A Wave Along a String

solved TravellingWave

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?

Inputs

wave number

k

80 rad/m

Outputs

wavelength

λ

How it's solved

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 web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

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

solution.rb

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

playground.rb

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 it drew on

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)