Wave Properties and the Wave Equation

Linking wave speed, frequency, and wavelength in one compact relationship

On 26 December 2004, oceanographic sensors picked up the Indian Ocean tsunami travelling at 800 km/h (222 m/s) across deep water with wave crests spaced roughly 250 km apart. The time between one crest and the next passing a fixed buoy was about 18.7 minutes. You can verify this directly: if $v = 222$ m/s and $\lambda = 250{,}000$ m, then $f = v/\lambda = 8.9 \times 10^{-4}$ Hz, so period $T = 1/f = 1{,}124$ s $\approx 18.7$ min. Three measurable quantities, speed, wavelength, frequency, linked by one equation: $v = f\lambda$.

This is why the medium matters. A rope, air column, or water surface largely determines wave speed. The source changes frequency. The wave adjusts its wavelength in response.

The wave equation $v = f\lambda$ links wave speed (m/s), frequency (Hz) and wavelength (m); in the same medium $v$ is approximately constant so doubling frequency halves wavelength. Period $T = 1/f$ (s).

Pause, copy the highlighted definition and formula into your book before moving on.

Distance graphs show wavelength; time graphs show period, they look identical but mean different things

We just saw that $v = f\lambda$ links wave speed, frequency and wavelength. That raises a question: given a wave graph, how do you tell whether the horizontal axis gives wavelength or period? This card answers it → always check the axis label: distance → read $\lambda$; time → read $T$.

A displacement-distance graph is a snapshot across space. A displacement-time graph is a history of one point over time.

On a displacement-distance graph, the horizontal axis represents position, so the distance between two adjacent crests is the wavelength. On a displacement-time graph, the horizontal axis represents time, so the distance between two adjacent crests is the period. Amplitude is read the same way on both graphs.

Top: read wavelength across space. Bottom: read period across time. Always check the horizontal axis label first.

On a displacement-distance graph, the crest-to-crest spacing is the wavelength $\lambda$ (m); on a displacement-time graph, the crest-to-crest spacing is the period $T$ (s). Amplitude is the maximum displacement from equilibrium on either graph.

Add the highlighted graph-reading rule to your notes before the check below.

Phase difference and why the medium is the boss of wave speed

We just saw how to read $\lambda$ and $T$ from graphs. That raises a question: what happens to wavelength when a wave crosses into a different medium? This card answers it → frequency stays constant (set by source) while speed and wavelength change together via $\lambda = v/f$.

Two points are in phase if they are at the same stage of oscillation. Points separated by one full wavelength (or period) are in phase. Points separated by half a wavelength are in antiphase.

Wave speed is determined by the properties of the medium: tension and linear density for strings, temperature and composition for sound in gases, refractive index for light. When a wave crosses into a new medium, speed changes. If the source stays the same, frequency stays constant, which means wavelength must change: $\lambda = v/f$.

When a wave crosses a boundary, frequency remains constant (set by the source) and speed changes, so wavelength adjusts via $\lambda = v/f$. Two points separated by $\lambda$ (or $T$) are in phase; separated by $\lambda/2$ they are in antiphase.

Pause, write the highlighted boundary rule into your book.

Why tightening a guitar string raises its pitch, and why thick strings sound lower

We just saw that frequency is set by the source while speed and wavelength adjust across a boundary. That raises a question: for a wave on a string, what actually sets that speed? This card answers it → two medium properties, tension and linear mass density, combine in one formula, part of the NESA syllabus dot point on solving problems by modelling wave relationships.

A guitarist tightens a string with a tuning peg and the pitch rises immediately, without touching the string's length or plucking harder. The string's linear mass density, $\mu$ (mass per unit length, in kg/m), and its tension, $T$ (in newtons), are the two medium properties that set the wave speed on that string.

These two properties combine as $v = \sqrt{T/\mu}$. Increasing tension increases wave speed. Increasing mass per unit length (a thicker or denser string) decreases wave speed. For a string of fixed length vibrating in a given harmonic, wavelength is fixed by the boundary conditions ($\lambda_n = 2L/n$, see L11), so a faster wave speed means a higher frequency ($f = v/\lambda$).

Wave speed on a string is $v = \sqrt{T/\mu}$, where $T$ is tension (N) and $\mu$ is linear mass density, mass per unit length (kg/m). Increasing tension increases wave speed (and frequency, for fixed length); increasing linear density decreases wave speed (and frequency). This is why tightening a guitar string raises its pitch, and why thicker strings sound lower at the same tension.

Pause, copy the highlighted formula and its two effects into your book before the check below.