A Harmonic Wave's Speed

Q. A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t + 0.018 x + π/4) where x and y are in cm and t in s. The positive direction of x is from left to right. What is the speed of its propagation, in m/s? expects 20 m/s

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Q. A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t $ω = 36$ rad/s + 0.018 x $k = 0.018$ rad/cm + π/4) where x and y are in cm and t in s. The positive direction of x is from left to right. What is the speed of its propagation, in m/s?

The playground

Outputs

speed

v

The solution

Solved with TravellingWave

Given

$ω = 36$ rad/s from “36 t”
$k = 0.018$ rad/cm from “0.018 x”

Find $v$ .

Working

$k = 0.018$ rad/cm $= 1.8$ m⁻¹
angular frequency and period(wave) $ω = \frac{2 \cdot π}{T}$
$36 = \frac{2 \cdot 3.14159265}{T}$
$T = 0.174532925$ s
frequency and period(wave) $ν = \frac{1}{T}$
$ν = \frac{1}{0.174532925}$
$ν = 5.72957795$ Hz
wave number and wavelength(wave) $k = \frac{2 \cdot π}{λ}$
$1.8 = \frac{2 \cdot 3.14159265}{λ}$
$λ = 3.4906585$ m
speed of a wave(wave) $v = ν \cdot λ$
$v = 5.72957795 \cdot 3.4906585$
$v = 20.0 m/s$
$v \approx 20$ m/sto 2 significant figures

Conclusion

A crest keeps its phase fixed, so it moves the angular frequency over the wave number each second; with t and x added, it runs from right to left.

The Ruby

1require_relative "../physics"23question :a_harmonic_wave_s_speed do4  kind :numerical5  describes "A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t + 0.018 x + π/4) where x " \6            "and y are in cm and t in s. The positive direction of x is from left to right. %{speed}, in m/s?"7  asks speed: "What is the speed of its propagation"8end

1require_relative "../questions/a_harmonic_wave_s_speed"2require_relative "../science/waves/travelling_wave"34solve :a_harmonic_wave_s_speed, using: TravellingWave do5  objects :wave67  deduce_from "36 t", wave.angular_frequency => "36 rad/s"8  deduce_from "0.018 x", wave.wave_number => "0.018 rad/cm"910  speed = wave.wave_speed11  answer speed:, in: "m/s", figures: 21213  conclusion "A crest keeps its phase fixed, so it moves the angular frequency over the wave number each " \14             "second; with t and x added, it runs from right to left."15end

1require_relative "../solutions/a_harmonic_wave_s_speed"23play_with :a_harmonic_wave_s_speed do4  title "A Harmonic Wave's Speed"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)