Wave Speed Calculator

A wave advances one wavelength in each period, so its speed is wavelength times frequency:

$v = f\lambda$

Symbols and SI units:

• $v$ — wave speed, metres per second (m/s)
• $f$ — frequency, hertz (Hz), i.e. cycles per second
• $\lambda$ — wavelength, metres (m)

Because the period is $T = 1/f$, the same relation can be written $v = \lambda/T$ — useful when a problem quotes a period in seconds instead of a frequency.

When it applies: to any periodic travelling wave — sound, light, water ripples, waves on a rope.

The key idea that trips people up: the speed is set by the medium, not by the source. Turning a tuning fork up in frequency does not speed the sound up; the wavelength shortens instead, keeping $f\lambda$ constant. The frequency is what the source fixes.

The assumption people forget: $v = f\lambda$ describes a wave in a single uniform medium. Cross into a new medium and $v$ and $\lambda$ both change while $f$ stays put.

If the wave speed is not given, it comes from a property of the medium.

Transverse wave on a string:

$v = \sqrt{\frac{T}{\mu}}$

with $T$ the tension in newtons (N) and $\mu$ the linear mass density in kilograms per metre (kg/m). Tightening a guitar string raises $v$, and with $\lambda$ fixed by the string length, the pitch rises.

Sound in air: about $343$ m/s at $20$ °C, rising roughly $0.6$ m/s per °C.

Light in vacuum: $c = 2.998 \times 10^8$ m/s; in a medium of refractive index $n$, $v = c/n$.

Rearranged, the same formula gives the other two unknowns:

$f = \frac{v}{\lambda}, \qquad \lambda = \frac{v}{f}$

The assumption people forget: $\mu$ is mass per unit length, not a density in kg/m³. Divide a string's total mass by its total length.

• Using $3 \times 10^8$ m/s for sound — sound in air is about $343$ m/s, six orders of magnitude slower.
• Believing a louder or higher-pitched wave travels faster — amplitude and frequency do not change the speed; only the medium does.
• Leaving prefixes unconverted — $2.4$ GHz is $2.4 \times 10^9$ Hz and $600$ nm is $6.0 \times 10^{-7}$ m. Convert before multiplying.
• Confusing frequency with period — they are reciprocals, so a $0.02$ s period is $50$ Hz, not $0.02$ Hz.
• Using kg/m³ in $v = \sqrt{T/\mu}$ — that formula needs linear density in kg/m.
• Assuming the wavelength is fixed when a wave enters a new medium — the frequency is what carries over unchanged.