AkCalculators

Wave Speed — v = f·λ and waves on strings (v = √(T/μ)) explained

Wave speed is a fundamental property of waves: how quickly the wave pattern travels through a medium. Different formulas describe wave speed depending on the wave type and medium. The simplest and most universal relation for periodic waves is:

v = f × λ

where v is wave speed, f is frequency and λ is wavelength. This relation holds for mechanical waves, sound waves, and electromagnetic waves in a given medium. For specialized cases such as waves on strings, an expression derived from the physical model relates wave speed to tension and linear mass density:

v = √(T / μ)

Here T is the tension in the string (newtons) and μ is the linear mass density (kg per meter). This article explains both relations, shows how to compute missing quantities, gives worked examples, discusses measurement techniques and practical considerations, and highlights common pitfalls.

1. The relation v = f·λ — origin and interpretation

Consider a periodic wave with crests spaced λ apart and the wave pattern moving at speed v. A fixed point in the medium sees successive crests pass with frequency f — the number of crests per second. If crests are λ meters apart and f crests pass each second, the wave pattern must move at v = distance per second = f × λ. This is purely geometric and independent of the wave’s physical origin — it applies to water waves, sound waves, and light waves (when using phase velocity in a medium).

2. Use cases and unit consistency

When using v = f·λ always use consistent units: f in hertz (Hz = s⁻¹), λ in meters (m), giving v in meters per second (m/s). For sound in air, typical numbers at 20°C: v ≈ 343 m/s, so a 440 Hz tone (A4) has λ ≈ 343/440 ≈ 0.78 m. If you prefer centimeters or millimeters, convert values to SI before using formulas to avoid errors.

3. Solving the basic relation

The equation is algebraic; rearrange to compute any one variable:

• v = f × λ
• f = v / λ
• λ = v / f

Use these rearrangements when two quantities are known. Our calculator accepts any two and computes the third automatically, providing a step-by-step explanation suitable for lab notes.

4. Waves on a string — deriving v = √(T / μ)

For a stretched string under tension T, consider a small element of the string and apply Newton's second law. The transverse restoring force from tension accelerates the element; for small transverse displacements and using the wave ansatz, the wave equation emerges with wave speed v = √(T / μ). Intuitively, higher tension increases speed while heavier strings (larger μ) slow waves down. This is why tightening a guitar string raises pitch: increased tension increases wave speed and hence frequency for a given string length and mode.

5. Solving string-mode formulas

Again, algebraic rearrangements allow solving for any one unknown:

• v = √(T / μ)
• T = μ · v²
• μ = T / v²

When using these forms be careful with units: T in N (kg·m/s²), μ in kg/m, gives v in m/s.

6. Worked examples — basic mode

Example A — sound wavelength: Given v = 343 m/s and f = 1000 Hz, λ = v / f = 0.343 m (34.3 cm). This is the wavelength of a 1 kHz tone in air at 20°C.

Example B — radio wave frequency: For λ = 1 m (AM radio band), in free space v ≈ 3.0×10⁸ m/s so f = v / λ ≈ 3.0×10⁸ Hz or 300 MHz.

7. Worked examples — string mode

Example C — guitar string: A guitar string has μ = 0.006 kg/m (approx) and tension T = 80 N. Wave speed v = √(80 / 0.006) ≈ √(13333.33) ≈ 115.47 m/s. The fundamental frequency for an open string of length L = 0.65 m is f₁ = v / (2L) ≈ 88.82 Hz (approx low E string range depending on actual μ and T).

Example D — adjusting pitch: To raise the fundamental frequency, increase T (tighten) — f scales with √T assuming μ and L constant.

8. Measurement tips and experimental setup

Measuring v via f and λ: For water waves, measure wavelength visually (distance between crests) and frequency by counting crests over time. For sound, measure frequency with a tone generator and wavelength via microphone arrays or by measuring phase differences.

Measuring string μ: Measure a section of the string’s mass (on a scale) and divide by its length. Often μ is given in grams/m; convert to kg/m for SI formulas.

Measuring tension: Tension can be measured with force sensors or inferred from weight and pulley setups (T equals hanging mass × g in a simple pulley arrangement). Always subtract frictional effects and measure carefully for accurate μ or v calculations.

9. Dispersion and medium dependence

While v = f·λ always holds as a kinematic identity, the dependence of v on f (dispersion) occurs when different frequencies travel at different speeds (v = v(f)). In nondispersive media (ideal air at moderate conditions) sound speed is roughly constant across audible frequencies; in dispersive media (water waves of different wavelengths, waveguides, optical fibers) v can depend on λ and f—then v = f·λ still holds but v and λ are frequency-dependent.

10. Common pitfalls and troubleshooting

• Wrong units: Mixing Hz with kHz or cm with meters causes errors. Convert to SI first.
• Infinite or zero values: If you enter zero for λ or f, the formula breaks. Represent 'infinite' distances or very large values with appropriate numeric approximations if needed.
• Sign conventions: For wave speed magnitude, use positive values. For longitudinal waves direction can be signed, but this calculator handles positive magnitudes for speed.

11. Practical applications

Wave speed formulas are used across physics and engineering: acoustics (room acoustics, speaker design), seismology (wave propagation in the Earth), telecommunications (signal propagation), musical instruments (tuning and string design), and instrumentation (measuring material properties by wave speed).

12. Summary

The wave relations v = f·λ and v = √(T/μ) are simple yet powerful. They let you compute any missing variable from two known quantities, and form the basis for more advanced wave mechanics. Use the calculator above to compute speeds, frequencies, wavelengths, tensions, or linear densities quickly and export results for lab notebooks.