave refraction?
- Wave refraction is the bending of a wave's direction of travel when it crosses a boundary between two different media, caused by a change in the wave's speed.
- What properties of a wave change during refraction?
- The wave speed (v) and the wavelength (λ) change when crossing the boundary. The wave frequency (f) remains strictly constant.
- What is Snell's Law for waves?
- Snell's Law relates the angles and speeds of waves: sin(θ₁) / sin(θ₂) = v₁ / v₂ = λ₁ / λ₂ = n₁₂, where θ₁ and θ₂ are the angles relative to the normal.
- How does water depth affect wave speed in a ripple tank?
- Waves travel faster in deeper water and slower in shallower water. Consequently, when waves transition from deep to shallow water, they slow down and bend toward the normal.
- What is the critical angle for waves?
- The critical angle is the angle of incidence in a slower medium for which the angle of refraction in the faster medium is exactly 90°. It is calculated as θ_c = arcsin(v₁ / v₂).
- What is total internal reflection for waves?
- It is a phenomenon where waves striking a boundary from a slower to a faster medium at an angle greater than the critical angle reflect entirely back into the slower medium, with no wave entering the faster medium.