A Wave Along a String

Q. A wave travelling along a string is described by y(x, t) = 0.005 sin(80.0 x – 3.0 t), in which the numerical constants are in SI units (0.005 m, 80.0 rad/m and 3.0 rad/s). What is the wavelength, in cm? expects 7.85 cm

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

Q. A wave travelling along a string is described by y(x, t) = 0.005 sin(80.0 x – 3.0 t), in which the numerical constants are in SI units (0.005 m, 80.0 rad/m $k = 80.0$ rad/m and 3.0 rad/s). What is the wavelength, in cm?

The playground

Inputs

wave number

k

80 rad/m

Outputs

wavelength

λ

The solution

Solved with TravellingWave

Given

$k = 80.0$ rad/m

Find $λ$ .

Working

$k = 80.0$ rad/m $= 80$ m⁻¹
wave number and wavelength(wave) $k = \frac{2 \cdot π}{λ}$
$80 = \frac{2 \cdot 3.14159265}{λ}$
$λ = 7.854$ cm
$λ = 0.0785398163$ m $= 7.85398163$ cm
$λ \approx 7.85$ cmto 3 significant figures

Conclusion

The number multiplying x counts radians of phase per metre, and one wavelength is a full turn of phase.

The Ruby

1require_relative "../physics"23question :a_wave_along_a_string do4  kind :numerical5  describes "A wave travelling along a string is described by y(x, t) = 0.005 sin(80.0 x – 3.0 t), in which the " \6            "numerical constants are in SI units (0.005 m, %{wave_number} and 3.0 rad/s). %{wavelength}, in cm?"7  given wave_number: "80.0 rad/m"8  asks wavelength: "What is the wavelength"9end

1require_relative "../questions/a_wave_along_a_string"2require_relative "../science/waves/travelling_wave"34solve :a_wave_along_a_string, using: TravellingWave do5  objects :wave67  take wave_number: wave.wave_number89  wavelength = wave.wavelength10  answer wavelength:, in: "cm", figures: 31112  conclusion "The number multiplying x counts radians of phase per metre, and one wavelength is a full " \13             "turn of phase."14end

1require_relative "../solutions/a_wave_along_a_string"23play_with :a_wave_along_a_string do4  title "A Wave Along a String"5end

The laws

quantity	written	units	within	from
pi	$π$	—	0 … ∞	TravellingWave
amplitude	$a$	m	—	TravellingWave
wave number	$k$	m⁻¹	—	TravellingWave
angular frequency	$ω$	rad/s	—	TravellingWave
wavelength	$λ$	m	0 … ∞	TravellingWave
period	$T$	s	—	TravellingWave
frequency	$ν$	Hz	0 … ∞	TravellingWave
wave speed	$v$	m/s	—	TravellingWave
separation	$Δ x$	m	—	TravellingWave
phase difference	$Δ φ$	—	—	TravellingWave
wave number of ground	$k_{g}$	m⁻¹	—
wavelength of ground	$λ_{g}$	m	0 … ∞
wave number of wave	$k$	m⁻¹	—
wavelength of wave	$λ$	m	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	—
period of ground	$T_{g}$	s	—
angular frequency of wave	$ω$	rad/s	—
period of wave	$T$	s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
frequency of wave	$ν$	Hz	0 … ∞
wave speed of ground	$v_{g}$	m/s	—
wave speed of wave	$v$	m/s	—
phase difference of ground	$Δ φ_{g}$	—	—
separation of ground	$Δ x_{g}$	m	—
phase difference of wave	$Δ φ$	—	—
separation of wave	$Δ x$	m	—
amplitude of wave	$a$	m	—

k_{w} = \frac{2 \cdot π}{λ_{w}}

wave number and wavelength

ω_{w} = \frac{2 \cdot π}{T_{w}}

angular frequency and period

ν_{w} = \frac{1}{T_{w}}

frequency and period

v_{w} = ν_{w} \cdot λ_{w}

speed of a wave

Δ φ_{w} = k_{w} \cdot Δ x_{w}

phase across a separation

k_{g} = \frac{2 \cdot π}{λ_{g}}

wave number and wavelength(ground)

k = \frac{2 \cdot π}{λ}

wave number and wavelength(wave)

ω_{g} = \frac{2 \cdot π}{T_{g}}

angular frequency and period(ground)

ω = \frac{2 \cdot π}{T}

angular frequency and period(wave)

ν_{g} = \frac{1}{T_{g}}

frequency and period(ground)

ν = \frac{1}{T}

frequency and period(wave)

v_{g} = ν_{g} \cdot λ_{g}

speed of a wave(ground)

v = ν \cdot λ

speed of a wave(wave)

Δ φ_{g} = k_{g} \cdot Δ x_{g}

phase across a separation(ground)

Δ φ = k \cdot Δ x

phase across a separation(wave)