Interactive physics simulator
Wave Speed: v = f λ
Master the kinetics of wave propagation. Explore how medium properties dictate propagation speed, run tension and density string labs, and study shallow water ripple tank velocity dynamics.
Wave Speed Laboratory
Modify frequency, wavelength, or medium properties. Observe how speed changes dynamically.
Live Wave Telemetry
- Wave Speed (v)
- 3.00 m/s
- Frequency (f)
- 1.00 Hz
- Wavelength (λ)
- 3.00 m
- Wave Period (T)
- 1.00 s
- Medium State
- Uniform
Introduction to Wave Speed
In physics, the wave speed (v) measures the rate at which a wave disturbance propagates through a medium. Unlike the local oscillation speed of individual particles in the medium, wave speed describes the linear velocity of the overall wave profile as it transports energy through space.
The relationship between wave speed, frequency, and wavelength is governed by the universal wave equation:
Where:
- v (Wave Speed): The rate of wave propagation, measured in meters per second (m/s).
- f (Frequency): The rate of source oscillations, measured in Hertz (Hz or cycles/s).
- λ (Wavelength): The spatial length of one complete wave cycle, measured in meters (m).
What Actually Dictates Wave Speed?
A common point of confusion for students is thinking that generating higher frequency waves makes them travel faster. **This is incorrect.**
Wave speed is determined **solely by the physical and mechanical properties of the medium**:
- On a string: Wave speed is governed by string tension T and linear mass density μ:
v = √(T / μ) Tightening the string increases speed; using a heavier, thicker string slows it down. - In shallow water: Wave speed is governed by gravity g and water depth d:
v = √(g · d) Waves travel faster in deep water and slow down as they approach the shallow shore. - In sound waves: Speed is determined by elasticity and density: sound travels faster in solids than liquids, and faster in liquids than gases.
Therefore, if you change the frequency f of a source, the wave speed v remains constant (as long as the medium doesn't change). Instead, the wavelength λ adjusts inversely (λ = v/f).
Wavelength vs. Speed
Because speed is fixed by the medium, any changes in the frequency of the source directly alter the spacing between waves.
- High Frequency: The source vibrates rapidly. Because the wave propagates at a constant speed, the waves are pushed out close together, yielding a **short wavelength**.
- Low Frequency: The source vibrates slowly. The waves have time to stretch out as they propagate, yielding a **long wavelength**.
Solved Examples
A transverse wave on a string has a frequency of 100 Hz. If the distance between two consecutive crests is measured to be 0.45 meters, calculate the speed of the wave along the string. Show all steps.
View Step-by-Step Solution
- Identify variables: frequency f = 100 Hz, and wavelength λ = 0.45 m (since wavelength is the distance between consecutive crests).
- Use the wave speed equation: v = f · λ.
- Substitute values: v = 100 Hz · 0.45 m.
- Calculate the result: v = 45 m/s.
- The speed of the wave along the string is 45 m/s.
**Final Answer:** v = 45 m/s
A 5.0-meter-long metal wire has a mass of 0.05 kg and is stretched under a tension of 80 N. Calculate: (a) the linear mass density of the wire, (b) the velocity of a transverse wave on this wire, and (c) the wavelength if the wave frequency is 20 Hz.
View Step-by-Step Solution
- Part (a): Linear mass density μ is defined as mass per unit length: μ = m / L.
- μ = 0.05 kg / 5.0 m = 0.01 kg/m.
- Part (b): The velocity of a transverse wave on a string depends on tension T and linear mass density μ: v = √(T / μ).
- v = √(80 N / 0.01 kg/m) = √(8000) ≈ 89.44 m/s.
- Part (c): Rearrange the wave equation to solve for wavelength λ: λ = v / f.
- λ = 89.44 m/s / 20 Hz ≈ 4.47 meters.
- The linear density is 0.01 kg/m, the wave speed is ≈ 89.4 m/s, and the wavelength is ≈ 4.47 m.
