A Clamped String's Component Waves

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 speed of each wave, in m/s? expects 1.8 × 10² m/s

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The web of concepts

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The equations it used

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Check an answer

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) $ω = 60$ rev/s 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 speed of each wave, in m/s?

The playground

Outputs

speed

v

The solution

Solved with TravellingWave

Given

$k = 120$ deg/m from “sin (2π/3 x)”
$ω = 60$ rev/s from “cos (120 πt)”

Find $v$ .

Working

$k = 120$ deg/m $= 2.0943951$ m⁻¹
$ω = 60$ rev/s $= 376.991118$ rad/s
angular frequency and period(wave) $ω = \frac{2 \cdot π}{T}$
$376.991118 = \frac{2 \cdot 3.14159265}{T}$
$T = 0.0166666667$ s
frequency and period(wave) $ν = \frac{1}{T}$
$ν = \frac{1}{0.0166666667}$
$ν = 60$ Hz
wave number and wavelength(wave) $k = \frac{2 \cdot π}{λ}$
$2.0943951 = \frac{2 \cdot 3.14159265}{λ}$
$λ = 3$ m
speed of a wave(wave) $v = ν \cdot λ$
$v = 60 \cdot 3$
$v = 180.0 m/s$
$v \approx 1.8 \times 10^{2}$ m/sto 2 significant figures

Conclusion

A standing wave is two equal waves running opposite ways with the same wave number and angular frequency, so each moves at the angular frequency over the wave number.

The Ruby

1require_relative "../physics"23question :a_clamped_string_s_component_waves 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. %{speed}, in m/s?"8  asks speed: "What is the speed of each wave"9end

1require_relative "../questions/a_clamped_string_s_component_waves"2require_relative "../science/waves/travelling_wave"34solve :a_clamped_string_s_component_waves, using: TravellingWave do5  objects :wave67  deduce_from "sin (2π/3 x)", wave.wave_number => "120 deg/m"8  deduce_from "cos (120 πt)", wave.angular_frequency => "60 rev/s"910  speed = wave.wave_speed11  answer speed:, in: "m/s", figures: 21213  conclusion "A standing wave is two equal waves running opposite ways with the same wave number and angular " \14             "frequency, so each moves at the angular frequency over the wave number."15end

1require_relative "../solutions/a_clamped_string_s_component_waves"23play_with :a_clamped_string_s_component_waves do4  title "A Clamped String's Component Waves"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)