Sine Waves

Shake the end in SHM and every point copies it, a little later.

12 minUpdated 21 October 2026sine wavewavelengthfrequencywave numberparticle velocitywave velocity

Lesson at a glanceShow the key points

If the end moves as A sin ωt, the wave is y = A sin ω(t − x/v) (15.6): every particle performs SHM of amplitude A and frequency ν, with a phase lag ωx/v.
Particle velocity ∂y/∂t = Aω cos ω(t − x/v) (15.7) is completely different from the constant wave velocity v.
Wavelength λ = vT (15.8): the smallest separation of particles in the same phase; v = νλ (15.9); wave number k = 2π/λ.
Crests (positive displacement) and troughs (negative) are λ apart.
Equivalent forms: A sin(ωt − kx), A sin 2π(t/T − x/λ), A sin k(vt − x); a phase constant φ can be added (15.13).

Keep moving the end of the string up and down in simple harmonic motion and the whole string comes alive with a sine wave, the shape behind almost every wave in physics.

From a shaking end to a sine wave

If the end x = 0 performs SHM, $f(t) = A\sin\omega t$ (15.5), with period T = 2π/ω and frequency ν = 1/T, the wave that travels out is

Sine wave(15.6)

y = A\sin\omega\Big(t - \frac xv\Big)

Every particle of the string performs SHM with the same amplitude and frequency, but each copies the motion of a particle to its left after a delay x/v. When one particle reaches its top, a particle just ahead is still rising and one just behind is already coming down.

The velocity of the particle at x (x held fixed, so a partial derivative) is

\frac{\partial y}{\partial t} = A\omega\cos\omega\Big(t - \frac xv\Big) \qquad (15.7)

Wavelength, frequency and speed

A particle a distance vT further along copies the first after a delay of exactly one period, which makes no difference: they move in perfect step (same phase). This separation is the wavelength.

Definition · Wavelength

The smallest separation of two particles vibrating in the same phase:

\lambda = vT

(15.8). Consecutive crests (positive displacement) or troughs (negative) are λ apart.

The wave relation(15.9)

v = \nu\lambda, \qquad k = \frac{2\pi}{\lambda}\ \text{(wave number)}

Interactive

A sine wave on a string

Every point does SHM with the same amplitude and frequency, each a little behind the one before it. Points one wavelength apart (P and Q) move in perfect step. The wave moves at v = νλ; the particles' own up-and-down speed (arrow at P) is a different thing entirely.

v = \nu\lambda

1.50m/s

k = 2\pi/\lambda

2.09rad/m

\omega = 2\pi\nu

3.14rad/s

particle P's velocity now-0.94m/s

Amplitude

A

0.60m

Wavelength

\lambda

3.0m

Frequency

\nu

0.50Hz

Other ways to write it

y = A\sin(\omega t - kx) = A\sin2\pi\Big(\frac tT - \frac x\lambda\Big) = A\sin k(vt - x) \qquad (15.10\text{–}15.12)

These all assume t = 0 when the end crosses its mean position going up. A different choice of t = 0 adds a phase constant: $y = A\sin[\omega(t - x/v) + \phi]$ (15.13). With φ = π/2 (end at its top at t = 0) this becomes $A\cos\omega(t - x/v)$ ; with φ = π, $y = A\sin(kx - \omega t)$ (15.15).

Worked example

Reading a wave equation

A wave on a string is y = (3.0 mm) sin[(2.0 cm⁻¹)x − (40 s⁻¹)t]. Find its amplitude, wave number, wavelength, frequency, period and speed, and the greatest speed of a particle.

What to remember

1 $y = A\sin\omega(t - x/v)$ : every particle does SHM, each lagging the one before.
2Particle velocity ∂y/∂t ≠ wave velocity v.
3 $\lambda = vT$ , $v = \nu\lambda$ , $k = 2\pi/\lambda$ , $\omega = 2\pi\nu$ , so $v = \omega/k$ .
4Equivalent forms A sin(ωt − kx), A sin 2π(t/T − x/λ); a phase constant depends on the choice of t = 0.