**Final Answer:** μ = 0.01 kg/m, v ≈ 89.4 m/s, λ ≈ 4.47 m
A water wave generator in a ripple tank oscillates at a frequency of 5.0 Hz. The waves travel at a speed of 0.35 m/s in deep water. As the waves travel into a shallow section, the speed drops to 0.20 m/s. Calculate: (a) the wavelength in the deep water, (b) the frequency of the waves in the shallow water, and (c) the wavelength in the shallow water.
View Step-by-Step Solution
- Part (a): Solve for wavelength λ in deep water using the wave equation: λ = v / f.
- λdeep = 0.35 m/s / 5.0 Hz = 0.07 meters (or 7.0 cm).
- Part (b): The frequency of a wave is determined strictly by its source. When entering a new medium, its frequency does not change. Therefore, fshallow = fdeep = 5.0 Hz.
- Part (c): Solve for wavelength λ in shallow water: λ = v / f.
- λshallow = 0.20 m/s / 5.0 Hz = 0.04 meters (or 4.0 cm).
- The deep wavelength is 7.0 cm, the shallow frequency is 5.0 Hz, and the shallow wavelength is 4.0 cm.
**Final Answer:** λdeep = 7.0 cm, f = 5.0 Hz, λshallow = 4.0 cm
Common Misconceptions & Pitfalls
- Misconception: Creating louder sound waves makes them travel faster through the air.
**Reality:** No. Loudness corresponds to wave amplitude. Amplitude determines the energy content of the wave, but does not alter the propagation speed, which is fixed by air temperature and pressure. - Misconception: Waves travel faster in thick strings because they have more mass.
**Reality:** No. Heavier, thicker strings have higher inertia (higher linear density μ). Since speed is inversely proportional to density (v = √(T / μ)), thicker strings actually slow the wave speed down. - Misconception: The speed of light is constant in all materials.
**Reality:** The speed of light is only constant in a vacuum (c ≈ 3 × 108 m/s). When light enters a medium like glass or water, it interacts with the material's electric field, slowing down.
Practice Questions
Question 1
Explain why the speed of sound is faster in water (approx 1480 m/s) than in air (approx 343 m/s) even though water is much more dense than air.
Show Explanation
Wave speed is determined by both the elastic property (resistance to compression) and the inertial property (density) of the medium: v = √(Elasticity / Density). Water is roughly 800 times denser than air, which would tend to slow waves down. However, water is also about 15,000 times less compressible (it has a vastly larger Bulk Modulus) than air. Because the increase in elasticity far outweighs the increase in density, sound travels more than 4 times faster in water than in air.
Question 2
If you double the tension of a guitar string, by what factor does the wave speed increase? Show the formula.
Show Explanation
The speed of a transverse wave on a string is given by v = √(T / μ). If the tension T is doubled to 2T, the new speed is v' = √(2T / μ) = √2 · √(T / μ) = √2 · v. Therefore, the wave speed increases by a factor of √2 (approximately 1.41, or a 41% speed increase).
Question 3
Does changing the wave amplitude affect the speed of wave propagation? Explain.
Show Explanation
No. For linear mechanical waves, the speed of propagation is determined strictly by the mechanical properties of the medium (such as elasticity, tension, or density) and is independent of the wave's amplitude. A wave with a larger amplitude carries more energy, but it travels through space at the exact same speed as a small-amplitude wave in that same medium.
Frequently Asked Questions
- What is wave speed?
- Wave speed (v) is the rate at which a wave disturbance travels through a medium, measured in meters per second (m/s). It represents the distance a wave crest covers per unit time.
- What is the wave speed formula?
- The universal wave equation is v = f · λ, where v is speed, f is frequency, and λ is wavelength. It is also written as v = λ / T, where T is the wave period.
- Does frequency affect wave speed?
- No. If you increase the frequency of a wave generator, the wave speed remains constant because the speed is set by the medium. To compensate, the wavelength decreases proportionally (λ = v / f).
- How does tension affect wave speed on a string?
- Wave speed on a string is directly proportional to the square root of string tension (T) and inversely proportional to the square root of the linear mass density (μ): v = √(T / μ).
- Why do waves slow down in shallow water?
- For shallow water waves (where depth is less than half the wavelength), the wave speed is governed by gravity and depth: v ≈ √(g · d). As water depth (d) decreases, the drag against the bottom increases, slowing the wave down.