Mechanical waves

A wave, transverse and longitudinal waves, wavelength and wave speed, the relations $\lambda = vT$ and $v = \lambda\nu$.

Terms

Studied 0 of 5

Formulas

Wavelength through the period

\lambda = v\,T

Quantities
Symbol	Name	Unit
$\lambda$	wavelength	m
$v$	wave speed	m/s
$T$	period of oscillations	s

Solved for another quantity:

v = \dfrac{\lambda}{T}

T = \dfrac{\lambda}{v}

Note. In the time of one oscillation $T$ the wave covers the distance $\lambda$ with the speed $v$ . Example: a wave with the speed $v = 2\ \text{m/s}$ and the period $T = 3\ \text{s}$ has the wavelength $\lambda = 2 \cdot 3 = 6\ \text{m}$ .

Wave speed through the frequency

v = \lambda\,\nu

Quantities
Symbol	Name	Unit
$v$	wave speed	m/s
$\lambda$	wavelength	m
$\nu$	frequency of oscillations	Hz

Solved for another quantity:

\lambda = \dfrac{v}{\nu}

\nu = \dfrac{v}{\lambda}

Note. It follows from $\lambda = vT$ and $\nu = 1 / T$ . Example: sound of frequency $\nu = 170\ \text{Hz}$ in air ( $v = 340\ \text{m/s}$ ) has the wavelength $\lambda = \dfrac{340}{170} = 2\ \text{m}$ .

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

A travelling wave

The red dot is a particle of the medium: it oscillates up and down and does not move to the right with the wave. The scene is not to scale.

t

= 0 sPlayback speed

Graphs

The graph shows the wave at the chosen moment: the displacement of the particles of the medium from equilibrium against the distance. Click the graph to choose a particle (the red dot).

y(x)

v

4 m/s

\nu

2 Hz

T

0.5 s

\lambda

2 m

Formulas with your numbers

T = \dfrac{1}{\nu} = \dfrac{1}{2} = 0.5\ \text{s}

\lambda = vT = 4 \cdot 0.5 = 2\ \text{m}

v = \lambda\nu = 2 \cdot 2 = 4\ \text{m/s}

How to read the graphs

The $y(x)$ graph is a "photograph" of the wave at one moment of time: the horizontal axis is the distance $x$ along the direction of propagation, the vertical axis is the displacement $y$ of the particles of the medium from equilibrium. It is not the graph of the motion of a single particle.
The wavelength $\lambda$ is the distance between two neighbouring crests (or two neighbouring troughs) on the graph. A shorter wavelength puts more crests on the graph.
While the wave runs, the whole picture shifts to the right with the speed $v$ : in a period $T$ each crest moves by the distance $\lambda$ . Hence $\lambda = vT$ .
The red dot on the graph is a particle of the medium at the chosen place. It only oscillates about its equilibrium position and is not carried along the wave: a wave transfers energy, not matter.

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Terms

Studied 0 of 5

Formulas

Wavelength through the period

\lambda = v\,T

Quantities
Symbol	Name	Unit
$\lambda$	wavelength	m
$v$	wave speed	m/s
$T$	period of oscillations	s

Solved for another quantity:

v = \dfrac{\lambda}{T}

T = \dfrac{\lambda}{v}

Wave speed through the frequency

v = \lambda\,\nu

Quantities
Symbol	Name	Unit
$v$	wave speed	m/s
$\lambda$	wavelength	m
$\nu$	frequency of oscillations	Hz

Solved for another quantity:

\lambda = \dfrac{v}{\nu}

\nu = \dfrac{v}{\lambda}

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

A travelling wave

The red dot is a particle of the medium: it oscillates up and down and does not move to the right with the wave. The scene is not to scale.

t

= 0 sPlayback speed

Graphs

The graph shows the wave at the chosen moment: the displacement of the particles of the medium from equilibrium against the distance. Click the graph to choose a particle (the red dot).

y(x)

v

4 m/s

\nu

2 Hz

T

0.5 s

\lambda

2 m

Formulas with your numbers

T = \dfrac{1}{\nu} = \dfrac{1}{2} = 0.5\ \text{s}

\lambda = vT = 4 \cdot 0.5 = 2\ \text{m}

v = \lambda\nu = 2 \cdot 2 = 4\ \text{m/s}

How to read the graphs

The

y(x)

graph is a "photograph" of the wave at one moment of time: the horizontal axis is the distance

x

along the direction of propagation, the vertical axis is the displacement

y

of the particles of the medium from equilibrium. It is not the graph of the motion of a single particle.

The wavelength

\lambda

is the distance between two neighbouring crests (or two neighbouring troughs) on the graph. A shorter wavelength puts more crests on the graph.

While the wave runs, the whole picture shifts to the right with the speed

v

: in a period

T

each crest moves by the distance

\lambda

. Hence

\lambda = vT

The red dot on the graph is a particle of the medium at the chosen place. It only oscillates about its equilibrium position and is not carried along the wave: a wave transfers energy, not matter.