Check yourself

A wave has frequency 50 Hz and wavelength 4 m. Its speed is…

Two points on a sine wave are λ/2 apart. Their motions are…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchAmplitude, period, frequency and wavelengthKhan Academy PlayWaves IntroPhET · CU Boulder ReadMathematics of wavesOpenStax University Physics §16.2 ReadThe anatomy of a waveThe Physics Classroom ReadThe wave equation (v = fλ)The Physics Classroom

PreviousWave Motion & Pulses Next Wave Speed & Power on a String

Sine Waves

Shake the end in SHM and every point copies it, a little later.

12 minUpdated 21 October 2026sine wavewavelengthfrequencywave numberparticle velocitywave velocity

Lesson at a glanceShow the key points

If the end moves as A sin ωt, the wave is y = A sin ω(t − x/v) (15.6): every particle performs SHM of amplitude A and frequency ν, with a phase lag ωx/v.
Particle velocity ∂y/∂t = Aω cos ω(t − x/v) (15.7) is completely different from the constant wave velocity v.
Wavelength λ = vT (15.8): the smallest separation of particles in the same phase; v = νλ (15.9); wave number k = 2π/λ.
Crests (positive displacement) and troughs (negative) are λ apart.
Equivalent forms: A sin(ωt − kx), A sin 2π(t/T − x/λ), A sin k(vt − x); a phase constant φ can be added (15.13).

Keep moving the end of the string up and down in simple harmonic motion and the whole string comes alive with a sine wave, the shape behind almost every wave in physics.

From a shaking end to a sine wave

If the end x = 0 performs SHM, $f(t) = A\sin\omega t$ (15.5), with period T = 2π/ω and frequency ν = 1/T, the wave that travels out is

Sine wave(15.6)

y = A\sin\omega\Big(t - \frac xv\Big)

The velocity of the particle at x (x held fixed, so a partial derivative) is

\frac{\partial y}{\partial t} = A\omega\cos\omega\Big(t - \frac xv\Big) \qquad (15.7)

Wavelength, frequency and speed

A particle a distance vT further along copies the first after a delay of exactly one period, which makes no difference: they move in perfect step (same phase). This separation is the wavelength.

Definition · Wavelength

The smallest separation of two particles vibrating in the same phase:

\lambda = vT

(15.8). Consecutive crests (positive displacement) or troughs (negative) are λ apart.

The wave relation(15.9)

v = \nu\lambda, \qquad k = \frac{2\pi}{\lambda}\ \text{(wave number)}

Interactive

A sine wave on a string

v = \nu\lambda

1.50m/s

k = 2\pi/\lambda

2.09rad/m

\omega = 2\pi\nu

3.14rad/s

particle P's velocity now-0.94m/s

Amplitude

A

0.60m

Wavelength

\lambda

3.0m

Frequency

\nu

0.50Hz

Other ways to write it

y = A\sin(\omega t - kx) = A\sin2\pi\Big(\frac tT - \frac x\lambda\Big) = A\sin k(vt - x) \qquad (15.10\text{–}15.12)

Worked example

Reading a wave equation

A wave on a string is y = (3.0 mm) sin[(2.0 cm⁻¹)x − (40 s⁻¹)t]. Find its amplitude, wave number, wavelength, frequency, period and speed, and the greatest speed of a particle.

What to remember

1 $y = A\sin\omega(t - x/v)$ : every particle does SHM, each lagging the one before.
2Particle velocity ∂y/∂t ≠ wave velocity v.
3 $\lambda = vT$ , $v = \nu\lambda$ , $k = 2\pi/\lambda$ , $\omega = 2\pi\nu$ , so $v = \omega/k$ .
4Equivalent forms A sin(ωt − kx), A sin 2π(t/T − x/λ); a phase constant depends on the choice of t = 0.

Check yourself

A wave has frequency 50 Hz and wavelength 4 m. Its speed is…

Two points on a sine wave are λ/2 apart. Their motions are…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousWave Motion & Pulses Next Wave Speed & Power on a String

From a shaking end to a sine wave#

Wavelength, frequency and speed#

Other ways to write it#

Check yourself

Keep exploring

From a shaking end to a sine wave#

Wavelength, frequency and speed#

Other ways to write it#

Check yourself

Keep exploring

From a shaking end to a sine wave

Wavelength, frequency and speed

Other ways to write it

From a shaking end to a sine wave

Wavelength, frequency and speed

Other ways to write it