A Clamped String's Wavelength

Q. The transverse displacement of a string (clamped at its both ends) is given by y(x, t) = 0.06 sin (2π/3 x) cos (120 πt) where x and y are in m and t in s. Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength of each wave, in metres? expects 3.0 m

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The web of concepts

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Q. The transverse displacement of a string (clamped at its both ends) is given by y(x, t) = 0.06 sin (2π/3 x) $k = 120$ deg/m cos (120 πt) where x and y are in m and t in s. Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength of each wave, in metres?

The playground

Outputs

wavelength

λ

The solution

Solved with TravellingWave

Given

$k = 120$ deg/m from “sin (2π/3 x)”

Find $λ$ .

Working

$k = 120$ deg/m $= 2.0943951$ m⁻¹
wave number and wavelength(wave) $k = \frac{2 \cdot π}{λ}$
$2.0943951 = \frac{2 \cdot 3.14159265}{λ}$
$λ = 3.00 m$
$λ \approx 3.0$ mto 2 significant figures

Conclusion

The two running waves share the standing wave's wave number, and a full turn of phase along the string is one wavelength.

The Ruby

1require_relative "../physics"23question :a_clamped_string_s_wavelength do4  kind :numerical5  describes "The transverse displacement of a string (clamped at its both ends) is given by y(x, t) = 0.06 sin (2π/3 x) " \6            "cos (120 πt) where x and y are in m and t in s. Interpret the wave as a superposition of two waves " \7            "travelling in opposite directions. %{wavelength}, in metres?"8  asks wavelength: "What is the wavelength of each wave"9end

1require_relative "../questions/a_clamped_string_s_wavelength"2require_relative "../science/waves/travelling_wave"34solve :a_clamped_string_s_wavelength, using: TravellingWave do5  objects :wave67  deduce_from "sin (2π/3 x)", wave.wave_number => "120 deg/m"89  wavelength = wave.wavelength10  answer wavelength:, in: "m", figures: 21112  conclusion "The two running waves share the standing wave's wave number, and a full turn of phase along the " \13             "string is one wavelength."14end

1require_relative "../solutions/a_clamped_string_s_wavelength"23play_with :a_clamped_string_s_wavelength do4  title "A Clamped String's Wavelength"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)